From 98b6f570d3354a7ba81016f540052ce375162434 Mon Sep 17 00:00:00 2001
From: tlunet <tlunet@users.noreply.github.com>
Date: Fri, 20 Sep 2024 16:56:29 +0000
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diff --git a/.doctrees/projects/Hamiltonian.doctree b/.doctrees/projects/Hamiltonian.doctree
index 3bb59c7c0e22b278492271aa26003e63ad191131..c38fed2850e184aedaf064d8df0e3dbf319c5aa2 100644
GIT binary patch
delta 33
mcmbO{mu2!?mJO%>v6vW`8E!uRPm&SLntcAB_*Q1d348$So($*!

delta 33
mcmbO{mu2!?mJO%>v6vcKnr}Y;Pm&SLntcAB_*Q1d348$T_zeC4

diff --git a/.doctrees/projects/doc_fput.doctree b/.doctrees/projects/doc_fput.doctree
index a14264dd3cc5eaae6a442fe7dc10c85316c157ca..03c684229ed3bcfba5fc7c85b0cd84368338a83c 100644
GIT binary patch
delta 37
kcmdnClyTcq#tmVvEG7nKhLisri*C+$Wno0{dfc>l0R3hReE<Le

delta 37
kcmdnClyTcq#tmVvET)E*=9B*$i*C+$Wno0{dfc>l0RHR@lK=n!

diff --git a/.doctrees/projects/parallelSDC_reloaded.doctree b/.doctrees/projects/parallelSDC_reloaded.doctree
index dc6392d06be5524856d94dee95248d2aa33b25eb..c037ebb6511ce2d911866ce5567238893d7fbad3 100644
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diff --git a/.doctrees/pySDC/implementations.problem_classes.HeatEquation_Chebychev.doctree b/.doctrees/pySDC/implementations.problem_classes.HeatEquation_Chebychev.doctree
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diff --git a/.doctrees/pySDC/implementations.problem_classes.doctree b/.doctrees/pySDC/implementations.problem_classes.doctree
index aa28e90506636e30dfb132cb9c45b590239e76b6..e1a5dadb856293065a0ee4c7bf1588b1b96821f7 100644
GIT binary patch
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DGOjvS

literal 0
HcmV?d00001

diff --git a/.doctrees/tutorial/doc_step_2_C.doctree b/.doctrees/tutorial/doc_step_2_C.doctree
index bff1ea6f755359e37a6de2e84a1fdd7c86831641..39e321ee71194d0e9c9f97e1d9b2333f869daf95 100644
GIT binary patch
delta 1468
zcmeH`&q@MO6vnYtB$N^+h?A|#YSKG%@2F#m%g9P$5>49l1Z~>5DJ(FEHs$BC@)ZJG
zwsfs;Q1}>as`(S>J^2EG+wVK)H)p<cZoB7e_k1tskp-QtXsN8OXAP$EG)F$CbUTsq
z%miZts#2tSH=nzkrJ_yQL~*q~NpGzk5IK$D?yIb27>4mHbmDw&G$6D@6Wf-8%k?lR
zPrK#c)-pc#Lb7LisOJJFI(K^pB8rQB!ZSVInJV`P=CS;hC=R#e9>F|b-}V9t>N&gy
z?yCxJi^S6$W&_ERkmeJb!)xHwjNycZ#$y;5H!Y1!rZWT$C$K0!wD9~WG^Pq20pl7w
zmXu|`+QgS7#<>s2O*|D8ydRMfId|a$K|adu%^C9MgzyIOS4ge!82vwb%4Rsxklz=k
xt0zk_yk+E-3d3PJ(U2ZnAI4*-*UDJRE=qAuPy6?YPfuHV8~^LHyY<@R><>MbgqHvS

delta 1442
zcmbPogmKCd#tp?%{Dzha1{PMP=2iv<I>x4klkII~H_w*}VZtJzHCad24^7-?v%PIM
z2f8586z$1=Y8BYTC#n0O37bt8bl-v|W;ywxM&so98pf#NW=249G__`?V732)<R<_0
z6h%{S0hXKRg=UU9$ehN>=S^fL*LsVi>M@6ym#2fS2Vxd?n9O8fT{JxwhG6r~0p$er
z#JLcGI{A)9>*NXHvXeXg(6xY!YnZI>Z-Q#QCCI3T$qAP7lcNpLykluPS<tj`k|CPQ
zEG<C7z|fMJeBTh&8WRIUu$)a0svZ*q6R`My3~`Iej=`uQY+?v8<Gcx)8HOM;wjle~
zAfRnBcNm&7BZ&5MVa7;GCmDgl#e@$LR0cXGMw11A{?PZApS;IH6jjQ2vZG-mFjP=O
l%fuKITJn=CB10yNSqY<t<Y;OJX6ez?JmgaI=8DKU8~~TNd(!{_

diff --git a/.doctrees/tutorial/doc_step_3_B.doctree b/.doctrees/tutorial/doc_step_3_B.doctree
index 38a6ce65f95adc2650799b2aaf07f484d4d7c36e..ea82e9163795a4e4b70c2648e49e94320e13adc0 100644
GIT binary patch
delta 808
zcmeyhmhsnG#tp*Cd`1QehGteqMpmW<I+kXe6_tya(8MM$RINl4+RSL9#KDiM)qFCy
zzRu)0mvSuP$YMr1#zqE{C+crOQU?+<2Z}XLUgu_vDsF577N22gjN}HOdXvc$^qVF}
z8=)yTffyraj3#ahF=m%1nlWZzV`{z7l$%+AZFTTQ6E`;ni))#oiCY)}#TzESF*8AR
zy@e%Me7!lExFtxv{Nz`@8q7w;T$3+|X-u~DE1i7PLKw-HqY(m4nZW3#e}rtd_1nt<
E0J0j;S^xk5

delta 814
zcmeyhmhsnG#tp*Ce1?_^1{PMP=2iv<I+m836_tya(8MM$RINl4+RSL9#DOL@nbAgP
za;!@QHer5OAAW>IMmol(hLb1iZ$T5Y0E#tEUgu_vWCT#H+2ls=rpYr5(S*$>e>7K^
zyg@%{a<q{#s!8USV3XvG(Znsxz~Z|+(Znr{!Q!=Es8*O5OrD_MG}*x$Rk?`)#276z
zG;u?aF%6U7n3?b*d)8!fKhTu*<|ZhPG%*6HmY@8}SA*HOm}~L{F^$Q#ex;LdS_mU~
Qe>6@8I8HX(`t9Wa0G%k+lmGw#

diff --git a/.doctrees/tutorial/doc_step_4_D.doctree b/.doctrees/tutorial/doc_step_4_D.doctree
index 5044636127426a8e87f7c10de6c8cbbb78afaece..45ee975f1579b449d7e796b291c6038f149692ae 100644
GIT binary patch
delta 109
zcmdnkz__u2af80GmYJS`frWvwxuuDMuC9WDf|;I~iG_)&iK!`sZ?0!<XliI=WNN(m
RqNxPa<U&(3()1*70RU~n7`6Za

delta 109
zcmdnkz__u2af80GmZ_elv5B#Xsild6uC9WDf|;I~fq|u&r74teu4iUuU~XYxY`XcP
QsRYyHLQ^x+^dxWr0CYwe)&Kwi

diff --git a/.doctrees/tutorial/doc_step_7_A.doctree b/.doctrees/tutorial/doc_step_7_A.doctree
index 51f903250408b3dbb624bcfda6e849c2f8fdc025..eddbe77bc7149b24bd3ce078dfb6d30ebd3e609a 100644
GIT binary patch
delta 175
zcmcb!j`7Yq#tj_y%*MtBo7cM=GqM<4nwd{-a1`16%R>#sGd0@W;AqCkWMDYi(LrQ$
zfwux9vw^84NXg`19{iIly|pLXy1ZdFH8a?J!PgQfX=u7R!_9?}%0}||2XFuYsvtFB

delta 165
zcmcb!j`7Yq#tj_y%$8<`o7cM=GqRW%SeQ(1a1`16%R>#svozV<;AqCkWMVRToxAMj
xN^fySCUe8djt(l5E4{TR+q%4AHnTL{e8Ja}k=fG1cyorE3y?uA(|G&?H~^YiG5!Dm

diff --git a/.doctrees/tutorial/doc_step_7_B.doctree b/.doctrees/tutorial/doc_step_7_B.doctree
index 39f329f9e6cd43b9c53bcf7cecfddf5998669b6d..af68052b9c6701c3322ccc713bfe3ae162917858 100644
GIT binary patch
delta 165
zcmaFf!}z?1al=~!7Gq;mlg%Fu^cYzT%*~A_H=1ZKZm?uzF*YzY0dXcbSZYnaX?2~+
w(0Fp7vCd=xTkXl5HaA$z%}fn|@?n#$ZEvuc85kLEerjd~)J84S#O(q&0FuEpkpKVy

delta 149
zcmaFf!}z?1al=~!786rrgUufe^ca~9EzBo3nCL8Suw-O5F)#wLCpTDXO}=S$o!QdJ
oY;&WD9wUpTfra7ZMw76~*0wiT%#96AHa|79Vx-&@ak~Hx0B_JQa{vGU

diff --git a/.doctrees/tutorial/doc_step_7_C.doctree b/.doctrees/tutorial/doc_step_7_C.doctree
index 44521058f3e033e5bffe0dc0089480f1b70cc57d..b8f4ad11f8b8a0d27284d19a62a8ccfeb6a153fa 100644
GIT binary patch
delta 136
zcmX>)o$=^&#tnYv%m(HblRwyKPk!sbzIna14I`_O9z@7w^AC$=CT2q;i^(5tw4n0#
zj39x{A8mRWH}A7&WMnb7G%x|mYffHcFSNPNj**ec+;Z|k2T#1RI+Nd9aBgmLcIE&8
D9Of^h

delta 137
zcmX>)o$=^&#tnYvEEWdF29rP9Xia|Wz`l9CwGAViksegcc=Hd7W+o;hlg)wVd5o+E
zdX`2elLO7QC%?Dg+^k|7#khH&JtHHtg^|JJ4?qLf*b8m0vtwjrvM`!_(7_X*qz>4K
ICTC|30EXu+&j0`b

diff --git a/.doctrees/tutorial/doc_step_7_D.doctree b/.doctrees/tutorial/doc_step_7_D.doctree
index 11816a7efa6e17dffc8e2e4cdea351c26fae2edb..32f3f68fb2a9994e05a89335a2674adc378b0161 100644
GIT binary patch
delta 574
zcmeH@JxT*n6orW<gG>r%jFJ!#3@Iiv$=tmEcOO`!uu^mjYLQkjwU8|cX94aY3X;;j
zXl3Ife1n!Q!S391Ip4jU?P|MvIWAHr>0MaXLgS+t<GjcdO~MGqZNZ5oFO|nrl!2tw
z9_y-)^(;fT&Qn9qdbg~l!a|I&nDlnEh`A#?*9Ul*9PS$Je&=!Gh=mW4)vSAXoMjE(
z8-e|@xnvw$T$bDoq>M8WYf{{g`q<F0&16-K_HjMW@Kxp=MM7KL6ocsGxG1uYCbkZX
zDkPRT2pM1HL`CTo-YQNd*}%5J#&3r>5reiSV{j_C#kiYtOJgJKkNG}5`>*EDYF<CZ
G)!sJ=`IMUg

delta 584
zcmeH@Jxat-6ona`i4qIV=-?0$l@ys|CNJ;4pS&9oOkt%cxB|7X6`Wdd2^MD`qastf
z1Pgbdm4z<Cr(^35EboE)osTmePsjJ0X-LB9nl1{d@RlYxNTYy-Hi~eba$p1-O?Vmf
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zmf+R{jZm!F1KG1HZ-b+Zmq?Hz?WsD%)y_I5B5qru^pDF{*~MoTb+jNyI7|CDEh8MJ
zD;*ZLCNz2djn$RHP0j&{iiq))bLbl?qdeB2kukPqziot<*_GU~xEyjzVIu7;d>!up
N@ALOQkMHt$@e9CXm<Ip=

diff --git a/.doctrees/tutorial/doc_step_8_C.doctree b/.doctrees/tutorial/doc_step_8_C.doctree
index ce12ed14f073486ea67c6ed649f21f0037cefe5f..c8c8603ee532bbbfba5c2565e2a9396053c817d5 100644
GIT binary patch
delta 328
zcmZ3piD|<oCYA=)sj3@U<g;0B2{ABC)^Cj4oLQH^#BXY5X>M+42?7=-W@eKU$`dx5
zH*N(=7+DyYnHicGfPlHhWQF!^o6i=;GYJ?OnwnS`8(D%S3@j`*E3}I;^BWo&8W~#}
zn}ba<Gv1s~F2XEeVqj=wVPOeYYG`3(0yKT=X4$?(CO$(GOCwW=*(L@?K=Fvl|7$r0
MtJ^j|p5Vy=0J-W~ZvX%Q

delta 328
zcmdn6iD}IyCYA=)sfrs}<g*!XZ04#v$i!!0VQFq=W?=vX#wJFS6U!4eTQzQF;x{p{
zFflbZFb7E(n3@135;w=UY+>Rzw6rubv#^Bdur!~XP@Xh7z9n??&GvXEegg{w3qu1V
zORz2z)6I$HBFqBDmX>Cg78a%;K_hc>kR@9;EA%BY@fuhfnwwZy0)e@SIZ!x)b4mt#
S?34__h(YPP&Ce%zasU9VzF&6$

diff --git a/.doctrees/tutorial/step_2.doctree b/.doctrees/tutorial/step_2.doctree
index 4fa64e7b0dcf4ff7402df3f404712981e469029f..a1c572dfe592c265829b94ff6e77c48554f097e9 100644
GIT binary patch
delta 1518
zcmeH{u}cDB7{)2l!ogbP2r`{Z<h<|G^PF5GXmV&uw@8O_iiSW3gC-B2xy(5dQHv-l
zTkEKkh!hX^rv8GU^DfHo<NkthecpS1@B2O9`(3f?Dt29s`pKLmhNAJLBqd{Fn8*b4
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R-+%vr_;>rBZp{q$z261B=V|}|

delta 1548
zcmeH_ze@sP7{{s5!ogbN5d1Zl$erHlos(+>O%83kMLJkhGz1zrw50Hz%bX(-&n$vU
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diff --git a/.doctrees/tutorial/step_3.doctree b/.doctrees/tutorial/step_3.doctree
index 683c7ce698e57e1cc4d1663e05b7355eac13cc43..41d427aa133a315a662c0b020eef697a58fa62c4 100644
GIT binary patch
delta 792
zcmZoV!rXL(d4p9spOJxrp_!GDk(H@|j-}aV#oi(&G_lDG`zz6f44ZW}S66*w=SS6U
zKDnW3F%~hM$#KozNa991#zqE{C(hi0CT2dFv8{3PpB7_Oabpv(`2JQjaT7zJIGPzI
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x2~^!MdHq5YRMnP~+7u>R^k^^|O-}6Dgd{f_3Gnm?3{*4vMZ)~eHy3Hz0{}$y*c$)<

delta 810
zcmZoV!rXL(d4p9spP{9KfrXW+xs`!|j-};h#oi(&G_lDG`ztYpv^Q5(eP_oaK6zCW
zHszCJntk~ZmKo_7n;K4@ICBe{n8jqqw#LbST8vS}%}l}K`&*5XYy_HNKKbK9g~<wS
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diff --git a/.doctrees/tutorial/step_4.doctree b/.doctrees/tutorial/step_4.doctree
index 810053bab7123297ac839e4b8a13756c7c3c9bb2..0841637a9d0e82d28eff2f79e7efd52f84adf9dc 100644
GIT binary patch
delta 126
zcmeBs$=vsnc|+z?Ei*j>0}BIVb4wEiU0nqQ1v5P}6AKel6H`+N-(1h!(A3b#$kcdp
bz;oltA6`UkmV0)Xkvx69jGL8TWpe=l-B2Y$

delta 126
zcmeBs$=vsnc|+z?EmJ*9V-sT&Q%e&CU0nqQ1v5P}0|QGlOH(M{T+htRz}&*X*mQEh
abK}V$UPNq`dv=(SJbk^4o0VT>a{&P5v?XEy

diff --git a/.doctrees/tutorial/step_7.doctree b/.doctrees/tutorial/step_7.doctree
index 8587814f2a3cae9d8809822c5cf69d01a2611fb8..01ccc1f4f011b17236ba392eba701a1f72b408f5 100644
GIT binary patch
delta 1038
zcmeH@J!@1!6oxr>hbvqxZZzR)p)N!S-0Yp5?+;g^;-;`s6q1k?LsFy=#9tr;|9}Lh
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X2M+R}eyk1M{}=khg<hQ$u#^1+e*-AX

delta 1044
zcmeH@O=}cE5QeGDu)^YD6BAZJWI-V?+3D@B?wRg!{lv||i=vQ(tQeAm2SNM=Lhuhr
zpbvEwU1H+F4_Xk+$zKpX8FCPRfl-1V7#6JAD0<P8xq0gMt$O>ZqPF|-c0Ycy|3Xyc
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diff --git a/.doctrees/tutorial/step_8.doctree b/.doctrees/tutorial/step_8.doctree
index 5ed3e570c6a524bf99e2ca2063e1320f24ec9bf1..a81dadd1103ba84f498762f52a437731bd78c2a0 100644
GIT binary patch
delta 339
zcmeC&#L~Nog{6UY>er1d#?KjVZI=0Ukcr>a%+lQ4&=LeJOw7zC2Yg7_?EQBuP{PQ<
zz|73h!~g`$Er1f+wjO1SXA&?nG&Qj>HnIdu7+6?r4)`F-%x7q1Xk=_@Yz{WbY_k7{
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RN`_d(;C15m?>vk`JOGeyWvBoE

delta 337
zcmeC(#L~Hmg{6UY>c@>N#?KjVY?k|VkcrR0!qVK#%)$T&j7^Lt2YyJ{?Duyo6TgXp
zg^8)TfjLORz|;gNv2E)K#&{+HLrY5|GYd<I4oh?6&4C|8nfVPY3@i){j4Z**O-zBx
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diff --git a/_images/coverage-badge.svg b/_images/coverage-badge.svg
index 5cb6ec5c45..69273e9c6e 100644
--- a/_images/coverage-badge.svg
+++ b/_images/coverage-badge.svg
@@ -1 +1 @@
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\ No newline at end of file
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\ No newline at end of file
diff --git a/_images/step_1_accuracy_test_coll.png b/_images/step_1_accuracy_test_coll.png
index 8f118fc93630519e448a67d02db1ceb7cf6820b4..8adcff1b9405304db0486f3b70bad28f8de8a042 100644
GIT binary patch
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diff --git a/_modules/core/controller.html b/_modules/core/controller.html
index 2ac57a03ac..d1d51b5e8b 100644
--- a/_modules/core/controller.html
+++ b/_modules/core/controller.html
@@ -88,7 +88,7 @@ <h1>Source code for core.controller</h1><div class="highlight"><pre>
         <span class="n">user_hooks</span> <span class="o">=</span> <span class="n">controller_params</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s1">&#39;hook_class&#39;</span><span class="p">,</span> <span class="p">[])</span>
         <span class="n">hook_classes</span> <span class="o">+=</span> <span class="n">user_hooks</span> <span class="k">if</span> <span class="nb">type</span><span class="p">(</span><span class="n">user_hooks</span><span class="p">)</span> <span class="o">==</span> <span class="nb">list</span> <span class="k">else</span> <span class="p">[</span><span class="n">user_hooks</span><span class="p">]</span>
         <span class="p">[</span><span class="bp">self</span><span class="o">.</span><span class="n">add_hook</span><span class="p">(</span><span class="n">hook</span><span class="p">)</span> <span class="k">for</span> <span class="n">hook</span> <span class="ow">in</span> <span class="n">hook_classes</span><span class="p">]</span>
-        <span class="n">controller_params</span><span class="p">[</span><span class="s1">&#39;hook_class&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="n">controller_params</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s1">&#39;hook_class&#39;</span><span class="p">,</span> <span class="n">hook_classes</span><span class="p">)</span>
+        <span class="n">controller_params</span><span class="p">[</span><span class="s1">&#39;hook_class&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="n">hook_classes</span>
 
         <span class="k">for</span> <span class="n">hook</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">hooks</span><span class="p">:</span>
             <span class="n">hook</span><span class="o">.</span><span class="n">pre_setup</span><span class="p">(</span><span class="n">step</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">level_number</span><span class="o">=</span><span class="kc">None</span><span class="p">)</span>
diff --git a/_modules/helpers/spectral_helper.html b/_modules/helpers/spectral_helper.html
new file mode 100644
index 0000000000..6a7bacae7f
--- /dev/null
+++ b/_modules/helpers/spectral_helper.html
@@ -0,0 +1,2337 @@
+<!DOCTYPE html>
+
+<html lang="en" data-content_root="../../">
+  <head>
+    <meta charset="utf-8" />
+    <meta name="viewport" content="width=device-width, initial-scale=1.0" />
+    <title>helpers.spectral_helper &#8212; pySDC 5.5.1 documentation</title>
+    <link rel="stylesheet" type="text/css" href="../../_static/pygments.css?v=fa44fd50" />
+    <link rel="stylesheet" type="text/css" href="../../_static/classic.css?v=def86cc0" />
+    <link rel="stylesheet" type="text/css" href="../../_static/custom.css?v=c40552b6" />
+    
+    <script src="../../_static/documentation_options.js?v=07fd27c7"></script>
+    <script src="../../_static/doctools.js?v=9a2dae69"></script>
+    <script src="../../_static/sphinx_highlight.js?v=dc90522c"></script>
+    
+    <link rel="index" title="Index" href="../../genindex.html" />
+    <link rel="search" title="Search" href="../../search.html" /> 
+  </head><body>
+    <div class="related" role="navigation" aria-label="Related">
+      <h3>Navigation</h3>
+      <ul>
+        <li class="right" style="margin-right: 10px">
+          <a href="../../genindex.html" title="General Index"
+             accesskey="I">index</a></li>
+        <li class="right" >
+          <a href="../../py-modindex.html" title="Python Module Index"
+             >modules</a> |</li>
+        <li class="nav-item nav-item-0"><a href="../../index.html">pySDC 5.5.1 documentation</a> &#187;</li>
+          <li class="nav-item nav-item-1"><a href="../index.html" accesskey="U">Module code</a> &#187;</li>
+        <li class="nav-item nav-item-this"><a href="">helpers.spectral_helper</a></li> 
+      </ul>
+    </div>  
+
+    <div class="document">
+      <div class="documentwrapper">
+        <div class="bodywrapper">
+          <div class="body" role="main">
+            
+  <h1>Source code for helpers.spectral_helper</h1><div class="highlight"><pre>
+<span></span><span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
+<span class="kn">import</span> <span class="nn">scipy</span>
+<span class="kn">from</span> <span class="nn">pySDC.implementations.datatype_classes.mesh</span> <span class="kn">import</span> <span class="n">mesh</span>
+<span class="kn">from</span> <span class="nn">scipy.special</span> <span class="kn">import</span> <span class="n">factorial</span>
+
+
+<div class="viewcode-block" id="SpectralHelper1D">
+<a class="viewcode-back" href="../../pySDC/helpers.spectral_helper.html#helpers.spectral_helper.SpectralHelper1D">[docs]</a>
+<span class="k">class</span> <span class="nc">SpectralHelper1D</span><span class="p">:</span>
+<span class="w">    </span><span class="sd">&quot;&quot;&quot;</span>
+<span class="sd">    Abstract base class for 1D spectral discretizations. Defines a common interface with parameters and functions that</span>
+<span class="sd">    all bases need to have.</span>
+
+<span class="sd">    When implementing new bases, please take care to use the modules that are supplied as class attributes to enable</span>
+<span class="sd">    the code for GPUs.</span>
+
+<span class="sd">    Attributes:</span>
+<span class="sd">        N (int): Resolution</span>
+<span class="sd">        x0 (float): Coordinate of left boundary</span>
+<span class="sd">        x1 (float): Coordinate of right boundary</span>
+<span class="sd">        L (float): Length of the domain</span>
+<span class="sd">        useGPU (bool): Whether to use GPUs</span>
+
+<span class="sd">    &quot;&quot;&quot;</span>
+
+    <span class="n">fft_lib</span> <span class="o">=</span> <span class="n">scipy</span><span class="o">.</span><span class="n">fft</span>
+    <span class="n">sparse_lib</span> <span class="o">=</span> <span class="n">scipy</span><span class="o">.</span><span class="n">sparse</span>
+    <span class="n">linalg</span> <span class="o">=</span> <span class="n">scipy</span><span class="o">.</span><span class="n">sparse</span><span class="o">.</span><span class="n">linalg</span>
+    <span class="n">xp</span> <span class="o">=</span> <span class="n">np</span>
+
+    <span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">N</span><span class="p">,</span> <span class="n">x0</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">x1</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">useGPU</span><span class="o">=</span><span class="kc">False</span><span class="p">):</span>
+<span class="w">        </span><span class="sd">&quot;&quot;&quot;</span>
+<span class="sd">        Constructor</span>
+
+<span class="sd">        Args:</span>
+<span class="sd">            N (int): Resolution</span>
+<span class="sd">            x0 (float): Coordinate of left boundary</span>
+<span class="sd">            x1 (float): Coordinate of right boundary</span>
+<span class="sd">            useGPU (bool): Whether to use GPUs</span>
+<span class="sd">        &quot;&quot;&quot;</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">N</span> <span class="o">=</span> <span class="n">N</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">x0</span> <span class="o">=</span> <span class="n">x0</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">x1</span> <span class="o">=</span> <span class="n">x1</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">L</span> <span class="o">=</span> <span class="n">x1</span> <span class="o">-</span> <span class="n">x0</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">useGPU</span> <span class="o">=</span> <span class="n">useGPU</span>
+
+        <span class="k">if</span> <span class="n">useGPU</span><span class="p">:</span>
+            <span class="bp">self</span><span class="o">.</span><span class="n">setup_GPU</span><span class="p">()</span>
+
+<div class="viewcode-block" id="SpectralHelper1D.setup_GPU">
+<a class="viewcode-back" href="../../pySDC/helpers.spectral_helper.html#helpers.spectral_helper.SpectralHelper1D.setup_GPU">[docs]</a>
+    <span class="nd">@classmethod</span>
+    <span class="k">def</span> <span class="nf">setup_GPU</span><span class="p">(</span><span class="bp">cls</span><span class="p">):</span>
+<span class="w">        </span><span class="sd">&quot;&quot;&quot;switch to GPU modules&quot;&quot;&quot;</span>
+        <span class="kn">import</span> <span class="nn">cupy</span> <span class="k">as</span> <span class="nn">cp</span>
+        <span class="kn">import</span> <span class="nn">cupyx.scipy.sparse</span> <span class="k">as</span> <span class="nn">sparse_lib</span>
+        <span class="kn">import</span> <span class="nn">cupyx.scipy.sparse.linalg</span> <span class="k">as</span> <span class="nn">linalg</span>
+        <span class="kn">import</span> <span class="nn">cupyx.scipy.fft</span> <span class="k">as</span> <span class="nn">fft_lib</span>
+        <span class="kn">from</span> <span class="nn">pySDC.implementations.datatype_classes.cupy_mesh</span> <span class="kn">import</span> <span class="n">cupy_mesh</span>
+
+        <span class="bp">cls</span><span class="o">.</span><span class="n">xp</span> <span class="o">=</span> <span class="n">cp</span>
+        <span class="bp">cls</span><span class="o">.</span><span class="n">sparse_lib</span> <span class="o">=</span> <span class="n">sparse_lib</span>
+        <span class="bp">cls</span><span class="o">.</span><span class="n">linalg</span> <span class="o">=</span> <span class="n">linalg</span>
+        <span class="bp">cls</span><span class="o">.</span><span class="n">fft_lib</span> <span class="o">=</span> <span class="n">fft_lib</span></div>
+
+
+<div class="viewcode-block" id="SpectralHelper1D.get_Id">
+<a class="viewcode-back" href="../../pySDC/helpers.spectral_helper.html#helpers.spectral_helper.SpectralHelper1D.get_Id">[docs]</a>
+    <span class="k">def</span> <span class="nf">get_Id</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
+<span class="w">        </span><span class="sd">&quot;&quot;&quot;</span>
+<span class="sd">        Get identity matrix</span>
+
+<span class="sd">        Returns:</span>
+<span class="sd">            sparse diagonal identity matrix</span>
+<span class="sd">        &quot;&quot;&quot;</span>
+        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">sparse_lib</span><span class="o">.</span><span class="n">eye</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">N</span><span class="p">)</span></div>
+
+
+<div class="viewcode-block" id="SpectralHelper1D.get_zero">
+<a class="viewcode-back" href="../../pySDC/helpers.spectral_helper.html#helpers.spectral_helper.SpectralHelper1D.get_zero">[docs]</a>
+    <span class="k">def</span> <span class="nf">get_zero</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
+<span class="w">        </span><span class="sd">&quot;&quot;&quot;</span>
+<span class="sd">        Get a matrix with all zeros of the correct size.</span>
+
+<span class="sd">        Returns:</span>
+<span class="sd">            sparse matrix with zeros everywhere</span>
+<span class="sd">        &quot;&quot;&quot;</span>
+        <span class="k">return</span> <span class="mi">0</span> <span class="o">*</span> <span class="bp">self</span><span class="o">.</span><span class="n">get_Id</span><span class="p">()</span></div>
+
+
+<div class="viewcode-block" id="SpectralHelper1D.get_differentiation_matrix">
+<a class="viewcode-back" href="../../pySDC/helpers.spectral_helper.html#helpers.spectral_helper.SpectralHelper1D.get_differentiation_matrix">[docs]</a>
+    <span class="k">def</span> <span class="nf">get_differentiation_matrix</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
+        <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">()</span></div>
+
+
+<div class="viewcode-block" id="SpectralHelper1D.get_integration_matrix">
+<a class="viewcode-back" href="../../pySDC/helpers.spectral_helper.html#helpers.spectral_helper.SpectralHelper1D.get_integration_matrix">[docs]</a>
+    <span class="k">def</span> <span class="nf">get_integration_matrix</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
+        <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">()</span></div>
+
+
+<div class="viewcode-block" id="SpectralHelper1D.get_wavenumbers">
+<a class="viewcode-back" href="../../pySDC/helpers.spectral_helper.html#helpers.spectral_helper.SpectralHelper1D.get_wavenumbers">[docs]</a>
+    <span class="k">def</span> <span class="nf">get_wavenumbers</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
+<span class="w">        </span><span class="sd">&quot;&quot;&quot;</span>
+<span class="sd">        Get the grid in spectral space</span>
+<span class="sd">        &quot;&quot;&quot;</span>
+        <span class="k">raise</span> <span class="ne">NotImplementedError</span></div>
+
+
+<div class="viewcode-block" id="SpectralHelper1D.get_empty_operator_matrix">
+<a class="viewcode-back" href="../../pySDC/helpers.spectral_helper.html#helpers.spectral_helper.SpectralHelper1D.get_empty_operator_matrix">[docs]</a>
+    <span class="k">def</span> <span class="nf">get_empty_operator_matrix</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">S</span><span class="p">,</span> <span class="n">O</span><span class="p">):</span>
+<span class="w">        </span><span class="sd">&quot;&quot;&quot;</span>
+<span class="sd">        Return a matrix of operators to be filled with the connections between the solution components.</span>
+
+<span class="sd">        Args:</span>
+<span class="sd">            S (int): Number of components in the solution</span>
+<span class="sd">            O (sparse matrix): Zero matrix used for initialization</span>
+
+<span class="sd">        Returns:</span>
+<span class="sd">            list of lists containing sparse zeros</span>
+<span class="sd">        &quot;&quot;&quot;</span>
+        <span class="k">return</span> <span class="p">[[</span><span class="n">O</span> <span class="k">for</span> <span class="n">_</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">S</span><span class="p">)]</span> <span class="k">for</span> <span class="n">_</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">S</span><span class="p">)]</span></div>
+
+
+<div class="viewcode-block" id="SpectralHelper1D.get_basis_change_matrix">
+<a class="viewcode-back" href="../../pySDC/helpers.spectral_helper.html#helpers.spectral_helper.SpectralHelper1D.get_basis_change_matrix">[docs]</a>
+    <span class="k">def</span> <span class="nf">get_basis_change_matrix</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
+<span class="w">        </span><span class="sd">&quot;&quot;&quot;</span>
+<span class="sd">        Some spectral discretization change the basis during differentiation. This method can be used to transfer</span>
+<span class="sd">        between the various bases.</span>
+
+<span class="sd">        This method accepts arbitrary arguments that may not be used in order to provide an easy interface for multi-</span>
+<span class="sd">        dimensional bases. For instance, you may combine an FFT discretization with an ultraspherical discretization.</span>
+<span class="sd">        The FFT discretization will always be in the same base, but the ultraspherical discretization uses a different</span>
+<span class="sd">        base for every derivative. You can then ask all bases for transfer matrices from one ultraspherical derivative</span>
+<span class="sd">        base to the next. The FFT discretization will ignore this and return an identity while the ultraspherical</span>
+<span class="sd">        discretization will return the desired matrix. After a Kronecker product, you get the 2D version of the matrix</span>
+<span class="sd">        you want. This is what the `SpectralHelper` does when you call the method of the same name on it.</span>
+
+<span class="sd">        Returns:</span>
+<span class="sd">            sparse bases change matrix</span>
+<span class="sd">        &quot;&quot;&quot;</span>
+        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">sparse_lib</span><span class="o">.</span><span class="n">eye</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">N</span><span class="p">)</span></div>
+
+
+<div class="viewcode-block" id="SpectralHelper1D.get_BC">
+<a class="viewcode-back" href="../../pySDC/helpers.spectral_helper.html#helpers.spectral_helper.SpectralHelper1D.get_BC">[docs]</a>
+    <span class="k">def</span> <span class="nf">get_BC</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">kind</span><span class="p">):</span>
+<span class="w">        </span><span class="sd">&quot;&quot;&quot;</span>
+<span class="sd">        To facilitate boundary conditions (BCs) we use either a basis where all functions satisfy the BCs automatically,</span>
+<span class="sd">        as is the case in FFT basis for periodic BCs, or boundary bordering. In boundary bordering, specific lines in</span>
+<span class="sd">        the matrix are replaced by the boundary conditions as obtained by this method.</span>
+
+<span class="sd">        Args:</span>
+<span class="sd">            kind (str): The type of BC you want to implement please refer to the implementations of this method in the</span>
+<span class="sd">            individual 1D bases for what is implemented</span>
+
+<span class="sd">        Returns:</span>
+<span class="sd">            self.xp.array: Boundary condition</span>
+<span class="sd">        &quot;&quot;&quot;</span>
+        <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span><span class="sa">f</span><span class="s1">&#39;No boundary conditions of </span><span class="si">{</span><span class="n">kind</span><span class="si">=!r}</span><span class="s1"> implemented!&#39;</span><span class="p">)</span></div>
+
+
+<div class="viewcode-block" id="SpectralHelper1D.get_filter_matrix">
+<a class="viewcode-back" href="../../pySDC/helpers.spectral_helper.html#helpers.spectral_helper.SpectralHelper1D.get_filter_matrix">[docs]</a>
+    <span class="k">def</span> <span class="nf">get_filter_matrix</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">kmin</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span> <span class="n">kmax</span><span class="o">=</span><span class="kc">None</span><span class="p">):</span>
+<span class="w">        </span><span class="sd">&quot;&quot;&quot;</span>
+<span class="sd">        Get a bandpass filter.</span>
+
+<span class="sd">        Args:</span>
+<span class="sd">            kmin (int): Lower limit of the bandpass filter</span>
+<span class="sd">            kmax (int): Upper limit of the bandpass filter</span>
+
+<span class="sd">        Returns:</span>
+<span class="sd">            sparse matrix</span>
+<span class="sd">        &quot;&quot;&quot;</span>
+
+        <span class="n">k</span> <span class="o">=</span> <span class="nb">abs</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">get_wavenumbers</span><span class="p">())</span>
+
+        <span class="n">kmax</span> <span class="o">=</span> <span class="nb">max</span><span class="p">(</span><span class="n">k</span><span class="p">)</span> <span class="k">if</span> <span class="n">kmax</span> <span class="ow">is</span> <span class="kc">None</span> <span class="k">else</span> <span class="n">kmax</span>
+
+        <span class="n">mask</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">xp</span><span class="o">.</span><span class="n">logical_or</span><span class="p">(</span><span class="n">k</span> <span class="o">&gt;=</span> <span class="n">kmax</span><span class="p">,</span> <span class="n">k</span> <span class="o">&lt;</span> <span class="n">kmin</span><span class="p">)</span>
+
+        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">useGPU</span><span class="p">:</span>
+            <span class="n">Id</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">get_Id</span><span class="p">()</span><span class="o">.</span><span class="n">get</span><span class="p">()</span>
+        <span class="k">else</span><span class="p">:</span>
+            <span class="n">Id</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">get_Id</span><span class="p">()</span>
+        <span class="n">F</span> <span class="o">=</span> <span class="n">Id</span><span class="o">.</span><span class="n">tolil</span><span class="p">()</span>
+        <span class="n">F</span><span class="p">[:,</span> <span class="n">mask</span><span class="p">]</span> <span class="o">=</span> <span class="mi">0</span>
+        <span class="k">return</span> <span class="n">F</span><span class="o">.</span><span class="n">tocsc</span><span class="p">()</span></div>
+
+
+<div class="viewcode-block" id="SpectralHelper1D.get_1dgrid">
+<a class="viewcode-back" href="../../pySDC/helpers.spectral_helper.html#helpers.spectral_helper.SpectralHelper1D.get_1dgrid">[docs]</a>
+    <span class="k">def</span> <span class="nf">get_1dgrid</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
+<span class="w">        </span><span class="sd">&quot;&quot;&quot;</span>
+<span class="sd">        Get the grid in physical space</span>
+
+<span class="sd">        Returns:</span>
+<span class="sd">            self.xp.array: Grid</span>
+<span class="sd">        &quot;&quot;&quot;</span>
+        <span class="k">raise</span> <span class="ne">NotImplementedError</span></div>
+</div>
+
+
+
+<div class="viewcode-block" id="ChebychevHelper">
+<a class="viewcode-back" href="../../pySDC/helpers.spectral_helper.html#helpers.spectral_helper.ChebychevHelper">[docs]</a>
+<span class="k">class</span> <span class="nc">ChebychevHelper</span><span class="p">(</span><span class="n">SpectralHelper1D</span><span class="p">):</span>
+<span class="w">    </span><span class="sd">&quot;&quot;&quot;</span>
+<span class="sd">    The Chebychev base consists of special kinds of polynomials, with the main advantage that you can easily transform</span>
+<span class="sd">    between physical and spectral space by discrete cosine transform.</span>
+<span class="sd">    The differentiation in the Chebychev T base is dense, but can be preconditioned to yield a differentiation operator</span>
+<span class="sd">    that moves to Chebychev U basis during differentiation, which is sparse. When using this technique, problems need to</span>
+<span class="sd">    be formulated in first order formulation.</span>
+
+<span class="sd">    This implementation is largely based on the Dedalus paper (arXiv:1905.10388).</span>
+<span class="sd">    &quot;&quot;&quot;</span>
+
+    <span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="n">transform_type</span><span class="o">=</span><span class="s1">&#39;fft&#39;</span><span class="p">,</span> <span class="n">x0</span><span class="o">=-</span><span class="mi">1</span><span class="p">,</span> <span class="n">x1</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
+<span class="w">        </span><span class="sd">&quot;&quot;&quot;</span>
+<span class="sd">        Constructor.</span>
+<span class="sd">        Please refer to the parent class for additional arguments. Notably, you have to supply a resolution `N` and you</span>
+<span class="sd">        may choose to run on GPUs via the `useGPU` argument.</span>
+
+<span class="sd">        Args:</span>
+<span class="sd">            transform_type (&#39;fft&#39; or &#39;dct&#39;): Either use DCT functions directly implemented in the transform library or</span>
+<span class="sd">                                             use the FFT from the library to compute the DCT</span>
+<span class="sd">            x0 (float): Coordinate of left boundary. Note that only -1 is currently implented</span>
+<span class="sd">            x1 (float): Coordinate of right boundary. Note that only +1 is currently implented</span>
+<span class="sd">        &quot;&quot;&quot;</span>
+        <span class="k">assert</span> <span class="n">x0</span> <span class="o">==</span> <span class="o">-</span><span class="mi">1</span>
+        <span class="k">assert</span> <span class="n">x1</span> <span class="o">==</span> <span class="mi">1</span>
+        <span class="nb">super</span><span class="p">()</span><span class="o">.</span><span class="fm">__init__</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="n">x0</span><span class="o">=</span><span class="n">x0</span><span class="p">,</span> <span class="n">x1</span><span class="o">=</span><span class="n">x1</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">transform_type</span> <span class="o">=</span> <span class="n">transform_type</span>
+
+        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">transform_type</span> <span class="o">==</span> <span class="s1">&#39;fft&#39;</span><span class="p">:</span>
+            <span class="bp">self</span><span class="o">.</span><span class="n">get_fft_utils</span><span class="p">()</span>
+
+        <span class="bp">self</span><span class="o">.</span><span class="n">cache</span> <span class="o">=</span> <span class="p">{}</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">norm</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">get_norm</span><span class="p">()</span>
+
+<div class="viewcode-block" id="ChebychevHelper.get_1dgrid">
+<a class="viewcode-back" href="../../pySDC/helpers.spectral_helper.html#helpers.spectral_helper.ChebychevHelper.get_1dgrid">[docs]</a>
+    <span class="k">def</span> <span class="nf">get_1dgrid</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
+<span class="w">        </span><span class="sd">&#39;&#39;&#39;</span>
+<span class="sd">        Generates a 1D grid with Chebychev points. These are clustered at the boundary. You need this kind of grid to</span>
+<span class="sd">        use discrete cosine transformation (DCT) to get the Chebychev representation. If you want a different grid, you</span>
+<span class="sd">        need to do an affine transformation before any Chebychev business.</span>
+
+<span class="sd">        Returns:</span>
+<span class="sd">            numpy.ndarray: 1D grid</span>
+<span class="sd">        &#39;&#39;&#39;</span>
+        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">xp</span><span class="o">.</span><span class="n">cos</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">pi</span> <span class="o">/</span> <span class="bp">self</span><span class="o">.</span><span class="n">N</span> <span class="o">*</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">xp</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">N</span><span class="p">)</span> <span class="o">+</span> <span class="mf">0.5</span><span class="p">))</span></div>
+
+
+<div class="viewcode-block" id="ChebychevHelper.get_wavenumbers">
+<a class="viewcode-back" href="../../pySDC/helpers.spectral_helper.html#helpers.spectral_helper.ChebychevHelper.get_wavenumbers">[docs]</a>
+    <span class="k">def</span> <span class="nf">get_wavenumbers</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
+<span class="w">        </span><span class="sd">&quot;&quot;&quot;Get the domain in spectral space&quot;&quot;&quot;</span>
+        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">xp</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">N</span><span class="p">)</span></div>
+
+
+<div class="viewcode-block" id="ChebychevHelper.get_conv">
+<a class="viewcode-back" href="../../pySDC/helpers.spectral_helper.html#helpers.spectral_helper.ChebychevHelper.get_conv">[docs]</a>
+    <span class="k">def</span> <span class="nf">get_conv</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">name</span><span class="p">,</span> <span class="n">N</span><span class="o">=</span><span class="kc">None</span><span class="p">):</span>
+<span class="w">        </span><span class="sd">&#39;&#39;&#39;</span>
+<span class="sd">        Get conversion matrix between different kinds of polynomials. The supported kinds are</span>
+<span class="sd">         - T: Chebychev polynomials of first kind</span>
+<span class="sd">         - U: Chebychev polynomials of second kind</span>
+<span class="sd">         - D: Dirichlet recombination.</span>
+
+<span class="sd">        You get the desired matrix by choosing a name as ``A2B``. I.e. ``T2U`` for the conversion matrix from T to U.</span>
+<span class="sd">        Once generates matrices are cached. So feel free to call the method as often as you like.</span>
+
+<span class="sd">        Args:</span>
+<span class="sd">         name (str): Conversion code, e.g. &#39;T2U&#39;</span>
+<span class="sd">         N (int): Size of the matrix (optional)</span>
+
+<span class="sd">        Returns:</span>
+<span class="sd">            scipy.sparse: Sparse conversion matrix</span>
+<span class="sd">        &#39;&#39;&#39;</span>
+        <span class="k">if</span> <span class="n">name</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">cache</span><span class="o">.</span><span class="n">keys</span><span class="p">()</span> <span class="ow">and</span> <span class="ow">not</span> <span class="n">N</span><span class="p">:</span>
+            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">cache</span><span class="p">[</span><span class="n">name</span><span class="p">]</span>
+
+        <span class="n">N</span> <span class="o">=</span> <span class="n">N</span> <span class="k">if</span> <span class="n">N</span> <span class="k">else</span> <span class="bp">self</span><span class="o">.</span><span class="n">N</span>
+        <span class="n">sp</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">sparse_lib</span>
+        <span class="n">xp</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">xp</span>
+
+        <span class="k">def</span> <span class="nf">get_forward_conv</span><span class="p">(</span><span class="n">name</span><span class="p">):</span>
+            <span class="k">if</span> <span class="n">name</span> <span class="o">==</span> <span class="s1">&#39;T2U&#39;</span><span class="p">:</span>
+                <span class="n">mat</span> <span class="o">=</span> <span class="p">(</span><span class="n">sp</span><span class="o">.</span><span class="n">eye</span><span class="p">(</span><span class="n">N</span><span class="p">)</span> <span class="o">-</span> <span class="n">sp</span><span class="o">.</span><span class="n">diags</span><span class="p">(</span><span class="n">xp</span><span class="o">.</span><span class="n">ones</span><span class="p">(</span><span class="n">N</span> <span class="o">-</span> <span class="mi">2</span><span class="p">),</span> <span class="n">offsets</span><span class="o">=+</span><span class="mi">2</span><span class="p">))</span> <span class="o">/</span> <span class="mf">2.0</span>
+                <span class="n">mat</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">]</span> <span class="o">*=</span> <span class="mi">2</span>
+            <span class="k">elif</span> <span class="n">name</span> <span class="o">==</span> <span class="s1">&#39;D2T&#39;</span><span class="p">:</span>
+                <span class="n">mat</span> <span class="o">=</span> <span class="n">sp</span><span class="o">.</span><span class="n">eye</span><span class="p">(</span><span class="n">N</span><span class="p">)</span> <span class="o">-</span> <span class="n">sp</span><span class="o">.</span><span class="n">diags</span><span class="p">(</span><span class="n">xp</span><span class="o">.</span><span class="n">ones</span><span class="p">(</span><span class="n">N</span> <span class="o">-</span> <span class="mi">2</span><span class="p">),</span> <span class="n">offsets</span><span class="o">=+</span><span class="mi">2</span><span class="p">)</span>
+            <span class="k">elif</span> <span class="n">name</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">==</span> <span class="n">name</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]:</span>
+                <span class="n">mat</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">sparse_lib</span><span class="o">.</span><span class="n">eye</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">N</span><span class="p">)</span>
+            <span class="k">else</span><span class="p">:</span>
+                <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span><span class="sa">f</span><span class="s1">&#39;Don</span><span class="se">\&#39;</span><span class="s1">t have conversion matrix </span><span class="si">{</span><span class="n">name</span><span class="si">!r}</span><span class="s1">&#39;</span><span class="p">)</span>
+            <span class="k">return</span> <span class="n">mat</span>
+
+        <span class="k">try</span><span class="p">:</span>
+            <span class="n">mat</span> <span class="o">=</span> <span class="n">get_forward_conv</span><span class="p">(</span><span class="n">name</span><span class="p">)</span>
+        <span class="k">except</span> <span class="ne">NotImplementedError</span> <span class="k">as</span> <span class="n">E</span><span class="p">:</span>
+            <span class="k">try</span><span class="p">:</span>
+                <span class="n">fwd</span> <span class="o">=</span> <span class="n">get_forward_conv</span><span class="p">(</span><span class="n">name</span><span class="p">[::</span><span class="o">-</span><span class="mi">1</span><span class="p">])</span>
+                <span class="kn">import</span> <span class="nn">scipy.sparse</span> <span class="k">as</span> <span class="nn">sp</span>
+
+                <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">sparse_lib</span> <span class="o">==</span> <span class="n">sp</span><span class="p">:</span>
+                    <span class="n">mat</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">sparse_lib</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">inv</span><span class="p">(</span><span class="n">fwd</span><span class="o">.</span><span class="n">tocsc</span><span class="p">())</span>
+                <span class="k">else</span><span class="p">:</span>
+                    <span class="n">mat</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">sparse_lib</span><span class="o">.</span><span class="n">csc_matrix</span><span class="p">(</span><span class="n">sp</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">inv</span><span class="p">(</span><span class="n">fwd</span><span class="o">.</span><span class="n">tocsc</span><span class="p">()</span><span class="o">.</span><span class="n">get</span><span class="p">()))</span>
+            <span class="k">except</span> <span class="ne">NotImplementedError</span><span class="p">:</span>
+                <span class="k">raise</span> <span class="ne">NotImplementedError</span> <span class="kn">from</span> <span class="nn">E</span>
+
+        <span class="bp">self</span><span class="o">.</span><span class="n">cache</span><span class="p">[</span><span class="n">name</span><span class="p">]</span> <span class="o">=</span> <span class="n">mat</span>
+        <span class="k">return</span> <span class="n">mat</span></div>
+
+
+<div class="viewcode-block" id="ChebychevHelper.get_basis_change_matrix">
+<a class="viewcode-back" href="../../pySDC/helpers.spectral_helper.html#helpers.spectral_helper.ChebychevHelper.get_basis_change_matrix">[docs]</a>
+    <span class="k">def</span> <span class="nf">get_basis_change_matrix</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">conv</span><span class="o">=</span><span class="s1">&#39;T2T&#39;</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
+<span class="w">        </span><span class="sd">&quot;&quot;&quot;</span>
+<span class="sd">        As the differentiation matrix in Chebychev-T base is dense but is sparse when simultaneously changing base to</span>
+<span class="sd">        Chebychev-U, you may need a basis change matrix to transfer the other matrices as well. This function returns a</span>
+<span class="sd">        conversion matrix from `ChebychevHelper.get_conv`. Not that `**kwargs` are used to absorb arguments for other</span>
+<span class="sd">        bases, see documentation of `SpectralHelper1D.get_basis_change_matrix`.</span>
+
+<span class="sd">        Args:</span>
+<span class="sd">            conv (str): Conversion code, i.e. T2U</span>
+
+<span class="sd">        Returns:</span>
+<span class="sd">            Sparse conversion matrix</span>
+<span class="sd">        &quot;&quot;&quot;</span>
+        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">get_conv</span><span class="p">(</span><span class="n">conv</span><span class="p">)</span></div>
+
+
+<div class="viewcode-block" id="ChebychevHelper.get_integration_matrix">
+<a class="viewcode-back" href="../../pySDC/helpers.spectral_helper.html#helpers.spectral_helper.ChebychevHelper.get_integration_matrix">[docs]</a>
+    <span class="k">def</span> <span class="nf">get_integration_matrix</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">lbnd</span><span class="o">=</span><span class="mi">0</span><span class="p">):</span>
+<span class="w">        </span><span class="sd">&quot;&quot;&quot;</span>
+<span class="sd">        Get matrix for integration</span>
+
+<span class="sd">        Args:</span>
+<span class="sd">            lbnd (float): Lower bound for integration, only 0 is currently implemented</span>
+
+<span class="sd">        Returns:</span>
+<span class="sd">           Sparse integration matrix</span>
+<span class="sd">        &quot;&quot;&quot;</span>
+        <span class="n">S</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">sparse_lib</span><span class="o">.</span><span class="n">diags</span><span class="p">(</span><span class="mi">1</span> <span class="o">/</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">xp</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">N</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span> <span class="o">+</span> <span class="mi">1</span><span class="p">),</span> <span class="n">offsets</span><span class="o">=-</span><span class="mi">1</span><span class="p">)</span> <span class="o">@</span> <span class="bp">self</span><span class="o">.</span><span class="n">get_conv</span><span class="p">(</span><span class="s1">&#39;T2U&#39;</span><span class="p">)</span>
+        <span class="n">n</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">xp</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">N</span><span class="p">)</span>
+        <span class="k">if</span> <span class="n">lbnd</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
+            <span class="n">S</span> <span class="o">=</span> <span class="n">S</span><span class="o">.</span><span class="n">tocsc</span><span class="p">()</span>
+            <span class="n">S</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">::</span><span class="mi">2</span><span class="p">]</span> <span class="o">=</span> <span class="p">(</span>
+                <span class="p">(</span><span class="n">n</span> <span class="o">/</span> <span class="p">(</span><span class="mi">2</span> <span class="o">*</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">xp</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">N</span><span class="p">)</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)))[</span><span class="mi">1</span><span class="p">::</span><span class="mi">2</span><span class="p">]</span>
+                <span class="o">*</span> <span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span> <span class="o">**</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">xp</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">N</span> <span class="o">//</span> <span class="mi">2</span><span class="p">))</span>
+                <span class="o">/</span> <span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">append</span><span class="p">([</span><span class="mi">1</span><span class="p">],</span> <span class="bp">self</span><span class="o">.</span><span class="n">xp</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">N</span> <span class="o">//</span> <span class="mi">2</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span> <span class="o">+</span> <span class="mi">1</span><span class="p">))</span>
+            <span class="p">)</span>
+        <span class="k">else</span><span class="p">:</span>
+            <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span><span class="sa">f</span><span class="s1">&#39;This function allows to integrate only from x=0, you attempted from x=</span><span class="si">{</span><span class="n">lbnd</span><span class="si">}</span><span class="s1">.&#39;</span><span class="p">)</span>
+        <span class="k">return</span> <span class="n">S</span></div>
+
+
+<div class="viewcode-block" id="ChebychevHelper.get_differentiation_matrix">
+<a class="viewcode-back" href="../../pySDC/helpers.spectral_helper.html#helpers.spectral_helper.ChebychevHelper.get_differentiation_matrix">[docs]</a>
+    <span class="k">def</span> <span class="nf">get_differentiation_matrix</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">p</span><span class="o">=</span><span class="mi">1</span><span class="p">):</span>
+<span class="w">        </span><span class="sd">&#39;&#39;&#39;</span>
+<span class="sd">        Keep in mind that the T2T differentiation matrix is dense.</span>
+
+<span class="sd">        Args:</span>
+<span class="sd">            p (int): Derivative you want to compute</span>
+
+<span class="sd">        Returns:</span>
+<span class="sd">            numpy.ndarray: Differentiation matrix</span>
+<span class="sd">        &#39;&#39;&#39;</span>
+        <span class="n">D</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">xp</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="bp">self</span><span class="o">.</span><span class="n">N</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">N</span><span class="p">))</span>
+        <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">N</span><span class="p">):</span>
+            <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">j</span><span class="p">):</span>
+                <span class="n">D</span><span class="p">[</span><span class="n">k</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="mi">2</span> <span class="o">*</span> <span class="n">j</span> <span class="o">*</span> <span class="p">((</span><span class="n">j</span> <span class="o">-</span> <span class="n">k</span><span class="p">)</span> <span class="o">%</span> <span class="mi">2</span><span class="p">)</span>
+
+        <span class="n">D</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="p">:]</span> <span class="o">/=</span> <span class="mi">2</span>
+        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">sparse_lib</span><span class="o">.</span><span class="n">csc_matrix</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">xp</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">matrix_power</span><span class="p">(</span><span class="n">D</span><span class="p">,</span> <span class="n">p</span><span class="p">))</span></div>
+
+
+<div class="viewcode-block" id="ChebychevHelper.get_norm">
+<a class="viewcode-back" href="../../pySDC/helpers.spectral_helper.html#helpers.spectral_helper.ChebychevHelper.get_norm">[docs]</a>
+    <span class="k">def</span> <span class="nf">get_norm</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">N</span><span class="o">=</span><span class="kc">None</span><span class="p">):</span>
+<span class="w">        </span><span class="sd">&#39;&#39;&#39;</span>
+<span class="sd">        Get normalization for converting Chebychev coefficients and DCT</span>
+
+<span class="sd">        Args:</span>
+<span class="sd">            N (int, optional): Resolution</span>
+
+<span class="sd">        Returns:</span>
+<span class="sd">            self.xp.array: Normalization</span>
+<span class="sd">        &#39;&#39;&#39;</span>
+        <span class="n">N</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">N</span> <span class="k">if</span> <span class="n">N</span> <span class="ow">is</span> <span class="kc">None</span> <span class="k">else</span> <span class="n">N</span>
+        <span class="n">norm</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">xp</span><span class="o">.</span><span class="n">ones</span><span class="p">(</span><span class="n">N</span><span class="p">)</span> <span class="o">/</span> <span class="n">N</span>
+        <span class="n">norm</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">/=</span> <span class="mi">2</span>
+        <span class="k">return</span> <span class="n">norm</span></div>
+
+
+<div class="viewcode-block" id="ChebychevHelper.get_fft_shuffle">
+<a class="viewcode-back" href="../../pySDC/helpers.spectral_helper.html#helpers.spectral_helper.ChebychevHelper.get_fft_shuffle">[docs]</a>
+    <span class="k">def</span> <span class="nf">get_fft_shuffle</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">forward</span><span class="p">,</span> <span class="n">N</span><span class="p">):</span>
+<span class="w">        </span><span class="sd">&quot;&quot;&quot;</span>
+<span class="sd">        In order to more easily parallelize using distributed FFT libraries, we express the DCT via an FFT following</span>
+<span class="sd">        doi.org/10.1109/TASSP.1980.1163351. The idea is based on reshuffling the data to be periodic and rotating it</span>
+<span class="sd">        in the complex plane. This function returns a mask to do the shuffling.</span>
+
+<span class="sd">        Args:</span>
+<span class="sd">            forward (bool): Whether you want the shuffle for forward transform or backward transform</span>
+<span class="sd">            N (int): size of the grid</span>
+
+<span class="sd">        Returns:</span>
+<span class="sd">            self.xp.array: Use as mask</span>
+<span class="sd">        &quot;&quot;&quot;</span>
+        <span class="n">xp</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">xp</span>
+        <span class="k">if</span> <span class="n">forward</span><span class="p">:</span>
+            <span class="k">return</span> <span class="n">xp</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">xp</span><span class="o">.</span><span class="n">arange</span><span class="p">((</span><span class="n">N</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> <span class="o">//</span> <span class="mi">2</span><span class="p">)</span> <span class="o">*</span> <span class="mi">2</span><span class="p">,</span> <span class="o">-</span><span class="n">xp</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="n">N</span> <span class="o">//</span> <span class="mi">2</span><span class="p">)</span> <span class="o">*</span> <span class="mi">2</span> <span class="o">-</span> <span class="mi">1</span> <span class="o">-</span> <span class="n">N</span> <span class="o">%</span> <span class="mi">2</span><span class="p">)</span>
+        <span class="k">else</span><span class="p">:</span>
+            <span class="n">mask</span> <span class="o">=</span> <span class="n">xp</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="n">N</span><span class="p">,</span> <span class="n">dtype</span><span class="o">=</span><span class="nb">int</span><span class="p">)</span>
+            <span class="n">mask</span><span class="p">[:</span> <span class="n">N</span> <span class="o">-</span> <span class="n">N</span> <span class="o">%</span> <span class="mi">2</span> <span class="p">:</span> <span class="mi">2</span><span class="p">]</span> <span class="o">=</span> <span class="n">xp</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="n">N</span> <span class="o">//</span> <span class="mi">2</span><span class="p">)</span>
+            <span class="n">mask</span><span class="p">[</span><span class="mi">1</span><span class="p">::</span><span class="mi">2</span><span class="p">]</span> <span class="o">=</span> <span class="n">N</span> <span class="o">-</span> <span class="n">xp</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="n">N</span> <span class="o">//</span> <span class="mi">2</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span>
+            <span class="n">mask</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">N</span> <span class="o">//</span> <span class="mi">2</span>
+            <span class="k">return</span> <span class="n">mask</span></div>
+
+
+<div class="viewcode-block" id="ChebychevHelper.get_fft_shift">
+<a class="viewcode-back" href="../../pySDC/helpers.spectral_helper.html#helpers.spectral_helper.ChebychevHelper.get_fft_shift">[docs]</a>
+    <span class="k">def</span> <span class="nf">get_fft_shift</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">forward</span><span class="p">,</span> <span class="n">N</span><span class="p">):</span>
+<span class="w">        </span><span class="sd">&quot;&quot;&quot;</span>
+<span class="sd">        As described in the docstring for `get_fft_shuffle`, we need to rotate in the complex plane in order to use FFT for DCT.</span>
+
+<span class="sd">        Args:</span>
+<span class="sd">            forward (bool): Whether you want the rotation for forward transform or backward transform</span>
+<span class="sd">            N (int): size of the grid</span>
+
+<span class="sd">        Returns:</span>
+<span class="sd">            self.xp.array: Rotation</span>
+<span class="sd">        &quot;&quot;&quot;</span>
+        <span class="n">k</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">get_wavenumbers</span><span class="p">()</span>
+        <span class="n">norm</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">get_norm</span><span class="p">()</span>
+        <span class="n">xp</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">xp</span>
+        <span class="k">if</span> <span class="n">forward</span><span class="p">:</span>
+            <span class="k">return</span> <span class="mi">2</span> <span class="o">*</span> <span class="n">xp</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="n">j</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">pi</span> <span class="o">*</span> <span class="n">k</span> <span class="o">/</span> <span class="p">(</span><span class="mi">2</span> <span class="o">*</span> <span class="n">N</span><span class="p">)</span> <span class="o">+</span> <span class="mi">0</span><span class="n">j</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">pi</span> <span class="o">/</span> <span class="mi">4</span><span class="p">)</span> <span class="o">*</span> <span class="n">norm</span>
+        <span class="k">else</span><span class="p">:</span>
+            <span class="n">shift</span> <span class="o">=</span> <span class="n">xp</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="mi">1</span><span class="n">j</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">pi</span> <span class="o">*</span> <span class="n">k</span> <span class="o">/</span> <span class="p">(</span><span class="mi">2</span> <span class="o">*</span> <span class="n">N</span><span class="p">))</span>
+            <span class="n">shift</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="mf">0.5</span>
+            <span class="k">return</span> <span class="n">shift</span> <span class="o">/</span> <span class="n">norm</span></div>
+
+
+<div class="viewcode-block" id="ChebychevHelper.get_fft_utils">
+<a class="viewcode-back" href="../../pySDC/helpers.spectral_helper.html#helpers.spectral_helper.ChebychevHelper.get_fft_utils">[docs]</a>
+    <span class="k">def</span> <span class="nf">get_fft_utils</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
+<span class="w">        </span><span class="sd">&quot;&quot;&quot;</span>
+<span class="sd">        Get the required utilities for using FFT to do DCT as described in the docstring for `get_fft_shuffle` and keep</span>
+<span class="sd">        them cached.</span>
+<span class="sd">        &quot;&quot;&quot;</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">fft_utils</span> <span class="o">=</span> <span class="p">{</span>
+            <span class="s1">&#39;fwd&#39;</span><span class="p">:</span> <span class="p">{},</span>
+            <span class="s1">&#39;bck&#39;</span><span class="p">:</span> <span class="p">{},</span>
+        <span class="p">}</span>
+
+        <span class="c1"># forwards transform</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">fft_utils</span><span class="p">[</span><span class="s1">&#39;fwd&#39;</span><span class="p">][</span><span class="s1">&#39;shuffle&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">get_fft_shuffle</span><span class="p">(</span><span class="kc">True</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">N</span><span class="p">)</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">fft_utils</span><span class="p">[</span><span class="s1">&#39;fwd&#39;</span><span class="p">][</span><span class="s1">&#39;shift&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">get_fft_shift</span><span class="p">(</span><span class="kc">True</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">N</span><span class="p">)</span>
+
+        <span class="c1"># backwards transform</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">fft_utils</span><span class="p">[</span><span class="s1">&#39;bck&#39;</span><span class="p">][</span><span class="s1">&#39;shuffle&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">get_fft_shuffle</span><span class="p">(</span><span class="kc">False</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">N</span><span class="p">)</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">fft_utils</span><span class="p">[</span><span class="s1">&#39;bck&#39;</span><span class="p">][</span><span class="s1">&#39;shift&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">get_fft_shift</span><span class="p">(</span><span class="kc">False</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">N</span><span class="p">)</span>
+
+        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">fft_utils</span></div>
+
+
+<div class="viewcode-block" id="ChebychevHelper.transform">
+<a class="viewcode-back" href="../../pySDC/helpers.spectral_helper.html#helpers.spectral_helper.ChebychevHelper.transform">[docs]</a>
+    <span class="k">def</span> <span class="nf">transform</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">axis</span><span class="o">=-</span><span class="mi">1</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
+<span class="w">        </span><span class="sd">&quot;&quot;&quot;</span>
+<span class="sd">        1D DCT along axis. `kwargs` will be passed on to the FFT library.</span>
+
+<span class="sd">        Args:</span>
+<span class="sd">            u: Data you want to transform</span>
+<span class="sd">            axis (int): Axis you want to transform along</span>
+
+<span class="sd">        Returns:</span>
+<span class="sd">            Data in spectral space</span>
+<span class="sd">        &quot;&quot;&quot;</span>
+        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">transform_type</span><span class="o">.</span><span class="n">lower</span><span class="p">()</span> <span class="o">==</span> <span class="s1">&#39;dct&#39;</span><span class="p">:</span>
+            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">fft_lib</span><span class="o">.</span><span class="n">dct</span><span class="p">(</span><span class="n">u</span><span class="p">,</span> <span class="n">axis</span><span class="o">=</span><span class="n">axis</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span> <span class="o">*</span> <span class="bp">self</span><span class="o">.</span><span class="n">norm</span>
+        <span class="k">elif</span> <span class="bp">self</span><span class="o">.</span><span class="n">transform_type</span><span class="o">.</span><span class="n">lower</span><span class="p">()</span> <span class="o">==</span> <span class="s1">&#39;fft&#39;</span><span class="p">:</span>
+            <span class="n">result</span> <span class="o">=</span> <span class="n">u</span><span class="o">.</span><span class="n">copy</span><span class="p">()</span>
+
+            <span class="n">shuffle</span> <span class="o">=</span> <span class="p">[</span><span class="nb">slice</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">s</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span> <span class="k">for</span> <span class="n">s</span> <span class="ow">in</span> <span class="n">u</span><span class="o">.</span><span class="n">shape</span><span class="p">]</span>
+            <span class="n">shuffle</span><span class="p">[</span><span class="n">axis</span><span class="p">]</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">fft_utils</span><span class="p">[</span><span class="s1">&#39;fwd&#39;</span><span class="p">][</span><span class="s1">&#39;shuffle&#39;</span><span class="p">]</span>
+
+            <span class="n">v</span> <span class="o">=</span> <span class="n">u</span><span class="p">[(</span><span class="o">*</span><span class="n">shuffle</span><span class="p">,)]</span>
+
+            <span class="n">V</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">fft_lib</span><span class="o">.</span><span class="n">fft</span><span class="p">(</span><span class="n">v</span><span class="p">,</span> <span class="n">axis</span><span class="o">=</span><span class="n">axis</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
+
+            <span class="n">expansion</span> <span class="o">=</span> <span class="p">[</span><span class="n">np</span><span class="o">.</span><span class="n">newaxis</span> <span class="k">for</span> <span class="n">_</span> <span class="ow">in</span> <span class="n">u</span><span class="o">.</span><span class="n">shape</span><span class="p">]</span>
+            <span class="n">expansion</span><span class="p">[</span><span class="n">axis</span><span class="p">]</span> <span class="o">=</span> <span class="nb">slice</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">u</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="n">axis</span><span class="p">],</span> <span class="mi">1</span><span class="p">)</span>
+
+            <span class="n">V</span> <span class="o">*=</span> <span class="bp">self</span><span class="o">.</span><span class="n">fft_utils</span><span class="p">[</span><span class="s1">&#39;fwd&#39;</span><span class="p">][</span><span class="s1">&#39;shift&#39;</span><span class="p">][(</span><span class="o">*</span><span class="n">expansion</span><span class="p">,)]</span>
+
+            <span class="n">result</span><span class="o">.</span><span class="n">real</span><span class="p">[</span><span class="o">...</span><span class="p">]</span> <span class="o">=</span> <span class="n">V</span><span class="o">.</span><span class="n">real</span><span class="p">[</span><span class="o">...</span><span class="p">]</span>
+            <span class="k">return</span> <span class="n">result</span>
+        <span class="k">else</span><span class="p">:</span>
+            <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span><span class="sa">f</span><span class="s1">&#39;Please choose a transform type from fft and dct, not </span><span class="si">{</span><span class="bp">self</span><span class="o">.</span><span class="n">transform_type</span><span class="si">=}</span><span class="s1">&#39;</span><span class="p">)</span></div>
+
+
+<div class="viewcode-block" id="ChebychevHelper.itransform">
+<a class="viewcode-back" href="../../pySDC/helpers.spectral_helper.html#helpers.spectral_helper.ChebychevHelper.itransform">[docs]</a>
+    <span class="k">def</span> <span class="nf">itransform</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">axis</span><span class="o">=-</span><span class="mi">1</span><span class="p">):</span>
+<span class="w">        </span><span class="sd">&quot;&quot;&quot;</span>
+<span class="sd">        1D inverse DCT along axis.</span>
+
+<span class="sd">        Args:</span>
+<span class="sd">            u: Data you want to transform</span>
+<span class="sd">            axis (int): Axis you want to transform along</span>
+
+<span class="sd">        Returns:</span>
+<span class="sd">            Data in physical space</span>
+<span class="sd">        &quot;&quot;&quot;</span>
+        <span class="k">assert</span> <span class="bp">self</span><span class="o">.</span><span class="n">norm</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">==</span> <span class="n">u</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="n">axis</span><span class="p">]</span>
+
+        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">transform_type</span> <span class="o">==</span> <span class="s1">&#39;dct&#39;</span><span class="p">:</span>
+            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">fft_lib</span><span class="o">.</span><span class="n">idct</span><span class="p">(</span><span class="n">u</span> <span class="o">/</span> <span class="bp">self</span><span class="o">.</span><span class="n">norm</span><span class="p">,</span> <span class="n">axis</span><span class="o">=</span><span class="n">axis</span><span class="p">)</span>
+        <span class="k">elif</span> <span class="bp">self</span><span class="o">.</span><span class="n">transform_type</span> <span class="o">==</span> <span class="s1">&#39;fft&#39;</span><span class="p">:</span>
+            <span class="n">result</span> <span class="o">=</span> <span class="n">u</span><span class="o">.</span><span class="n">copy</span><span class="p">()</span>
+
+            <span class="n">expansion</span> <span class="o">=</span> <span class="p">[</span><span class="n">np</span><span class="o">.</span><span class="n">newaxis</span> <span class="k">for</span> <span class="n">_</span> <span class="ow">in</span> <span class="n">u</span><span class="o">.</span><span class="n">shape</span><span class="p">]</span>
+            <span class="n">expansion</span><span class="p">[</span><span class="n">axis</span><span class="p">]</span> <span class="o">=</span> <span class="nb">slice</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">u</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="n">axis</span><span class="p">],</span> <span class="mi">1</span><span class="p">)</span>
+
+            <span class="n">v</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">fft_lib</span><span class="o">.</span><span class="n">ifft</span><span class="p">(</span><span class="n">u</span> <span class="o">*</span> <span class="bp">self</span><span class="o">.</span><span class="n">fft_utils</span><span class="p">[</span><span class="s1">&#39;bck&#39;</span><span class="p">][</span><span class="s1">&#39;shift&#39;</span><span class="p">][(</span><span class="o">*</span><span class="n">expansion</span><span class="p">,)],</span> <span class="n">axis</span><span class="o">=</span><span class="n">axis</span><span class="p">)</span>
+
+            <span class="n">shuffle</span> <span class="o">=</span> <span class="p">[</span><span class="nb">slice</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">s</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span> <span class="k">for</span> <span class="n">s</span> <span class="ow">in</span> <span class="n">u</span><span class="o">.</span><span class="n">shape</span><span class="p">]</span>
+            <span class="n">shuffle</span><span class="p">[</span><span class="n">axis</span><span class="p">]</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">fft_utils</span><span class="p">[</span><span class="s1">&#39;bck&#39;</span><span class="p">][</span><span class="s1">&#39;shuffle&#39;</span><span class="p">]</span>
+            <span class="n">V</span> <span class="o">=</span> <span class="n">v</span><span class="p">[(</span><span class="o">*</span><span class="n">shuffle</span><span class="p">,)]</span>
+
+            <span class="n">result</span><span class="o">.</span><span class="n">real</span><span class="p">[</span><span class="o">...</span><span class="p">]</span> <span class="o">=</span> <span class="n">V</span><span class="o">.</span><span class="n">real</span><span class="p">[</span><span class="o">...</span><span class="p">]</span>
+            <span class="k">return</span> <span class="n">result</span>
+        <span class="k">else</span><span class="p">:</span>
+            <span class="k">raise</span> <span class="ne">NotImplementedError</span></div>
+
+
+<div class="viewcode-block" id="ChebychevHelper.get_BC">
+<a class="viewcode-back" href="../../pySDC/helpers.spectral_helper.html#helpers.spectral_helper.ChebychevHelper.get_BC">[docs]</a>
+    <span class="k">def</span> <span class="nf">get_BC</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">kind</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
+<span class="w">        </span><span class="sd">&quot;&quot;&quot;</span>
+<span class="sd">        Get boundary condition row for boundary bordering. `kwargs` will be passed on to implementations of the BC of</span>
+<span class="sd">        the kind you choose. Specifically, `x` for `&#39;dirichlet&#39;` boundary condition, which is the coordinate at which to</span>
+<span class="sd">        set the BC.</span>
+
+<span class="sd">        Args:</span>
+<span class="sd">            kind (&#39;integral&#39; or &#39;dirichlet&#39;): Kind of boundary condition you want</span>
+<span class="sd">        &quot;&quot;&quot;</span>
+        <span class="k">if</span> <span class="n">kind</span><span class="o">.</span><span class="n">lower</span><span class="p">()</span> <span class="o">==</span> <span class="s1">&#39;integral&#39;</span><span class="p">:</span>
+            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">get_integ_BC_row</span><span class="p">(</span><span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
+        <span class="k">elif</span> <span class="n">kind</span><span class="o">.</span><span class="n">lower</span><span class="p">()</span> <span class="o">==</span> <span class="s1">&#39;dirichlet&#39;</span><span class="p">:</span>
+            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">get_Dirichlet_BC_row</span><span class="p">(</span><span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
+        <span class="k">else</span><span class="p">:</span>
+            <span class="k">return</span> <span class="nb">super</span><span class="p">()</span><span class="o">.</span><span class="n">get_BC</span><span class="p">(</span><span class="n">kind</span><span class="p">)</span></div>
+
+
+<div class="viewcode-block" id="ChebychevHelper.get_integ_BC_row">
+<a class="viewcode-back" href="../../pySDC/helpers.spectral_helper.html#helpers.spectral_helper.ChebychevHelper.get_integ_BC_row">[docs]</a>
+    <span class="k">def</span> <span class="nf">get_integ_BC_row</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
+<span class="w">        </span><span class="sd">&quot;&quot;&quot;</span>
+<span class="sd">        Get a row for generating integral BCs with T polynomials.</span>
+<span class="sd">        It returns the values of the integrals of T polynomials over the entire interval.</span>
+
+<span class="sd">        Returns:</span>
+<span class="sd">            self.xp.ndarray: Row to put into a matrix</span>
+<span class="sd">        &quot;&quot;&quot;</span>
+        <span class="n">n</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">xp</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">N</span><span class="p">)</span> <span class="o">+</span> <span class="mi">1</span>
+        <span class="n">me</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">xp</span><span class="o">.</span><span class="n">zeros_like</span><span class="p">(</span><span class="n">n</span><span class="p">)</span><span class="o">.</span><span class="n">astype</span><span class="p">(</span><span class="nb">float</span><span class="p">)</span>
+        <span class="n">me</span><span class="p">[</span><span class="mi">2</span><span class="p">:]</span> <span class="o">=</span> <span class="p">((</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span> <span class="o">**</span> <span class="n">n</span><span class="p">[</span><span class="mi">1</span><span class="p">:</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> <span class="o">/</span> <span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">n</span><span class="p">[</span><span class="mi">1</span><span class="p">:</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="o">**</span> <span class="mi">2</span><span class="p">)</span>
+        <span class="n">me</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="mf">2.0</span>
+        <span class="k">return</span> <span class="n">me</span></div>
+
+
+<div class="viewcode-block" id="ChebychevHelper.get_Dirichlet_BC_row">
+<a class="viewcode-back" href="../../pySDC/helpers.spectral_helper.html#helpers.spectral_helper.ChebychevHelper.get_Dirichlet_BC_row">[docs]</a>
+    <span class="k">def</span> <span class="nf">get_Dirichlet_BC_row</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
+<span class="w">        </span><span class="sd">&quot;&quot;&quot;</span>
+<span class="sd">        Get a row for generating Dirichlet BCs at x with T polynomials.</span>
+<span class="sd">        It returns the values of the T polynomials at x.</span>
+
+<span class="sd">        Args:</span>
+<span class="sd">            x (float): Position of the boundary condition</span>
+
+<span class="sd">        Returns:</span>
+<span class="sd">            self.xp.ndarray: Row to put into a matrix</span>
+<span class="sd">        &quot;&quot;&quot;</span>
+        <span class="k">if</span> <span class="n">x</span> <span class="o">==</span> <span class="o">-</span><span class="mi">1</span><span class="p">:</span>
+            <span class="k">return</span> <span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span> <span class="o">**</span> <span class="bp">self</span><span class="o">.</span><span class="n">xp</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">N</span><span class="p">)</span>
+        <span class="k">elif</span> <span class="n">x</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
+            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">xp</span><span class="o">.</span><span class="n">ones</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">N</span><span class="p">)</span>
+        <span class="k">elif</span> <span class="n">x</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
+            <span class="n">n</span> <span class="o">=</span> <span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span> <span class="o">**</span> <span class="bp">self</span><span class="o">.</span><span class="n">xp</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">N</span><span class="p">))</span> <span class="o">/</span> <span class="mi">2</span>
+            <span class="n">n</span><span class="p">[</span><span class="mi">2</span><span class="p">::</span><span class="mi">4</span><span class="p">]</span> <span class="o">*=</span> <span class="o">-</span><span class="mi">1</span>
+            <span class="k">return</span> <span class="n">n</span>
+        <span class="k">else</span><span class="p">:</span>
+            <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span><span class="sa">f</span><span class="s1">&#39;Don</span><span class="se">\&#39;</span><span class="s1">t know how to generate Dirichlet BC</span><span class="se">\&#39;</span><span class="s1">s at </span><span class="si">{</span><span class="n">x</span><span class="si">=}</span><span class="s1">!&#39;</span><span class="p">)</span></div>
+
+
+<div class="viewcode-block" id="ChebychevHelper.get_Dirichlet_recombination_matrix">
+<a class="viewcode-back" href="../../pySDC/helpers.spectral_helper.html#helpers.spectral_helper.ChebychevHelper.get_Dirichlet_recombination_matrix">[docs]</a>
+    <span class="k">def</span> <span class="nf">get_Dirichlet_recombination_matrix</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
+<span class="w">        </span><span class="sd">&#39;&#39;&#39;</span>
+<span class="sd">        Get matrix for Dirichlet recombination, which changes the basis to have sparse boundary conditions.</span>
+<span class="sd">        This makes for a good right preconditioner.</span>
+
+<span class="sd">        Returns:</span>
+<span class="sd">            scipy.sparse: Sparse conversion matrix</span>
+<span class="sd">        &#39;&#39;&#39;</span>
+        <span class="n">N</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">N</span>
+        <span class="n">sp</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">sparse_lib</span>
+        <span class="n">xp</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">xp</span>
+
+        <span class="k">return</span> <span class="n">sp</span><span class="o">.</span><span class="n">eye</span><span class="p">(</span><span class="n">N</span><span class="p">)</span> <span class="o">-</span> <span class="n">sp</span><span class="o">.</span><span class="n">diags</span><span class="p">(</span><span class="n">xp</span><span class="o">.</span><span class="n">ones</span><span class="p">(</span><span class="n">N</span> <span class="o">-</span> <span class="mi">2</span><span class="p">),</span> <span class="n">offsets</span><span class="o">=+</span><span class="mi">2</span><span class="p">)</span></div>
+</div>
+
+
+
+<div class="viewcode-block" id="UltrasphericalHelper">
+<a class="viewcode-back" href="../../pySDC/helpers.spectral_helper.html#helpers.spectral_helper.UltrasphericalHelper">[docs]</a>
+<span class="k">class</span> <span class="nc">UltrasphericalHelper</span><span class="p">(</span><span class="n">ChebychevHelper</span><span class="p">):</span>
+<span class="w">    </span><span class="sd">&quot;&quot;&quot;</span>
+<span class="sd">    This implementation follows https://doi.org/10.1137/120865458.</span>
+<span class="sd">    The ultraspherical method works in Chebychev polynomials as well, but also uses various Gegenbauer polynomials.</span>
+<span class="sd">    The idea is that for every derivative of Chebychev T polynomials, there is a basis of Gegenbauer polynomials where the differentiation matrix is a single off-diagonal.</span>
+<span class="sd">    There are also conversion operators from one derivative basis to the next that are sparse.</span>
+
+<span class="sd">    This basis is used like this: For every equation that you have, look for the highest derivative and bump all matrices to the correct basis. If your highest derivative is 2 and you have an identity, it needs to get bumped from 0 to 1 and from 1 to 2. If you have a first derivative as well, it needs to be bumped from 1 to 2.</span>
+<span class="sd">    You don&#39;t need the same resulting basis in all equations. You just need to take care that you translate the right hand side to the correct basis as well.</span>
+<span class="sd">    &quot;&quot;&quot;</span>
+
+<div class="viewcode-block" id="UltrasphericalHelper.get_differentiation_matrix">
+<a class="viewcode-back" href="../../pySDC/helpers.spectral_helper.html#helpers.spectral_helper.UltrasphericalHelper.get_differentiation_matrix">[docs]</a>
+    <span class="k">def</span> <span class="nf">get_differentiation_matrix</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">p</span><span class="o">=</span><span class="mi">1</span><span class="p">):</span>
+<span class="w">        </span><span class="sd">&quot;&quot;&quot;</span>
+<span class="sd">        Notice that while sparse, this matrix is not diagonal, which means the inversion cannot be parallelized easily.</span>
+
+<span class="sd">        Args:</span>
+<span class="sd">            p (int): Order of the derivative</span>
+
+<span class="sd">        Returns:</span>
+<span class="sd">            sparse differentiation matrix</span>
+<span class="sd">        &quot;&quot;&quot;</span>
+        <span class="n">sp</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">sparse_lib</span>
+        <span class="n">xp</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">xp</span>
+        <span class="n">N</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">N</span>
+        <span class="n">l</span> <span class="o">=</span> <span class="n">p</span>
+        <span class="k">return</span> <span class="mi">2</span> <span class="o">**</span> <span class="p">(</span><span class="n">l</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span> <span class="o">*</span> <span class="n">factorial</span><span class="p">(</span><span class="n">l</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span> <span class="o">*</span> <span class="n">sp</span><span class="o">.</span><span class="n">diags</span><span class="p">(</span><span class="n">xp</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="n">N</span> <span class="o">-</span> <span class="n">l</span><span class="p">)</span> <span class="o">+</span> <span class="n">l</span><span class="p">,</span> <span class="n">offsets</span><span class="o">=</span><span class="n">l</span><span class="p">)</span></div>
+
+
+<div class="viewcode-block" id="UltrasphericalHelper.get_S">
+<a class="viewcode-back" href="../../pySDC/helpers.spectral_helper.html#helpers.spectral_helper.UltrasphericalHelper.get_S">[docs]</a>
+    <span class="k">def</span> <span class="nf">get_S</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">lmbda</span><span class="p">):</span>
+<span class="w">        </span><span class="sd">&quot;&quot;&quot;</span>
+<span class="sd">        Get matrix for bumping the derivative base by one from lmbda to lmbda + 1. This is the same language as in</span>
+<span class="sd">        https://doi.org/10.1137/120865458.</span>
+
+<span class="sd">        Args:</span>
+<span class="sd">            lmbda (int): Ingoing derivative base</span>
+
+<span class="sd">        Returns:</span>
+<span class="sd">            sparse matrix: Conversion from derivative base lmbda to lmbda + 1</span>
+<span class="sd">        &quot;&quot;&quot;</span>
+        <span class="n">N</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">N</span>
+
+        <span class="k">if</span> <span class="n">lmbda</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
+            <span class="n">sp</span> <span class="o">=</span> <span class="n">scipy</span><span class="o">.</span><span class="n">sparse</span>
+            <span class="n">mat</span> <span class="o">=</span> <span class="p">((</span><span class="n">sp</span><span class="o">.</span><span class="n">eye</span><span class="p">(</span><span class="n">N</span><span class="p">)</span> <span class="o">-</span> <span class="n">sp</span><span class="o">.</span><span class="n">diags</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">ones</span><span class="p">(</span><span class="n">N</span> <span class="o">-</span> <span class="mi">2</span><span class="p">),</span> <span class="n">offsets</span><span class="o">=+</span><span class="mi">2</span><span class="p">))</span> <span class="o">/</span> <span class="mf">2.0</span><span class="p">)</span><span class="o">.</span><span class="n">tolil</span><span class="p">()</span>
+            <span class="n">mat</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">]</span> <span class="o">*=</span> <span class="mi">2</span>
+        <span class="k">else</span><span class="p">:</span>
+            <span class="n">sp</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">sparse_lib</span>
+            <span class="n">xp</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">xp</span>
+            <span class="n">mat</span> <span class="o">=</span> <span class="n">sp</span><span class="o">.</span><span class="n">diags</span><span class="p">(</span><span class="n">lmbda</span> <span class="o">/</span> <span class="p">(</span><span class="n">lmbda</span> <span class="o">+</span> <span class="n">xp</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="n">N</span><span class="p">)))</span> <span class="o">-</span> <span class="n">sp</span><span class="o">.</span><span class="n">diags</span><span class="p">(</span>
+                <span class="n">lmbda</span> <span class="o">/</span> <span class="p">(</span><span class="n">lmbda</span> <span class="o">+</span> <span class="mi">2</span> <span class="o">+</span> <span class="n">xp</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="n">N</span> <span class="o">-</span> <span class="mi">2</span><span class="p">)),</span> <span class="n">offsets</span><span class="o">=+</span><span class="mi">2</span>
+            <span class="p">)</span>
+
+        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">sparse_lib</span><span class="o">.</span><span class="n">csc_matrix</span><span class="p">(</span><span class="n">mat</span><span class="p">)</span></div>
+
+
+<div class="viewcode-block" id="UltrasphericalHelper.get_basis_change_matrix">
+<a class="viewcode-back" href="../../pySDC/helpers.spectral_helper.html#helpers.spectral_helper.UltrasphericalHelper.get_basis_change_matrix">[docs]</a>
+    <span class="k">def</span> <span class="nf">get_basis_change_matrix</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">p_in</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span> <span class="n">p_out</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
+<span class="w">        </span><span class="sd">&quot;&quot;&quot;</span>
+<span class="sd">        Get a conversion matrix from derivative base `p_in` to `p_out`.</span>
+
+<span class="sd">        Args:</span>
+<span class="sd">            p_out (int): Resulting derivative base</span>
+<span class="sd">            p_in (int): Ingoing derivative base</span>
+<span class="sd">        &quot;&quot;&quot;</span>
+        <span class="n">mat_fwd</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">sparse_lib</span><span class="o">.</span><span class="n">eye</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">N</span><span class="p">)</span>
+        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">min</span><span class="p">([</span><span class="n">p_in</span><span class="p">,</span> <span class="n">p_out</span><span class="p">]),</span> <span class="nb">max</span><span class="p">([</span><span class="n">p_in</span><span class="p">,</span> <span class="n">p_out</span><span class="p">])):</span>
+            <span class="n">mat_fwd</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">get_S</span><span class="p">(</span><span class="n">i</span><span class="p">)</span> <span class="o">@</span> <span class="n">mat_fwd</span>
+
+        <span class="k">if</span> <span class="n">p_out</span> <span class="o">&gt;</span> <span class="n">p_in</span><span class="p">:</span>
+            <span class="k">return</span> <span class="n">mat_fwd</span>
+
+        <span class="k">else</span><span class="p">:</span>
+            <span class="c1"># We have to invert the matrix on CPU because the GPU equivalent is not implemented in CuPy at the time of writing.</span>
+            <span class="kn">import</span> <span class="nn">scipy.sparse</span> <span class="k">as</span> <span class="nn">sp</span>
+
+            <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">useGPU</span><span class="p">:</span>
+                <span class="n">mat_fwd</span> <span class="o">=</span> <span class="n">mat_fwd</span><span class="o">.</span><span class="n">get</span><span class="p">()</span>
+
+            <span class="n">mat_bck</span> <span class="o">=</span> <span class="n">sp</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">inv</span><span class="p">(</span><span class="n">mat_fwd</span><span class="o">.</span><span class="n">tocsc</span><span class="p">())</span>
+
+            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">sparse_lib</span><span class="o">.</span><span class="n">csc_matrix</span><span class="p">(</span><span class="n">mat_bck</span><span class="p">)</span></div>
+
+
+<div class="viewcode-block" id="UltrasphericalHelper.get_integration_matrix">
+<a class="viewcode-back" href="../../pySDC/helpers.spectral_helper.html#helpers.spectral_helper.UltrasphericalHelper.get_integration_matrix">[docs]</a>
+    <span class="k">def</span> <span class="nf">get_integration_matrix</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
+<span class="w">        </span><span class="sd">&quot;&quot;&quot;</span>
+<span class="sd">        Get an integration matrix. Please use `UltrasphericalHelper.get_integration_constant` afterwards to compute the</span>
+<span class="sd">        integration constant such that integration starts from x=-1.</span>
+
+<span class="sd">        Example:</span>
+
+<span class="sd">        .. code-block:: python</span>
+
+<span class="sd">            import numpy as np</span>
+<span class="sd">            from pySDC.helpers.spectral_helper import UltrasphericalHelper</span>
+
+<span class="sd">            N = 4</span>
+<span class="sd">            helper = UltrasphericalHelper(N)</span>
+<span class="sd">            coeffs = np.random.random(N)</span>
+<span class="sd">            coeffs[-1] = 0</span>
+
+<span class="sd">            poly = np.polynomial.Chebyshev(coeffs)</span>
+
+<span class="sd">            S = helper.get_integration_matrix()</span>
+<span class="sd">            U_hat = S @ coeffs</span>
+<span class="sd">            U_hat[0] = helper.get_integration_constant(U_hat, axis=-1)</span>
+
+<span class="sd">            assert np.allclose(poly.integ(lbnd=-1).coef[:-1], U_hat)</span>
+
+<span class="sd">        Returns:</span>
+<span class="sd">            sparse integration matrix</span>
+<span class="sd">        &quot;&quot;&quot;</span>
+        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">sparse_lib</span><span class="o">.</span><span class="n">diags</span><span class="p">(</span><span class="mi">1</span> <span class="o">/</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">xp</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">N</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span> <span class="o">+</span> <span class="mi">1</span><span class="p">),</span> <span class="n">offsets</span><span class="o">=-</span><span class="mi">1</span><span class="p">)</span> <span class="o">@</span> <span class="bp">self</span><span class="o">.</span><span class="n">get_basis_change_matrix</span><span class="p">(</span>
+            <span class="n">p_out</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">p_in</span><span class="o">=</span><span class="mi">0</span>
+        <span class="p">)</span></div>
+
+
+<div class="viewcode-block" id="UltrasphericalHelper.get_integration_constant">
+<a class="viewcode-back" href="../../pySDC/helpers.spectral_helper.html#helpers.spectral_helper.UltrasphericalHelper.get_integration_constant">[docs]</a>
+    <span class="k">def</span> <span class="nf">get_integration_constant</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">u_hat</span><span class="p">,</span> <span class="n">axis</span><span class="p">):</span>
+<span class="w">        </span><span class="sd">&quot;&quot;&quot;</span>
+<span class="sd">        Get integration constant for lower bound of -1. See documentation of `UltrasphericalHelper.get_integration_matrix` for details.</span>
+
+<span class="sd">        Args:</span>
+<span class="sd">            u_hat: Solution in spectral space</span>
+<span class="sd">            axis: Axis you want to integrate over</span>
+
+<span class="sd">        Returns:</span>
+<span class="sd">            Integration constant, has one less dimension than `u_hat`</span>
+<span class="sd">        &quot;&quot;&quot;</span>
+        <span class="n">slices</span> <span class="o">=</span> <span class="p">[</span>
+            <span class="kc">None</span><span class="p">,</span>
+        <span class="p">]</span> <span class="o">*</span> <span class="n">u_hat</span><span class="o">.</span><span class="n">ndim</span>
+        <span class="n">slices</span><span class="p">[</span><span class="n">axis</span><span class="p">]</span> <span class="o">=</span> <span class="nb">slice</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">u_hat</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="n">axis</span><span class="p">])</span>
+        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">xp</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">u_hat</span><span class="p">[(</span><span class="o">*</span><span class="n">slices</span><span class="p">,)]</span> <span class="o">*</span> <span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span> <span class="o">**</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">xp</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="n">u_hat</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="n">axis</span><span class="p">]</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)),</span> <span class="n">axis</span><span class="o">=</span><span class="n">axis</span><span class="p">)</span></div>
+</div>
+
+
+
+<div class="viewcode-block" id="FFTHelper">
+<a class="viewcode-back" href="../../pySDC/helpers.spectral_helper.html#helpers.spectral_helper.FFTHelper">[docs]</a>
+<span class="k">class</span> <span class="nc">FFTHelper</span><span class="p">(</span><span class="n">SpectralHelper1D</span><span class="p">):</span>
+    <span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="n">x0</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span> <span class="n">x1</span><span class="o">=</span><span class="mi">2</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">pi</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
+<span class="w">        </span><span class="sd">&quot;&quot;&quot;</span>
+<span class="sd">        Constructor.</span>
+<span class="sd">        Please refer to the parent class for additional arguments. Notably, you have to supply a resolution `N` and you</span>
+<span class="sd">        may choose to run on GPUs via the `useGPU` argument.</span>
+
+<span class="sd">        Args:</span>
+<span class="sd">            transform_type (&#39;fft&#39; or &#39;dct&#39;): Either use DCT functions directly implemented in the transform library or</span>
+<span class="sd">                                             use the FFT from the library to compute the DCT</span>
+<span class="sd">            x0 (float, optional): Coordinate of left boundary</span>
+<span class="sd">            x1 (float, optional): Coordinate of right boundary</span>
+<span class="sd">        &quot;&quot;&quot;</span>
+        <span class="nb">super</span><span class="p">()</span><span class="o">.</span><span class="fm">__init__</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="n">x0</span><span class="o">=</span><span class="n">x0</span><span class="p">,</span> <span class="n">x1</span><span class="o">=</span><span class="n">x1</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
+
+<div class="viewcode-block" id="FFTHelper.get_1dgrid">
+<a class="viewcode-back" href="../../pySDC/helpers.spectral_helper.html#helpers.spectral_helper.FFTHelper.get_1dgrid">[docs]</a>
+    <span class="k">def</span> <span class="nf">get_1dgrid</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
+<span class="w">        </span><span class="sd">&quot;&quot;&quot;</span>
+<span class="sd">        We use equally spaced points including the left boundary and not including the right one, which is the left boundary.</span>
+<span class="sd">        &quot;&quot;&quot;</span>
+        <span class="n">dx</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">L</span> <span class="o">/</span> <span class="bp">self</span><span class="o">.</span><span class="n">N</span>
+        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">xp</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">N</span><span class="p">)</span> <span class="o">*</span> <span class="n">dx</span> <span class="o">+</span> <span class="bp">self</span><span class="o">.</span><span class="n">x0</span></div>
+
+
+<div class="viewcode-block" id="FFTHelper.get_wavenumbers">
+<a class="viewcode-back" href="../../pySDC/helpers.spectral_helper.html#helpers.spectral_helper.FFTHelper.get_wavenumbers">[docs]</a>
+    <span class="k">def</span> <span class="nf">get_wavenumbers</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
+<span class="w">        </span><span class="sd">&quot;&quot;&quot;</span>
+<span class="sd">        Be careful that this ordering is very unintuitive.</span>
+<span class="sd">        &quot;&quot;&quot;</span>
+        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">xp</span><span class="o">.</span><span class="n">fft</span><span class="o">.</span><span class="n">fftfreq</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">N</span><span class="p">,</span> <span class="mf">1.0</span> <span class="o">/</span> <span class="bp">self</span><span class="o">.</span><span class="n">N</span><span class="p">)</span> <span class="o">*</span> <span class="mi">2</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">pi</span> <span class="o">/</span> <span class="bp">self</span><span class="o">.</span><span class="n">L</span></div>
+
+
+<div class="viewcode-block" id="FFTHelper.get_differentiation_matrix">
+<a class="viewcode-back" href="../../pySDC/helpers.spectral_helper.html#helpers.spectral_helper.FFTHelper.get_differentiation_matrix">[docs]</a>
+    <span class="k">def</span> <span class="nf">get_differentiation_matrix</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">p</span><span class="o">=</span><span class="mi">1</span><span class="p">):</span>
+<span class="w">        </span><span class="sd">&quot;&quot;&quot;</span>
+<span class="sd">        This matrix is diagonal, allowing to invert concurrently.</span>
+
+<span class="sd">        Args:</span>
+<span class="sd">            p (int): Order of the derivative</span>
+
+<span class="sd">        Returns:</span>
+<span class="sd">            sparse differentiation matrix</span>
+<span class="sd">        &quot;&quot;&quot;</span>
+        <span class="n">k</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">get_wavenumbers</span><span class="p">()</span>
+
+        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">useGPU</span><span class="p">:</span>
+            <span class="c1"># Have to raise the matrix to power p on CPU because the GPU equivalent is not implemented in CuPy at the time of writing.</span>
+            <span class="kn">import</span> <span class="nn">scipy.sparse</span> <span class="k">as</span> <span class="nn">sp</span>
+
+            <span class="n">D</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">sparse_lib</span><span class="o">.</span><span class="n">diags</span><span class="p">(</span><span class="mi">1</span><span class="n">j</span> <span class="o">*</span> <span class="n">k</span><span class="p">)</span><span class="o">.</span><span class="n">get</span><span class="p">()</span>
+            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">sparse_lib</span><span class="o">.</span><span class="n">csc_matrix</span><span class="p">(</span><span class="n">sp</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">matrix_power</span><span class="p">(</span><span class="n">D</span><span class="p">,</span> <span class="n">p</span><span class="p">))</span>
+        <span class="k">else</span><span class="p">:</span>
+            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">matrix_power</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">sparse_lib</span><span class="o">.</span><span class="n">diags</span><span class="p">(</span><span class="mi">1</span><span class="n">j</span> <span class="o">*</span> <span class="n">k</span><span class="p">),</span> <span class="n">p</span><span class="p">)</span></div>
+
+
+<div class="viewcode-block" id="FFTHelper.get_integration_matrix">
+<a class="viewcode-back" href="../../pySDC/helpers.spectral_helper.html#helpers.spectral_helper.FFTHelper.get_integration_matrix">[docs]</a>
+    <span class="k">def</span> <span class="nf">get_integration_matrix</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">p</span><span class="o">=</span><span class="mi">1</span><span class="p">):</span>
+<span class="w">        </span><span class="sd">&quot;&quot;&quot;</span>
+<span class="sd">        Get integration matrix to compute `p`-th integral over the entire domain.</span>
+
+<span class="sd">        Args:</span>
+<span class="sd">            p (int): Order of integral you want to compute</span>
+
+<span class="sd">        Returns:</span>
+<span class="sd">            sparse integration matrix</span>
+<span class="sd">        &quot;&quot;&quot;</span>
+        <span class="n">k</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">xp</span><span class="o">.</span><span class="n">array</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">get_wavenumbers</span><span class="p">(),</span> <span class="n">dtype</span><span class="o">=</span><span class="s1">&#39;complex128&#39;</span><span class="p">)</span>
+        <span class="n">k</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="mi">1</span><span class="n">j</span> <span class="o">*</span> <span class="bp">self</span><span class="o">.</span><span class="n">L</span>
+        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">matrix_power</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">sparse_lib</span><span class="o">.</span><span class="n">diags</span><span class="p">(</span><span class="mi">1</span> <span class="o">/</span> <span class="p">(</span><span class="mi">1</span><span class="n">j</span> <span class="o">*</span> <span class="n">k</span><span class="p">)),</span> <span class="n">p</span><span class="p">)</span></div>
+
+
+<div class="viewcode-block" id="FFTHelper.transform">
+<a class="viewcode-back" href="../../pySDC/helpers.spectral_helper.html#helpers.spectral_helper.FFTHelper.transform">[docs]</a>
+    <span class="k">def</span> <span class="nf">transform</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">axis</span><span class="o">=-</span><span class="mi">1</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
+<span class="w">        </span><span class="sd">&quot;&quot;&quot;</span>
+<span class="sd">        1D FFT along axis. `kwargs` are passed on to the FFT library.</span>
+
+<span class="sd">        Args:</span>
+<span class="sd">            u: Data you want to transform</span>
+<span class="sd">            axis (int): Axis you want to transform along</span>
+
+<span class="sd">        Returns:</span>
+<span class="sd">            transformed data</span>
+<span class="sd">        &quot;&quot;&quot;</span>
+        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">fft_lib</span><span class="o">.</span><span class="n">fft</span><span class="p">(</span><span class="n">u</span><span class="p">,</span> <span class="n">axis</span><span class="o">=</span><span class="n">axis</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span></div>
+
+
+<div class="viewcode-block" id="FFTHelper.itransform">
+<a class="viewcode-back" href="../../pySDC/helpers.spectral_helper.html#helpers.spectral_helper.FFTHelper.itransform">[docs]</a>
+    <span class="k">def</span> <span class="nf">itransform</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">axis</span><span class="o">=-</span><span class="mi">1</span><span class="p">):</span>
+<span class="w">        </span><span class="sd">&quot;&quot;&quot;</span>
+<span class="sd">        Inverse 1D FFT.</span>
+
+<span class="sd">        Args:</span>
+<span class="sd">            u: Data you want to transform</span>
+<span class="sd">            axis (int): Axis you want to transform along</span>
+
+<span class="sd">        Returns:</span>
+<span class="sd">            transformed data</span>
+<span class="sd">        &quot;&quot;&quot;</span>
+        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">fft_lib</span><span class="o">.</span><span class="n">ifft</span><span class="p">(</span><span class="n">u</span><span class="p">,</span> <span class="n">axis</span><span class="o">=</span><span class="n">axis</span><span class="p">)</span></div>
+
+
+<div class="viewcode-block" id="FFTHelper.get_BC">
+<a class="viewcode-back" href="../../pySDC/helpers.spectral_helper.html#helpers.spectral_helper.FFTHelper.get_BC">[docs]</a>
+    <span class="k">def</span> <span class="nf">get_BC</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">kind</span><span class="p">):</span>
+<span class="w">        </span><span class="sd">&quot;&quot;&quot;</span>
+<span class="sd">        Get a sort of boundary condition. You can use `kind=integral`, to fix the integral, or you can use `kind=Nyquist`.</span>
+<span class="sd">        The latter is not really a boundary condition, but is used to set the Nyquist mode to some value, preferably zero.</span>
+<span class="sd">        You should set the Nyquist mode zero when the solution in physical space is real and the resolution is even.</span>
+
+<span class="sd">        Args:</span>
+<span class="sd">            kind (&#39;integral&#39; or &#39;nyquist&#39;): Kind of BC</span>
+
+<span class="sd">        Returns:</span>
+<span class="sd">            self.xp.ndarray: Boundary condition row</span>
+<span class="sd">        &quot;&quot;&quot;</span>
+        <span class="k">if</span> <span class="n">kind</span><span class="o">.</span><span class="n">lower</span><span class="p">()</span> <span class="o">==</span> <span class="s1">&#39;integral&#39;</span><span class="p">:</span>
+            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">get_integ_BC_row</span><span class="p">()</span>
+        <span class="k">elif</span> <span class="n">kind</span><span class="o">.</span><span class="n">lower</span><span class="p">()</span> <span class="o">==</span> <span class="s1">&#39;nyquist&#39;</span><span class="p">:</span>
+            <span class="k">assert</span> <span class="p">(</span>
+                <span class="bp">self</span><span class="o">.</span><span class="n">N</span> <span class="o">%</span> <span class="mi">2</span> <span class="o">==</span> <span class="mi">0</span>
+            <span class="p">),</span> <span class="sa">f</span><span class="s1">&#39;Do not eliminate the Nyquist mode with odd resolution as it is fully resolved. You chose </span><span class="si">{</span><span class="bp">self</span><span class="o">.</span><span class="n">N</span><span class="si">}</span><span class="s1"> in this axis&#39;</span>
+            <span class="n">BC</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">xp</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">N</span><span class="p">)</span>
+            <span class="n">BC</span><span class="p">[</span><span class="bp">self</span><span class="o">.</span><span class="n">get_Nyquist_mode_index</span><span class="p">()]</span> <span class="o">=</span> <span class="mi">1</span>
+            <span class="k">return</span> <span class="n">BC</span>
+        <span class="k">else</span><span class="p">:</span>
+            <span class="k">return</span> <span class="nb">super</span><span class="p">()</span><span class="o">.</span><span class="n">get_BC</span><span class="p">(</span><span class="n">kind</span><span class="p">)</span></div>
+
+
+<div class="viewcode-block" id="FFTHelper.get_Nyquist_mode_index">
+<a class="viewcode-back" href="../../pySDC/helpers.spectral_helper.html#helpers.spectral_helper.FFTHelper.get_Nyquist_mode_index">[docs]</a>
+    <span class="k">def</span> <span class="nf">get_Nyquist_mode_index</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
+<span class="w">        </span><span class="sd">&quot;&quot;&quot;</span>
+<span class="sd">        Compute the index of the Nyquist mode, i.e. the mode with the lowest wavenumber, which doesn&#39;t have a positive</span>
+<span class="sd">        counterpart for even resolution. This means real waves of this wave number cannot be properly resolved and you</span>
+<span class="sd">        are best advised to set this mode zero if representing real functions on even-resolution grids is what you&#39;re</span>
+<span class="sd">        after.</span>
+
+<span class="sd">        Returns:</span>
+<span class="sd">            int: Index of the Nyquist mode</span>
+<span class="sd">        &quot;&quot;&quot;</span>
+        <span class="n">k</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">get_wavenumbers</span><span class="p">()</span>
+        <span class="n">Nyquist_mode</span> <span class="o">=</span> <span class="nb">min</span><span class="p">(</span><span class="n">k</span><span class="p">)</span>
+        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">xp</span><span class="o">.</span><span class="n">where</span><span class="p">(</span><span class="n">k</span> <span class="o">==</span> <span class="n">Nyquist_mode</span><span class="p">)[</span><span class="mi">0</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span></div>
+
+
+<div class="viewcode-block" id="FFTHelper.get_integ_BC_row">
+<a class="viewcode-back" href="../../pySDC/helpers.spectral_helper.html#helpers.spectral_helper.FFTHelper.get_integ_BC_row">[docs]</a>
+    <span class="k">def</span> <span class="nf">get_integ_BC_row</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
+<span class="w">        </span><span class="sd">&quot;&quot;&quot;</span>
+<span class="sd">        Only the 0-mode has non-zero integral with FFT basis in periodic BCs</span>
+<span class="sd">        &quot;&quot;&quot;</span>
+        <span class="n">me</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">xp</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">N</span><span class="p">)</span>
+        <span class="n">me</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">L</span> <span class="o">/</span> <span class="bp">self</span><span class="o">.</span><span class="n">N</span>
+        <span class="k">return</span> <span class="n">me</span></div>
+</div>
+
+
+
+<div class="viewcode-block" id="SpectralHelper">
+<a class="viewcode-back" href="../../pySDC/helpers.spectral_helper.html#helpers.spectral_helper.SpectralHelper">[docs]</a>
+<span class="k">class</span> <span class="nc">SpectralHelper</span><span class="p">:</span>
+<span class="w">    </span><span class="sd">&quot;&quot;&quot;</span>
+<span class="sd">    This class has three functions:</span>
+<span class="sd">      - Easily assemble matrices containing multiple equations</span>
+<span class="sd">      - Direct product of 1D bases to solve problems in more dimensions</span>
+<span class="sd">      - Distribute the FFTs to facilitate concurrency.</span>
+
+<span class="sd">    Attributes:</span>
+<span class="sd">        comm (mpi4py.Intracomm): MPI communicator</span>
+<span class="sd">        debug (bool): Perform additional checks at extra computational cost</span>
+<span class="sd">        useGPU (bool): Whether to use GPUs</span>
+<span class="sd">        axes (list): List of 1D bases</span>
+<span class="sd">        components (list): List of strings of the names of components in the equations</span>
+<span class="sd">        full_BCs (list): List of Dictionaries containing all information about the boundary conditions</span>
+<span class="sd">        BC_mat (list): List of lists of sparse matrices to put BCs into and eventually assemble the BC matrix from</span>
+<span class="sd">        BCs (sparse matrix): Matrix containing only the BCs</span>
+<span class="sd">        fft_cache (dict): Cache FFTs of various shapes here to facilitate padding and so on</span>
+<span class="sd">        BC_rhs_mask (self.xp.ndarray): Mask values that contain boundary conditions in the right hand side</span>
+<span class="sd">        BC_zero_index (self.xp.ndarray): Indeces of rows in the matrix that are replaced by BCs</span>
+<span class="sd">        BC_line_zero_matrix (sparse matrix): Matrix that zeros rows where we can then add the BCs in using `BCs`</span>
+<span class="sd">        rhs_BCs_hat (self.xp.ndarray): Boundary conditions in spectral space</span>
+<span class="sd">        global_shape (tuple): Global shape of the solution as in `mpi4py-fft`</span>
+<span class="sd">        local_slice (slice): Local slice of the solution as in `mpi4py-fft`</span>
+<span class="sd">        fft_obj: When using distributed FFTs, this will be a parallel transform object from `mpi4py-fft`</span>
+<span class="sd">        init (tuple): This is the same `init` that is used throughout the problem classes</span>
+<span class="sd">        init_forward (tuple): This is the equivalent of `init` in spectral space</span>
+<span class="sd">    &quot;&quot;&quot;</span>
+
+    <span class="n">xp</span> <span class="o">=</span> <span class="n">np</span>
+    <span class="n">fft_lib</span> <span class="o">=</span> <span class="n">scipy</span><span class="o">.</span><span class="n">fft</span>
+    <span class="n">sparse_lib</span> <span class="o">=</span> <span class="n">scipy</span><span class="o">.</span><span class="n">sparse</span>
+    <span class="n">linalg</span> <span class="o">=</span> <span class="n">scipy</span><span class="o">.</span><span class="n">sparse</span><span class="o">.</span><span class="n">linalg</span>
+    <span class="n">dtype</span> <span class="o">=</span> <span class="n">mesh</span>
+    <span class="n">fft_backend</span> <span class="o">=</span> <span class="s1">&#39;fftw&#39;</span>
+    <span class="n">fft_comm_backend</span> <span class="o">=</span> <span class="s1">&#39;MPI&#39;</span>
+
+<div class="viewcode-block" id="SpectralHelper.setup_GPU">
+<a class="viewcode-back" href="../../pySDC/helpers.spectral_helper.html#helpers.spectral_helper.SpectralHelper.setup_GPU">[docs]</a>
+    <span class="nd">@classmethod</span>
+    <span class="k">def</span> <span class="nf">setup_GPU</span><span class="p">(</span><span class="bp">cls</span><span class="p">):</span>
+<span class="w">        </span><span class="sd">&quot;&quot;&quot;switch to GPU modules&quot;&quot;&quot;</span>
+        <span class="kn">import</span> <span class="nn">cupy</span> <span class="k">as</span> <span class="nn">cp</span>
+        <span class="kn">import</span> <span class="nn">cupyx.scipy.sparse</span> <span class="k">as</span> <span class="nn">sparse_lib</span>
+        <span class="kn">import</span> <span class="nn">cupyx.scipy.sparse.linalg</span> <span class="k">as</span> <span class="nn">linalg</span>
+        <span class="kn">from</span> <span class="nn">pySDC.implementations.datatype_classes.cupy_mesh</span> <span class="kn">import</span> <span class="n">cupy_mesh</span>
+
+        <span class="bp">cls</span><span class="o">.</span><span class="n">xp</span> <span class="o">=</span> <span class="n">cp</span>
+        <span class="bp">cls</span><span class="o">.</span><span class="n">sparse_lib</span> <span class="o">=</span> <span class="n">sparse_lib</span>
+        <span class="bp">cls</span><span class="o">.</span><span class="n">linalg</span> <span class="o">=</span> <span class="n">linalg</span>
+
+        <span class="bp">cls</span><span class="o">.</span><span class="n">fft_backend</span> <span class="o">=</span> <span class="s1">&#39;cupy&#39;</span>
+        <span class="bp">cls</span><span class="o">.</span><span class="n">fft_comm_backend</span> <span class="o">=</span> <span class="s1">&#39;NCCL&#39;</span>
+
+        <span class="bp">cls</span><span class="o">.</span><span class="n">dtype</span> <span class="o">=</span> <span class="n">cupy_mesh</span></div>
+
+
+    <span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">comm</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">useGPU</span><span class="o">=</span><span class="kc">False</span><span class="p">,</span> <span class="n">debug</span><span class="o">=</span><span class="kc">False</span><span class="p">):</span>
+<span class="w">        </span><span class="sd">&quot;&quot;&quot;</span>
+<span class="sd">        Constructor</span>
+
+<span class="sd">        Args:</span>
+<span class="sd">            comm (mpi4py.Intracomm): MPI communicator</span>
+<span class="sd">            useGPU (bool): Whether to use GPUs</span>
+<span class="sd">            debug (bool): Perform additional checks at extra computational cost</span>
+<span class="sd">        &quot;&quot;&quot;</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">comm</span> <span class="o">=</span> <span class="n">comm</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">debug</span> <span class="o">=</span> <span class="n">debug</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">useGPU</span> <span class="o">=</span> <span class="n">useGPU</span>
+
+        <span class="k">if</span> <span class="n">useGPU</span><span class="p">:</span>
+            <span class="bp">self</span><span class="o">.</span><span class="n">setup_GPU</span><span class="p">()</span>
+
+        <span class="bp">self</span><span class="o">.</span><span class="n">axes</span> <span class="o">=</span> <span class="p">[]</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">components</span> <span class="o">=</span> <span class="p">[]</span>
+
+        <span class="bp">self</span><span class="o">.</span><span class="n">full_BCs</span> <span class="o">=</span> <span class="p">[]</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">BC_mat</span> <span class="o">=</span> <span class="kc">None</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">BCs</span> <span class="o">=</span> <span class="kc">None</span>
+
+        <span class="bp">self</span><span class="o">.</span><span class="n">fft_cache</span> <span class="o">=</span> <span class="p">{}</span>
+
+    <span class="nd">@property</span>
+    <span class="k">def</span> <span class="nf">u_init</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
+<span class="w">        </span><span class="sd">&quot;&quot;&quot;</span>
+<span class="sd">        Get empty data container in physical space</span>
+<span class="sd">        &quot;&quot;&quot;</span>
+        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">dtype</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">init</span><span class="p">)</span>
+
+    <span class="nd">@property</span>
+    <span class="k">def</span> <span class="nf">u_init_forward</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
+<span class="w">        </span><span class="sd">&quot;&quot;&quot;</span>
+<span class="sd">        Get empty data container in spectral space</span>
+<span class="sd">        &quot;&quot;&quot;</span>
+        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">dtype</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">init_forward</span><span class="p">)</span>
+
+    <span class="nd">@property</span>
+    <span class="k">def</span> <span class="nf">shape</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
+<span class="w">        </span><span class="sd">&quot;&quot;&quot;</span>
+<span class="sd">        Get shape of individual solution component</span>
+<span class="sd">        &quot;&quot;&quot;</span>
+        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">init</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">1</span><span class="p">:]</span>
+
+    <span class="nd">@property</span>
+    <span class="k">def</span> <span class="nf">ndim</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
+        <span class="k">return</span> <span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">axes</span><span class="p">)</span>
+
+    <span class="nd">@property</span>
+    <span class="k">def</span> <span class="nf">ncomponents</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
+        <span class="k">return</span> <span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">components</span><span class="p">)</span>
+
+    <span class="nd">@property</span>
+    <span class="k">def</span> <span class="nf">V</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
+<span class="w">        </span><span class="sd">&quot;&quot;&quot;</span>
+<span class="sd">        Get domain volume</span>
+<span class="sd">        &quot;&quot;&quot;</span>
+        <span class="k">return</span> <span class="n">np</span><span class="o">.</span><span class="n">prod</span><span class="p">([</span><span class="n">me</span><span class="o">.</span><span class="n">L</span> <span class="k">for</span> <span class="n">me</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">axes</span><span class="p">])</span>
+
+<div class="viewcode-block" id="SpectralHelper.add_axis">
+<a class="viewcode-back" href="../../pySDC/helpers.spectral_helper.html#helpers.spectral_helper.SpectralHelper.add_axis">[docs]</a>
+    <span class="k">def</span> <span class="nf">add_axis</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">base</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
+<span class="w">        </span><span class="sd">&quot;&quot;&quot;</span>
+<span class="sd">        Add an axis to the domain by deciding on suitable 1D base.</span>
+<span class="sd">        Arguments to the bases are forwarded using `*args` and `**kwargs`. Please refer to the documentation of the 1D</span>
+<span class="sd">        bases for possible arguments.</span>
+
+<span class="sd">        Args:</span>
+<span class="sd">            base (str): 1D spectral method</span>
+<span class="sd">        &quot;&quot;&quot;</span>
+        <span class="n">kwargs</span><span class="p">[</span><span class="s1">&#39;useGPU&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">useGPU</span>
+
+        <span class="k">if</span> <span class="n">base</span><span class="o">.</span><span class="n">lower</span><span class="p">()</span> <span class="ow">in</span> <span class="p">[</span><span class="s1">&#39;chebychov&#39;</span><span class="p">,</span> <span class="s1">&#39;chebychev&#39;</span><span class="p">,</span> <span class="s1">&#39;cheby&#39;</span><span class="p">,</span> <span class="s1">&#39;chebychovhelper&#39;</span><span class="p">]:</span>
+            <span class="n">kwargs</span><span class="p">[</span><span class="s1">&#39;transform_type&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="n">kwargs</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s1">&#39;transform_type&#39;</span><span class="p">,</span> <span class="s1">&#39;fft&#39;</span><span class="p">)</span>
+            <span class="bp">self</span><span class="o">.</span><span class="n">axes</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">ChebychevHelper</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">))</span>
+        <span class="k">elif</span> <span class="n">base</span><span class="o">.</span><span class="n">lower</span><span class="p">()</span> <span class="ow">in</span> <span class="p">[</span><span class="s1">&#39;fft&#39;</span><span class="p">,</span> <span class="s1">&#39;fourier&#39;</span><span class="p">,</span> <span class="s1">&#39;ffthelper&#39;</span><span class="p">]:</span>
+            <span class="bp">self</span><span class="o">.</span><span class="n">axes</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">FFTHelper</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">))</span>
+        <span class="k">elif</span> <span class="n">base</span><span class="o">.</span><span class="n">lower</span><span class="p">()</span> <span class="ow">in</span> <span class="p">[</span><span class="s1">&#39;ultraspherical&#39;</span><span class="p">,</span> <span class="s1">&#39;gegenbauer&#39;</span><span class="p">]:</span>
+            <span class="bp">self</span><span class="o">.</span><span class="n">axes</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">UltrasphericalHelper</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">))</span>
+        <span class="k">else</span><span class="p">:</span>
+            <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span><span class="sa">f</span><span class="s1">&#39;</span><span class="si">{</span><span class="n">base</span><span class="si">=!r}</span><span class="s1"> is not implemented!&#39;</span><span class="p">)</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">axes</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">xp</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">xp</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">axes</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">sparse_lib</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">sparse_lib</span></div>
+
+
+<div class="viewcode-block" id="SpectralHelper.add_component">
+<a class="viewcode-back" href="../../pySDC/helpers.spectral_helper.html#helpers.spectral_helper.SpectralHelper.add_component">[docs]</a>
+    <span class="k">def</span> <span class="nf">add_component</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">name</span><span class="p">):</span>
+<span class="w">        </span><span class="sd">&quot;&quot;&quot;</span>
+<span class="sd">        Add solution component(s).</span>
+
+<span class="sd">        Args:</span>
+<span class="sd">            name (str or list of strings): Name(s) of component(s)</span>
+<span class="sd">        &quot;&quot;&quot;</span>
+        <span class="k">if</span> <span class="nb">type</span><span class="p">(</span><span class="n">name</span><span class="p">)</span> <span class="ow">in</span> <span class="p">[</span><span class="nb">list</span><span class="p">,</span> <span class="nb">tuple</span><span class="p">]:</span>
+            <span class="k">for</span> <span class="n">me</span> <span class="ow">in</span> <span class="n">name</span><span class="p">:</span>
+                <span class="bp">self</span><span class="o">.</span><span class="n">add_component</span><span class="p">(</span><span class="n">me</span><span class="p">)</span>
+        <span class="k">elif</span> <span class="nb">type</span><span class="p">(</span><span class="n">name</span><span class="p">)</span> <span class="ow">in</span> <span class="p">[</span><span class="nb">str</span><span class="p">]:</span>
+            <span class="k">if</span> <span class="n">name</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">components</span><span class="p">:</span>
+                <span class="k">raise</span> <span class="ne">Exception</span><span class="p">(</span><span class="sa">f</span><span class="s1">&#39;</span><span class="si">{</span><span class="n">name</span><span class="si">=!r}</span><span class="s1"> is already added to this problem!&#39;</span><span class="p">)</span>
+            <span class="bp">self</span><span class="o">.</span><span class="n">components</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">name</span><span class="p">)</span>
+        <span class="k">else</span><span class="p">:</span>
+            <span class="k">raise</span> <span class="ne">NotImplementedError</span></div>
+
+
+<div class="viewcode-block" id="SpectralHelper.index">
+<a class="viewcode-back" href="../../pySDC/helpers.spectral_helper.html#helpers.spectral_helper.SpectralHelper.index">[docs]</a>
+    <span class="k">def</span> <span class="nf">index</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">name</span><span class="p">):</span>
+<span class="w">        </span><span class="sd">&quot;&quot;&quot;</span>
+<span class="sd">        Get the index of component `name`.</span>
+
+<span class="sd">        Args:</span>
+<span class="sd">            name (str or list of strings): Name(s) of component(s)</span>
+
+<span class="sd">        Returns:</span>
+<span class="sd">            int: Index of the component</span>
+<span class="sd">        &quot;&quot;&quot;</span>
+        <span class="k">if</span> <span class="nb">type</span><span class="p">(</span><span class="n">name</span><span class="p">)</span> <span class="ow">in</span> <span class="p">[</span><span class="nb">str</span><span class="p">,</span> <span class="nb">int</span><span class="p">]:</span>
+            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">components</span><span class="o">.</span><span class="n">index</span><span class="p">(</span><span class="n">name</span><span class="p">)</span>
+        <span class="k">elif</span> <span class="nb">type</span><span class="p">(</span><span class="n">name</span><span class="p">)</span> <span class="ow">in</span> <span class="p">[</span><span class="nb">list</span><span class="p">,</span> <span class="nb">tuple</span><span class="p">]:</span>
+            <span class="k">return</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">index</span><span class="p">(</span><span class="n">me</span><span class="p">)</span> <span class="k">for</span> <span class="n">me</span> <span class="ow">in</span> <span class="n">name</span><span class="p">)</span>
+        <span class="k">else</span><span class="p">:</span>
+            <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span><span class="sa">f</span><span class="s1">&#39;Don</span><span class="se">\&#39;</span><span class="s1">t know how to compute index for </span><span class="si">{</span><span class="nb">type</span><span class="p">(</span><span class="n">name</span><span class="p">)</span><span class="si">=}</span><span class="s1">&#39;</span><span class="p">)</span></div>
+
+
+<div class="viewcode-block" id="SpectralHelper.get_empty_operator_matrix">
+<a class="viewcode-back" href="../../pySDC/helpers.spectral_helper.html#helpers.spectral_helper.SpectralHelper.get_empty_operator_matrix">[docs]</a>
+    <span class="k">def</span> <span class="nf">get_empty_operator_matrix</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
+<span class="w">        </span><span class="sd">&quot;&quot;&quot;</span>
+<span class="sd">        Return a matrix of operators to be filled with the connections between the solution components.</span>
+
+<span class="sd">        Returns:</span>
+<span class="sd">            list containing sparse zeros</span>
+<span class="sd">        &quot;&quot;&quot;</span>
+        <span class="n">S</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">components</span><span class="p">)</span>
+        <span class="n">O</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">get_Id</span><span class="p">()</span> <span class="o">*</span> <span class="mi">0</span>
+        <span class="k">return</span> <span class="p">[[</span><span class="n">O</span> <span class="k">for</span> <span class="n">_</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">S</span><span class="p">)]</span> <span class="k">for</span> <span class="n">_</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">S</span><span class="p">)]</span></div>
+
+
+<div class="viewcode-block" id="SpectralHelper.get_BC">
+<a class="viewcode-back" href="../../pySDC/helpers.spectral_helper.html#helpers.spectral_helper.SpectralHelper.get_BC">[docs]</a>
+    <span class="k">def</span> <span class="nf">get_BC</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">axis</span><span class="p">,</span> <span class="n">kind</span><span class="p">,</span> <span class="n">line</span><span class="o">=-</span><span class="mi">1</span><span class="p">,</span> <span class="n">scalar</span><span class="o">=</span><span class="kc">False</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
+<span class="w">        </span><span class="sd">&quot;&quot;&quot;</span>
+<span class="sd">        Use this method for boundary bordering. It gets the respective matrix row and embeds it into a matrix.</span>
+<span class="sd">        Pay attention that if you have multiple BCs in a single equation, you need to put them in different lines.</span>
+<span class="sd">        Typically, the last line that does not contain a BC is the best choice.</span>
+<span class="sd">        Forward arguments for the boundary conditions using `kwargs`. Refer to documentation of 1D bases for details.</span>
+
+<span class="sd">        Args:</span>
+<span class="sd">            axis (int): Axis you want to add the BC to</span>
+<span class="sd">            kind (str): kind of BC, e.g. Dirichlet</span>
+<span class="sd">            line (int): Line you want the BC to go in</span>
+<span class="sd">            scalar (bool): Put the BC in all space positions in the other direction</span>
+
+<span class="sd">        Returns:</span>
+<span class="sd">            sparse matrix containing the BC</span>
+<span class="sd">        &quot;&quot;&quot;</span>
+        <span class="n">sp</span> <span class="o">=</span> <span class="n">scipy</span><span class="o">.</span><span class="n">sparse</span>
+
+        <span class="n">base</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">axes</span><span class="p">[</span><span class="n">axis</span><span class="p">]</span>
+
+        <span class="n">BC</span> <span class="o">=</span> <span class="n">sp</span><span class="o">.</span><span class="n">eye</span><span class="p">(</span><span class="n">base</span><span class="o">.</span><span class="n">N</span><span class="p">)</span><span class="o">.</span><span class="n">tolil</span><span class="p">()</span> <span class="o">*</span> <span class="mi">0</span>
+        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">useGPU</span><span class="p">:</span>
+            <span class="n">BC</span><span class="p">[</span><span class="n">line</span><span class="p">,</span> <span class="p">:]</span> <span class="o">=</span> <span class="n">base</span><span class="o">.</span><span class="n">get_BC</span><span class="p">(</span><span class="n">kind</span><span class="o">=</span><span class="n">kind</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span><span class="o">.</span><span class="n">get</span><span class="p">()</span>
+        <span class="k">else</span><span class="p">:</span>
+            <span class="n">BC</span><span class="p">[</span><span class="n">line</span><span class="p">,</span> <span class="p">:]</span> <span class="o">=</span> <span class="n">base</span><span class="o">.</span><span class="n">get_BC</span><span class="p">(</span><span class="n">kind</span><span class="o">=</span><span class="n">kind</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
+
+        <span class="n">ndim</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">axes</span><span class="p">)</span>
+        <span class="k">if</span> <span class="n">ndim</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
+            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">sparse_lib</span><span class="o">.</span><span class="n">csc_matrix</span><span class="p">(</span><span class="n">BC</span><span class="p">)</span>
+        <span class="k">elif</span> <span class="n">ndim</span> <span class="o">==</span> <span class="mi">2</span><span class="p">:</span>
+            <span class="n">axis2</span> <span class="o">=</span> <span class="p">(</span><span class="n">axis</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> <span class="o">%</span> <span class="n">ndim</span>
+
+            <span class="k">if</span> <span class="n">scalar</span><span class="p">:</span>
+                <span class="n">_Id</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">sparse_lib</span><span class="o">.</span><span class="n">diags</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">xp</span><span class="o">.</span><span class="n">append</span><span class="p">([</span><span class="mi">1</span><span class="p">],</span> <span class="bp">self</span><span class="o">.</span><span class="n">xp</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">axes</span><span class="p">[</span><span class="n">axis2</span><span class="p">]</span><span class="o">.</span><span class="n">N</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)))</span>
+            <span class="k">else</span><span class="p">:</span>
+                <span class="n">_Id</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">axes</span><span class="p">[</span><span class="n">axis2</span><span class="p">]</span><span class="o">.</span><span class="n">get_Id</span><span class="p">()</span>
+
+            <span class="n">Id</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">get_local_slice_of_1D_matrix</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">axes</span><span class="p">[</span><span class="n">axis2</span><span class="p">]</span><span class="o">.</span><span class="n">get_Id</span><span class="p">()</span> <span class="o">@</span> <span class="n">_Id</span><span class="p">,</span> <span class="n">axis</span><span class="o">=</span><span class="n">axis2</span><span class="p">)</span>
+
+            <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">useGPU</span><span class="p">:</span>
+                <span class="n">Id</span> <span class="o">=</span> <span class="n">Id</span><span class="o">.</span><span class="n">get</span><span class="p">()</span>
+
+            <span class="n">mats</span> <span class="o">=</span> <span class="p">[</span>
+                <span class="kc">None</span><span class="p">,</span>
+            <span class="p">]</span> <span class="o">*</span> <span class="n">ndim</span>
+            <span class="n">mats</span><span class="p">[</span><span class="n">axis</span><span class="p">]</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">get_local_slice_of_1D_matrix</span><span class="p">(</span><span class="n">BC</span><span class="p">,</span> <span class="n">axis</span><span class="o">=</span><span class="n">axis</span><span class="p">)</span>
+            <span class="n">mats</span><span class="p">[</span><span class="n">axis2</span><span class="p">]</span> <span class="o">=</span> <span class="n">Id</span>
+            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">sparse_lib</span><span class="o">.</span><span class="n">csc_matrix</span><span class="p">(</span><span class="n">sp</span><span class="o">.</span><span class="n">kron</span><span class="p">(</span><span class="o">*</span><span class="n">mats</span><span class="p">))</span></div>
+
+
+<div class="viewcode-block" id="SpectralHelper.remove_BC">
+<a class="viewcode-back" href="../../pySDC/helpers.spectral_helper.html#helpers.spectral_helper.SpectralHelper.remove_BC">[docs]</a>
+    <span class="k">def</span> <span class="nf">remove_BC</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">component</span><span class="p">,</span> <span class="n">equation</span><span class="p">,</span> <span class="n">axis</span><span class="p">,</span> <span class="n">kind</span><span class="p">,</span> <span class="n">line</span><span class="o">=-</span><span class="mi">1</span><span class="p">,</span> <span class="n">scalar</span><span class="o">=</span><span class="kc">False</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
+<span class="w">        </span><span class="sd">&quot;&quot;&quot;</span>
+<span class="sd">        Remove a BC from the matrix. This is useful e.g. when you add a non-scalar BC and then need to selectively</span>
+<span class="sd">        remove single BCs again, as in incompressible Navier-Stokes, for instance.</span>
+<span class="sd">        Forward arguments for the boundary conditions using `kwargs`. Refer to documentation of 1D bases for details.</span>
+
+<span class="sd">        Args:</span>
+<span class="sd">            component (str): Name of the component the BC should act on</span>
+<span class="sd">            equation (str): Name of the equation for the component you want to put the BC in</span>
+<span class="sd">            axis (int): Axis you want to add the BC to</span>
+<span class="sd">            kind (str): kind of BC, e.g. Dirichlet</span>
+<span class="sd">            v: Value of the BC</span>
+<span class="sd">            line (int): Line you want the BC to go in</span>
+<span class="sd">            scalar (bool): Put the BC in all space positions in the other direction</span>
+<span class="sd">        &quot;&quot;&quot;</span>
+        <span class="n">_BC</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">get_BC</span><span class="p">(</span><span class="n">axis</span><span class="o">=</span><span class="n">axis</span><span class="p">,</span> <span class="n">kind</span><span class="o">=</span><span class="n">kind</span><span class="p">,</span> <span class="n">line</span><span class="o">=</span><span class="n">line</span><span class="p">,</span> <span class="n">scalar</span><span class="o">=</span><span class="n">scalar</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">BC_mat</span><span class="p">[</span><span class="bp">self</span><span class="o">.</span><span class="n">index</span><span class="p">(</span><span class="n">equation</span><span class="p">)][</span><span class="bp">self</span><span class="o">.</span><span class="n">index</span><span class="p">(</span><span class="n">component</span><span class="p">)]</span> <span class="o">-=</span> <span class="n">_BC</span>
+
+        <span class="k">if</span> <span class="n">scalar</span><span class="p">:</span>
+            <span class="n">slices</span> <span class="o">=</span> <span class="p">[</span><span class="bp">self</span><span class="o">.</span><span class="n">index</span><span class="p">(</span><span class="n">equation</span><span class="p">)]</span> <span class="o">+</span> <span class="p">[</span>
+                <span class="mi">0</span><span class="p">,</span>
+            <span class="p">]</span> <span class="o">*</span> <span class="bp">self</span><span class="o">.</span><span class="n">ndim</span>
+            <span class="n">slices</span><span class="p">[</span><span class="n">axis</span> <span class="o">+</span> <span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">line</span>
+        <span class="k">else</span><span class="p">:</span>
+            <span class="n">slices</span> <span class="o">=</span> <span class="p">(</span>
+                <span class="p">[</span><span class="bp">self</span><span class="o">.</span><span class="n">index</span><span class="p">(</span><span class="n">equation</span><span class="p">)]</span>
+                <span class="o">+</span> <span class="p">[</span><span class="nb">slice</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">init</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">])</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">axis</span><span class="p">)]</span>
+                <span class="o">+</span> <span class="p">[</span><span class="n">line</span><span class="p">]</span>
+                <span class="o">+</span> <span class="p">[</span><span class="nb">slice</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">init</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">])</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">axis</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">axes</span><span class="p">))]</span>
+            <span class="p">)</span>
+        <span class="n">N</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">axes</span><span class="p">[</span><span class="n">axis</span><span class="p">]</span><span class="o">.</span><span class="n">N</span>
+        <span class="k">if</span> <span class="p">(</span><span class="n">N</span> <span class="o">+</span> <span class="n">line</span><span class="p">)</span> <span class="o">%</span> <span class="n">N</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">xp</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="n">N</span><span class="p">)[</span><span class="bp">self</span><span class="o">.</span><span class="n">local_slice</span><span class="p">[</span><span class="n">axis</span><span class="p">]]:</span>
+            <span class="bp">self</span><span class="o">.</span><span class="n">BC_rhs_mask</span><span class="p">[(</span><span class="o">*</span><span class="n">slices</span><span class="p">,)]</span> <span class="o">=</span> <span class="kc">False</span></div>
+
+
+<div class="viewcode-block" id="SpectralHelper.add_BC">
+<a class="viewcode-back" href="../../pySDC/helpers.spectral_helper.html#helpers.spectral_helper.SpectralHelper.add_BC">[docs]</a>
+    <span class="k">def</span> <span class="nf">add_BC</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">component</span><span class="p">,</span> <span class="n">equation</span><span class="p">,</span> <span class="n">axis</span><span class="p">,</span> <span class="n">kind</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">line</span><span class="o">=-</span><span class="mi">1</span><span class="p">,</span> <span class="n">scalar</span><span class="o">=</span><span class="kc">False</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
+<span class="w">        </span><span class="sd">&quot;&quot;&quot;</span>
+<span class="sd">        Add a BC to the matrix. Note that you need to convert the list of lists of BCs that this method generates to a</span>
+<span class="sd">        single sparse matrix by calling `setup_BCs` after adding/removing all BCs.</span>
+<span class="sd">        Forward arguments for the boundary conditions using `kwargs`. Refer to documentation of 1D bases for details.</span>
+
+<span class="sd">        Args:</span>
+<span class="sd">            component (str): Name of the component the BC should act on</span>
+<span class="sd">            equation (str): Name of the equation for the component you want to put the BC in</span>
+<span class="sd">            axis (int): Axis you want to add the BC to</span>
+<span class="sd">            kind (str): kind of BC, e.g. Dirichlet</span>
+<span class="sd">            v: Value of the BC</span>
+<span class="sd">            line (int): Line you want the BC to go in</span>
+<span class="sd">            scalar (bool): Put the BC in all space positions in the other direction</span>
+<span class="sd">        &quot;&quot;&quot;</span>
+        <span class="n">_BC</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">get_BC</span><span class="p">(</span><span class="n">axis</span><span class="o">=</span><span class="n">axis</span><span class="p">,</span> <span class="n">kind</span><span class="o">=</span><span class="n">kind</span><span class="p">,</span> <span class="n">line</span><span class="o">=</span><span class="n">line</span><span class="p">,</span> <span class="n">scalar</span><span class="o">=</span><span class="n">scalar</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">BC_mat</span><span class="p">[</span><span class="bp">self</span><span class="o">.</span><span class="n">index</span><span class="p">(</span><span class="n">equation</span><span class="p">)][</span><span class="bp">self</span><span class="o">.</span><span class="n">index</span><span class="p">(</span><span class="n">component</span><span class="p">)]</span> <span class="o">+=</span> <span class="n">_BC</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">full_BCs</span> <span class="o">+=</span> <span class="p">[</span>
+            <span class="p">{</span>
+                <span class="s1">&#39;component&#39;</span><span class="p">:</span> <span class="n">component</span><span class="p">,</span>
+                <span class="s1">&#39;equation&#39;</span><span class="p">:</span> <span class="n">equation</span><span class="p">,</span>
+                <span class="s1">&#39;axis&#39;</span><span class="p">:</span> <span class="n">axis</span><span class="p">,</span>
+                <span class="s1">&#39;kind&#39;</span><span class="p">:</span> <span class="n">kind</span><span class="p">,</span>
+                <span class="s1">&#39;v&#39;</span><span class="p">:</span> <span class="n">v</span><span class="p">,</span>
+                <span class="s1">&#39;line&#39;</span><span class="p">:</span> <span class="n">line</span><span class="p">,</span>
+                <span class="s1">&#39;scalar&#39;</span><span class="p">:</span> <span class="n">scalar</span><span class="p">,</span>
+                <span class="o">**</span><span class="n">kwargs</span><span class="p">,</span>
+            <span class="p">}</span>
+        <span class="p">]</span>
+
+        <span class="k">if</span> <span class="n">scalar</span><span class="p">:</span>
+            <span class="n">slices</span> <span class="o">=</span> <span class="p">[</span><span class="bp">self</span><span class="o">.</span><span class="n">index</span><span class="p">(</span><span class="n">equation</span><span class="p">)]</span> <span class="o">+</span> <span class="p">[</span>
+                <span class="mi">0</span><span class="p">,</span>
+            <span class="p">]</span> <span class="o">*</span> <span class="bp">self</span><span class="o">.</span><span class="n">ndim</span>
+            <span class="n">slices</span><span class="p">[</span><span class="n">axis</span> <span class="o">+</span> <span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">line</span>
+            <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">comm</span><span class="p">:</span>
+                <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">comm</span><span class="o">.</span><span class="n">rank</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
+                    <span class="bp">self</span><span class="o">.</span><span class="n">BC_rhs_mask</span><span class="p">[(</span><span class="o">*</span><span class="n">slices</span><span class="p">,)]</span> <span class="o">=</span> <span class="kc">True</span>
+            <span class="k">else</span><span class="p">:</span>
+                <span class="bp">self</span><span class="o">.</span><span class="n">BC_rhs_mask</span><span class="p">[(</span><span class="o">*</span><span class="n">slices</span><span class="p">,)]</span> <span class="o">=</span> <span class="kc">True</span>
+        <span class="k">else</span><span class="p">:</span>
+            <span class="n">slices</span> <span class="o">=</span> <span class="p">(</span>
+                <span class="p">[</span><span class="bp">self</span><span class="o">.</span><span class="n">index</span><span class="p">(</span><span class="n">equation</span><span class="p">)]</span>
+                <span class="o">+</span> <span class="p">[</span><span class="nb">slice</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">init</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">])</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">axis</span><span class="p">)]</span>
+                <span class="o">+</span> <span class="p">[</span><span class="n">line</span><span class="p">]</span>
+                <span class="o">+</span> <span class="p">[</span><span class="nb">slice</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">init</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">])</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">axis</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">axes</span><span class="p">))]</span>
+            <span class="p">)</span>
+            <span class="n">N</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">axes</span><span class="p">[</span><span class="n">axis</span><span class="p">]</span><span class="o">.</span><span class="n">N</span>
+            <span class="k">if</span> <span class="p">(</span><span class="n">N</span> <span class="o">+</span> <span class="n">line</span><span class="p">)</span> <span class="o">%</span> <span class="n">N</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">xp</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="n">N</span><span class="p">)[</span><span class="bp">self</span><span class="o">.</span><span class="n">local_slice</span><span class="p">[</span><span class="n">axis</span><span class="p">]]:</span>
+                <span class="n">slices</span><span class="p">[</span><span class="n">axis</span> <span class="o">+</span> <span class="mi">1</span><span class="p">]</span> <span class="o">-=</span> <span class="bp">self</span><span class="o">.</span><span class="n">local_slice</span><span class="p">[</span><span class="n">axis</span><span class="p">]</span><span class="o">.</span><span class="n">start</span>
+                <span class="bp">self</span><span class="o">.</span><span class="n">BC_rhs_mask</span><span class="p">[(</span><span class="o">*</span><span class="n">slices</span><span class="p">,)]</span> <span class="o">=</span> <span class="kc">True</span></div>
+
+
+<div class="viewcode-block" id="SpectralHelper.setup_BCs">
+<a class="viewcode-back" href="../../pySDC/helpers.spectral_helper.html#helpers.spectral_helper.SpectralHelper.setup_BCs">[docs]</a>
+    <span class="k">def</span> <span class="nf">setup_BCs</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
+<span class="w">        </span><span class="sd">&quot;&quot;&quot;</span>
+<span class="sd">        Convert the list of lists of BCs to the boundary condition operator.</span>
+<span class="sd">        Also, boundary bordering requires to zero out all other entries in the matrix in rows containing a boundary</span>
+<span class="sd">        condition. This method sets up a suitable sparse matrix to do this.</span>
+<span class="sd">        &quot;&quot;&quot;</span>
+        <span class="n">sp</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">sparse_lib</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">BCs</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">convert_operator_matrix_to_operator</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">BC_mat</span><span class="p">)</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">BC_zero_index</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">xp</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">prod</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">init</span><span class="p">[</span><span class="mi">0</span><span class="p">]))[</span><span class="bp">self</span><span class="o">.</span><span class="n">BC_rhs_mask</span><span class="o">.</span><span class="n">flatten</span><span class="p">()]</span>
+
+        <span class="n">diags</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">xp</span><span class="o">.</span><span class="n">ones</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">BCs</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
+        <span class="n">diags</span><span class="p">[</span><span class="bp">self</span><span class="o">.</span><span class="n">BC_zero_index</span><span class="p">]</span> <span class="o">=</span> <span class="mi">0</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">BC_line_zero_matrix</span> <span class="o">=</span> <span class="n">sp</span><span class="o">.</span><span class="n">diags</span><span class="p">(</span><span class="n">diags</span><span class="p">)</span>
+
+        <span class="c1"># prepare BCs in spectral space to easily add to the RHS</span>
+        <span class="n">rhs_BCs</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">put_BCs_in_rhs</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">u_init</span><span class="p">)</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">rhs_BCs_hat</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">transform</span><span class="p">(</span><span class="n">rhs_BCs</span><span class="p">)</span></div>
+
+
+<div class="viewcode-block" id="SpectralHelper.check_BCs">
+<a class="viewcode-back" href="../../pySDC/helpers.spectral_helper.html#helpers.spectral_helper.SpectralHelper.check_BCs">[docs]</a>
+    <span class="k">def</span> <span class="nf">check_BCs</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">u</span><span class="p">):</span>
+<span class="w">        </span><span class="sd">&quot;&quot;&quot;</span>
+<span class="sd">        Check that the solution satisfies the boundary conditions</span>
+
+<span class="sd">        Args:</span>
+<span class="sd">            u: The solution you want to check</span>
+<span class="sd">        &quot;&quot;&quot;</span>
+        <span class="k">assert</span> <span class="bp">self</span><span class="o">.</span><span class="n">ndim</span> <span class="o">&lt;</span> <span class="mi">3</span>
+        <span class="k">for</span> <span class="n">axis</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">ndim</span><span class="p">):</span>
+            <span class="n">BCs</span> <span class="o">=</span> <span class="p">[</span><span class="n">me</span> <span class="k">for</span> <span class="n">me</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">full_BCs</span> <span class="k">if</span> <span class="n">me</span><span class="p">[</span><span class="s2">&quot;axis&quot;</span><span class="p">]</span> <span class="o">==</span> <span class="n">axis</span> <span class="ow">and</span> <span class="ow">not</span> <span class="n">me</span><span class="p">[</span><span class="s2">&quot;scalar&quot;</span><span class="p">]]</span>
+
+            <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">BCs</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">:</span>
+                <span class="n">u_hat</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">transform</span><span class="p">(</span><span class="n">u</span><span class="p">,</span> <span class="n">axes</span><span class="o">=</span><span class="p">(</span><span class="n">axis</span> <span class="o">-</span> <span class="bp">self</span><span class="o">.</span><span class="n">ndim</span><span class="p">,))</span>
+                <span class="k">for</span> <span class="n">BC</span> <span class="ow">in</span> <span class="n">BCs</span><span class="p">:</span>
+                    <span class="n">kwargs</span> <span class="o">=</span> <span class="p">{</span>
+                        <span class="n">key</span><span class="p">:</span> <span class="n">value</span>
+                        <span class="k">for</span> <span class="n">key</span><span class="p">,</span> <span class="n">value</span> <span class="ow">in</span> <span class="n">BC</span><span class="o">.</span><span class="n">items</span><span class="p">()</span>
+                        <span class="k">if</span> <span class="n">key</span> <span class="ow">not</span> <span class="ow">in</span> <span class="p">[</span><span class="s1">&#39;component&#39;</span><span class="p">,</span> <span class="s1">&#39;equation&#39;</span><span class="p">,</span> <span class="s1">&#39;axis&#39;</span><span class="p">,</span> <span class="s1">&#39;v&#39;</span><span class="p">,</span> <span class="s1">&#39;line&#39;</span><span class="p">,</span> <span class="s1">&#39;scalar&#39;</span><span class="p">]</span>
+                    <span class="p">}</span>
+
+                    <span class="k">if</span> <span class="n">axis</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
+                        <span class="n">get</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">axes</span><span class="p">[</span><span class="n">axis</span><span class="p">]</span><span class="o">.</span><span class="n">get_BC</span><span class="p">(</span><span class="o">**</span><span class="n">kwargs</span><span class="p">)</span> <span class="o">@</span> <span class="n">u_hat</span><span class="p">[</span><span class="bp">self</span><span class="o">.</span><span class="n">index</span><span class="p">(</span><span class="n">BC</span><span class="p">[</span><span class="s1">&#39;component&#39;</span><span class="p">])]</span>
+                    <span class="k">elif</span> <span class="n">axis</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
+                        <span class="n">get</span> <span class="o">=</span> <span class="n">u_hat</span><span class="p">[</span><span class="bp">self</span><span class="o">.</span><span class="n">index</span><span class="p">(</span><span class="n">BC</span><span class="p">[</span><span class="s1">&#39;component&#39;</span><span class="p">])]</span> <span class="o">@</span> <span class="bp">self</span><span class="o">.</span><span class="n">axes</span><span class="p">[</span><span class="n">axis</span><span class="p">]</span><span class="o">.</span><span class="n">get_BC</span><span class="p">(</span><span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
+                    <span class="n">want</span> <span class="o">=</span> <span class="n">BC</span><span class="p">[</span><span class="s1">&#39;v&#39;</span><span class="p">]</span>
+                    <span class="k">assert</span> <span class="bp">self</span><span class="o">.</span><span class="n">xp</span><span class="o">.</span><span class="n">allclose</span><span class="p">(</span>
+                        <span class="n">get</span><span class="p">,</span> <span class="n">want</span>
+                    <span class="p">),</span> <span class="sa">f</span><span class="s1">&#39;Unexpected BC in </span><span class="si">{</span><span class="n">BC</span><span class="p">[</span><span class="s2">&quot;component&quot;</span><span class="p">]</span><span class="si">}</span><span class="s1"> in equation </span><span class="si">{</span><span class="n">BC</span><span class="p">[</span><span class="s2">&quot;equation&quot;</span><span class="p">]</span><span class="si">}</span><span class="s1">, line </span><span class="si">{</span><span class="n">BC</span><span class="p">[</span><span class="s2">&quot;line&quot;</span><span class="p">]</span><span class="si">}</span><span class="s1">! Got </span><span class="si">{</span><span class="n">get</span><span class="si">}</span><span class="s1">, wanted </span><span class="si">{</span><span class="n">want</span><span class="si">}</span><span class="s1">&#39;</span></div>
+
+
+<div class="viewcode-block" id="SpectralHelper.put_BCs_in_matrix">
+<a class="viewcode-back" href="../../pySDC/helpers.spectral_helper.html#helpers.spectral_helper.SpectralHelper.put_BCs_in_matrix">[docs]</a>
+    <span class="k">def</span> <span class="nf">put_BCs_in_matrix</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">A</span><span class="p">):</span>
+<span class="w">        </span><span class="sd">&quot;&quot;&quot;</span>
+<span class="sd">        Put the boundary conditions in a matrix by replacing rows with BCs.</span>
+<span class="sd">        &quot;&quot;&quot;</span>
+        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">BC_line_zero_matrix</span> <span class="o">@</span> <span class="n">A</span> <span class="o">+</span> <span class="bp">self</span><span class="o">.</span><span class="n">BCs</span></div>
+
+
+<div class="viewcode-block" id="SpectralHelper.put_BCs_in_rhs_hat">
+<a class="viewcode-back" href="../../pySDC/helpers.spectral_helper.html#helpers.spectral_helper.SpectralHelper.put_BCs_in_rhs_hat">[docs]</a>
+    <span class="k">def</span> <span class="nf">put_BCs_in_rhs_hat</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">rhs_hat</span><span class="p">):</span>
+<span class="w">        </span><span class="sd">&quot;&quot;&quot;</span>
+<span class="sd">        Put the BCs in the right hand side in spectral space for solving.</span>
+<span class="sd">        This function needs no transforms.</span>
+
+<span class="sd">        Args:</span>
+<span class="sd">            rhs_hat: Right hand side in spectral space</span>
+
+<span class="sd">        Returns:</span>
+<span class="sd">            rhs in spectral space with BCs</span>
+<span class="sd">        &quot;&quot;&quot;</span>
+        <span class="n">ndim</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">ndim</span>
+
+        <span class="k">for</span> <span class="n">axis</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">ndim</span><span class="p">):</span>
+            <span class="k">for</span> <span class="n">bc</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">full_BCs</span><span class="p">:</span>
+                <span class="n">slices</span> <span class="o">=</span> <span class="p">(</span>
+                    <span class="p">[</span><span class="nb">slice</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">init</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">])</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">axis</span><span class="p">)]</span>
+                    <span class="o">+</span> <span class="p">[</span><span class="n">bc</span><span class="p">[</span><span class="s1">&#39;line&#39;</span><span class="p">]]</span>
+                    <span class="o">+</span> <span class="p">[</span><span class="nb">slice</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">init</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">])</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">axis</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">axes</span><span class="p">))]</span>
+                <span class="p">)</span>
+                <span class="k">if</span> <span class="n">axis</span> <span class="o">==</span> <span class="n">bc</span><span class="p">[</span><span class="s1">&#39;axis&#39;</span><span class="p">]:</span>
+                    <span class="n">_slice</span> <span class="o">=</span> <span class="p">[</span><span class="bp">self</span><span class="o">.</span><span class="n">index</span><span class="p">(</span><span class="n">bc</span><span class="p">[</span><span class="s1">&#39;equation&#39;</span><span class="p">])]</span> <span class="o">+</span> <span class="n">slices</span>
+                    <span class="n">N</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">axes</span><span class="p">[</span><span class="n">axis</span><span class="p">]</span><span class="o">.</span><span class="n">N</span>
+                    <span class="k">if</span> <span class="p">(</span><span class="n">N</span> <span class="o">+</span> <span class="n">bc</span><span class="p">[</span><span class="s1">&#39;line&#39;</span><span class="p">])</span> <span class="o">%</span> <span class="n">N</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">xp</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="n">N</span><span class="p">)[</span><span class="bp">self</span><span class="o">.</span><span class="n">local_slice</span><span class="p">[</span><span class="n">axis</span><span class="p">]]:</span>
+                        <span class="n">_slice</span><span class="p">[</span><span class="n">axis</span> <span class="o">+</span> <span class="mi">1</span><span class="p">]</span> <span class="o">-=</span> <span class="bp">self</span><span class="o">.</span><span class="n">local_slice</span><span class="p">[</span><span class="n">axis</span><span class="p">]</span><span class="o">.</span><span class="n">start</span>
+                        <span class="n">rhs_hat</span><span class="p">[(</span><span class="o">*</span><span class="n">_slice</span><span class="p">,)]</span> <span class="o">=</span> <span class="mi">0</span>
+
+        <span class="k">return</span> <span class="n">rhs_hat</span> <span class="o">+</span> <span class="bp">self</span><span class="o">.</span><span class="n">rhs_BCs_hat</span></div>
+
+
+<div class="viewcode-block" id="SpectralHelper.put_BCs_in_rhs">
+<a class="viewcode-back" href="../../pySDC/helpers.spectral_helper.html#helpers.spectral_helper.SpectralHelper.put_BCs_in_rhs">[docs]</a>
+    <span class="k">def</span> <span class="nf">put_BCs_in_rhs</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">rhs</span><span class="p">):</span>
+<span class="w">        </span><span class="sd">&quot;&quot;&quot;</span>
+<span class="sd">        Put the BCs in the right hand side for solving.</span>
+<span class="sd">        This function will transform along each axis individually and add all BCs in that axis.</span>
+<span class="sd">        Consider `put_BCs_in_rhs_hat` to add BCs with no extra transforms needed.</span>
+
+<span class="sd">        Args:</span>
+<span class="sd">            rhs: Right hand side in physical space</span>
+
+<span class="sd">        Returns:</span>
+<span class="sd">            rhs in physical space with BCs</span>
+<span class="sd">        &quot;&quot;&quot;</span>
+        <span class="k">assert</span> <span class="n">rhs</span><span class="o">.</span><span class="n">ndim</span> <span class="o">&gt;</span> <span class="mi">1</span><span class="p">,</span> <span class="s1">&#39;rhs must not be flattened here!&#39;</span>
+
+        <span class="n">ndim</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">ndim</span>
+
+        <span class="k">for</span> <span class="n">axis</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">ndim</span><span class="p">):</span>
+            <span class="n">_rhs_hat</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">transform</span><span class="p">(</span><span class="n">rhs</span><span class="p">,</span> <span class="n">axes</span><span class="o">=</span><span class="p">(</span><span class="n">axis</span> <span class="o">-</span> <span class="n">ndim</span><span class="p">,))</span>
+
+            <span class="k">for</span> <span class="n">bc</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">full_BCs</span><span class="p">:</span>
+                <span class="n">slices</span> <span class="o">=</span> <span class="p">(</span>
+                    <span class="p">[</span><span class="nb">slice</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">init</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">])</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">axis</span><span class="p">)]</span>
+                    <span class="o">+</span> <span class="p">[</span><span class="n">bc</span><span class="p">[</span><span class="s1">&#39;line&#39;</span><span class="p">]]</span>
+                    <span class="o">+</span> <span class="p">[</span><span class="nb">slice</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">init</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">])</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">axis</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">axes</span><span class="p">))]</span>
+                <span class="p">)</span>
+                <span class="k">if</span> <span class="n">axis</span> <span class="o">==</span> <span class="n">bc</span><span class="p">[</span><span class="s1">&#39;axis&#39;</span><span class="p">]:</span>
+                    <span class="n">_slice</span> <span class="o">=</span> <span class="p">[</span><span class="bp">self</span><span class="o">.</span><span class="n">index</span><span class="p">(</span><span class="n">bc</span><span class="p">[</span><span class="s1">&#39;equation&#39;</span><span class="p">])]</span> <span class="o">+</span> <span class="n">slices</span>
+
+                    <span class="n">N</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">axes</span><span class="p">[</span><span class="n">axis</span><span class="p">]</span><span class="o">.</span><span class="n">N</span>
+                    <span class="k">if</span> <span class="p">(</span><span class="n">N</span> <span class="o">+</span> <span class="n">bc</span><span class="p">[</span><span class="s1">&#39;line&#39;</span><span class="p">])</span> <span class="o">%</span> <span class="n">N</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">xp</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="n">N</span><span class="p">)[</span><span class="bp">self</span><span class="o">.</span><span class="n">local_slice</span><span class="p">[</span><span class="n">axis</span><span class="p">]]:</span>
+                        <span class="n">_slice</span><span class="p">[</span><span class="n">axis</span> <span class="o">+</span> <span class="mi">1</span><span class="p">]</span> <span class="o">-=</span> <span class="bp">self</span><span class="o">.</span><span class="n">local_slice</span><span class="p">[</span><span class="n">axis</span><span class="p">]</span><span class="o">.</span><span class="n">start</span>
+
+                        <span class="n">_rhs_hat</span><span class="p">[(</span><span class="o">*</span><span class="n">_slice</span><span class="p">,)]</span> <span class="o">=</span> <span class="n">bc</span><span class="p">[</span><span class="s1">&#39;v&#39;</span><span class="p">]</span>
+
+            <span class="n">rhs</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">itransform</span><span class="p">(</span><span class="n">_rhs_hat</span><span class="p">,</span> <span class="n">axes</span><span class="o">=</span><span class="p">(</span><span class="n">axis</span> <span class="o">-</span> <span class="n">ndim</span><span class="p">,))</span>
+
+        <span class="k">return</span> <span class="n">rhs</span></div>
+
+
+<div class="viewcode-block" id="SpectralHelper.add_equation_lhs">
+<a class="viewcode-back" href="../../pySDC/helpers.spectral_helper.html#helpers.spectral_helper.SpectralHelper.add_equation_lhs">[docs]</a>
+    <span class="k">def</span> <span class="nf">add_equation_lhs</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">A</span><span class="p">,</span> <span class="n">equation</span><span class="p">,</span> <span class="n">relations</span><span class="p">):</span>
+<span class="w">        </span><span class="sd">&quot;&quot;&quot;</span>
+<span class="sd">        Add the left hand part (that you want to solve implicitly) of an equation to a list of lists of sparse matrices</span>
+<span class="sd">        that you will convert to an operator later.</span>
+
+<span class="sd">        Example:</span>
+<span class="sd">        Setup linear operator `L` for 1D heat equation using Chebychev method in first order form and T-to-U</span>
+<span class="sd">        preconditioning:</span>
+
+<span class="sd">        .. code-block:: python</span>
+<span class="sd">            helper = SpectralHelper()</span>
+
+<span class="sd">            helper.add_axis(base=&#39;chebychev&#39;, N=8)</span>
+<span class="sd">            helper.add_component([&#39;u&#39;, &#39;ux&#39;])</span>
+<span class="sd">            helper.setup_fft()</span>
+
+<span class="sd">            I = helper.get_Id()</span>
+<span class="sd">            Dx = helper.get_differentiation_matrix(axes=(0,))</span>
+<span class="sd">            T2U = helper.get_basis_change_matrix(&#39;T2U&#39;)</span>
+
+<span class="sd">            L_lhs = {</span>
+<span class="sd">                &#39;ux&#39;: {&#39;u&#39;: -T2U @ Dx, &#39;ux&#39;: T2U @ I},</span>
+<span class="sd">                &#39;u&#39;: {&#39;ux&#39;: -(T2U @ Dx)},</span>
+<span class="sd">            }</span>
+
+<span class="sd">            operator = helper.get_empty_operator_matrix()</span>
+<span class="sd">            for line, equation in L_lhs.items():</span>
+<span class="sd">                helper.add_equation_lhs(operator, line, equation)</span>
+
+<span class="sd">            L = helper.convert_operator_matrix_to_operator(operator)</span>
+
+<span class="sd">        Args:</span>
+<span class="sd">            A (list of lists of sparse matrices): The operator to be</span>
+<span class="sd">            equation (str): The equation of the component you want this in</span>
+<span class="sd">            relations: (dict): Relations between quantities</span>
+<span class="sd">        &quot;&quot;&quot;</span>
+        <span class="k">for</span> <span class="n">k</span><span class="p">,</span> <span class="n">v</span> <span class="ow">in</span> <span class="n">relations</span><span class="o">.</span><span class="n">items</span><span class="p">():</span>
+            <span class="n">A</span><span class="p">[</span><span class="bp">self</span><span class="o">.</span><span class="n">index</span><span class="p">(</span><span class="n">equation</span><span class="p">)][</span><span class="bp">self</span><span class="o">.</span><span class="n">index</span><span class="p">(</span><span class="n">k</span><span class="p">)]</span> <span class="o">=</span> <span class="n">v</span></div>
+
+
+<div class="viewcode-block" id="SpectralHelper.convert_operator_matrix_to_operator">
+<a class="viewcode-back" href="../../pySDC/helpers.spectral_helper.html#helpers.spectral_helper.SpectralHelper.convert_operator_matrix_to_operator">[docs]</a>
+    <span class="k">def</span> <span class="nf">convert_operator_matrix_to_operator</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">M</span><span class="p">):</span>
+<span class="w">        </span><span class="sd">&quot;&quot;&quot;</span>
+<span class="sd">        Promote the list of lists of sparse matrices to a single sparse matrix that can be used as linear operator.</span>
+<span class="sd">        See documentation of `SpectralHelper.add_equation_lhs` for an example.</span>
+
+<span class="sd">        Args:</span>
+<span class="sd">            M (list of lists of sparse matrices): The operator to be</span>
+
+<span class="sd">        Returns:</span>
+<span class="sd">            sparse linear operator</span>
+<span class="sd">        &quot;&quot;&quot;</span>
+        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">components</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
+            <span class="k">return</span> <span class="n">M</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span>
+        <span class="k">else</span><span class="p">:</span>
+            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">sparse_lib</span><span class="o">.</span><span class="n">bmat</span><span class="p">(</span><span class="n">M</span><span class="p">,</span> <span class="nb">format</span><span class="o">=</span><span class="s1">&#39;csc&#39;</span><span class="p">)</span></div>
+
+
+<div class="viewcode-block" id="SpectralHelper.get_wavenumbers">
+<a class="viewcode-back" href="../../pySDC/helpers.spectral_helper.html#helpers.spectral_helper.SpectralHelper.get_wavenumbers">[docs]</a>
+    <span class="k">def</span> <span class="nf">get_wavenumbers</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
+<span class="w">        </span><span class="sd">&quot;&quot;&quot;</span>
+<span class="sd">        Get grid in spectral space</span>
+<span class="sd">        &quot;&quot;&quot;</span>
+        <span class="n">grids</span> <span class="o">=</span> <span class="p">[</span><span class="bp">self</span><span class="o">.</span><span class="n">axes</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">.</span><span class="n">get_wavenumbers</span><span class="p">()[</span><span class="bp">self</span><span class="o">.</span><span class="n">local_slice</span><span class="p">[</span><span class="n">i</span><span class="p">]]</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">axes</span><span class="p">))][::</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
+        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">xp</span><span class="o">.</span><span class="n">meshgrid</span><span class="p">(</span><span class="o">*</span><span class="n">grids</span><span class="p">)</span></div>
+
+
+<div class="viewcode-block" id="SpectralHelper.get_grid">
+<a class="viewcode-back" href="../../pySDC/helpers.spectral_helper.html#helpers.spectral_helper.SpectralHelper.get_grid">[docs]</a>
+    <span class="k">def</span> <span class="nf">get_grid</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
+<span class="w">        </span><span class="sd">&quot;&quot;&quot;</span>
+<span class="sd">        Get grid in physical space</span>
+<span class="sd">        &quot;&quot;&quot;</span>
+        <span class="n">grids</span> <span class="o">=</span> <span class="p">[</span><span class="bp">self</span><span class="o">.</span><span class="n">axes</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">.</span><span class="n">get_1dgrid</span><span class="p">()[</span><span class="bp">self</span><span class="o">.</span><span class="n">local_slice</span><span class="p">[</span><span class="n">i</span><span class="p">]]</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">axes</span><span class="p">))][::</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
+        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">xp</span><span class="o">.</span><span class="n">meshgrid</span><span class="p">(</span><span class="o">*</span><span class="n">grids</span><span class="p">)</span></div>
+
+
+<div class="viewcode-block" id="SpectralHelper.get_fft">
+<a class="viewcode-back" href="../../pySDC/helpers.spectral_helper.html#helpers.spectral_helper.SpectralHelper.get_fft">[docs]</a>
+    <span class="k">def</span> <span class="nf">get_fft</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">axes</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">direction</span><span class="o">=</span><span class="s1">&#39;object&#39;</span><span class="p">,</span> <span class="n">padding</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">shape</span><span class="o">=</span><span class="kc">None</span><span class="p">):</span>
+<span class="w">        </span><span class="sd">&quot;&quot;&quot;</span>
+<span class="sd">        When using MPI, we use `PFFT` objects generated by mpi4py-fft</span>
+
+<span class="sd">        Args:</span>
+<span class="sd">            axes (tuple): Axes you want to transform over</span>
+<span class="sd">            direction (str): use &quot;forward&quot; or &quot;backward&quot; to get functions for performing the transforms or &quot;object&quot; to get the PFFT object</span>
+<span class="sd">            padding (tuple): Padding for dealiasing</span>
+<span class="sd">            shape (tuple): Shape of the transform</span>
+
+<span class="sd">        Returns:</span>
+<span class="sd">            transform</span>
+<span class="sd">        &quot;&quot;&quot;</span>
+        <span class="n">axes</span> <span class="o">=</span> <span class="nb">tuple</span><span class="p">(</span><span class="o">-</span><span class="n">i</span> <span class="o">-</span> <span class="mi">1</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">ndim</span><span class="p">))</span> <span class="k">if</span> <span class="n">axes</span> <span class="ow">is</span> <span class="kc">None</span> <span class="k">else</span> <span class="n">axes</span>
+        <span class="n">shape</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">global_shape</span><span class="p">[</span><span class="mi">1</span><span class="p">:]</span> <span class="k">if</span> <span class="n">shape</span> <span class="ow">is</span> <span class="kc">None</span> <span class="k">else</span> <span class="n">shape</span>
+        <span class="n">padding</span> <span class="o">=</span> <span class="p">(</span>
+            <span class="p">[</span>
+                <span class="mi">1</span><span class="p">,</span>
+            <span class="p">]</span>
+            <span class="o">*</span> <span class="bp">self</span><span class="o">.</span><span class="n">ndim</span>
+            <span class="k">if</span> <span class="n">padding</span> <span class="ow">is</span> <span class="kc">None</span>
+            <span class="k">else</span> <span class="n">padding</span>
+        <span class="p">)</span>
+        <span class="n">key</span> <span class="o">=</span> <span class="p">(</span><span class="n">axes</span><span class="p">,</span> <span class="n">direction</span><span class="p">,</span> <span class="nb">tuple</span><span class="p">(</span><span class="n">padding</span><span class="p">),</span> <span class="nb">tuple</span><span class="p">(</span><span class="n">shape</span><span class="p">))</span>
+
+        <span class="k">if</span> <span class="n">key</span> <span class="ow">not</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">fft_cache</span><span class="o">.</span><span class="n">keys</span><span class="p">():</span>
+            <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">comm</span> <span class="ow">is</span> <span class="kc">None</span><span class="p">:</span>
+                <span class="k">assert</span> <span class="n">np</span><span class="o">.</span><span class="n">allclose</span><span class="p">(</span><span class="n">padding</span><span class="p">,</span> <span class="mi">1</span><span class="p">),</span> <span class="s1">&#39;Zero padding is not implemented for non-MPI transforms&#39;</span>
+
+                <span class="k">if</span> <span class="n">direction</span> <span class="o">==</span> <span class="s1">&#39;forward&#39;</span><span class="p">:</span>
+                    <span class="bp">self</span><span class="o">.</span><span class="n">fft_cache</span><span class="p">[</span><span class="n">key</span><span class="p">]</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">xp</span><span class="o">.</span><span class="n">fft</span><span class="o">.</span><span class="n">fftn</span>
+                <span class="k">elif</span> <span class="n">direction</span> <span class="o">==</span> <span class="s1">&#39;backward&#39;</span><span class="p">:</span>
+                    <span class="bp">self</span><span class="o">.</span><span class="n">fft_cache</span><span class="p">[</span><span class="n">key</span><span class="p">]</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">xp</span><span class="o">.</span><span class="n">fft</span><span class="o">.</span><span class="n">ifftn</span>
+                <span class="k">elif</span> <span class="n">direction</span> <span class="o">==</span> <span class="s1">&#39;object&#39;</span><span class="p">:</span>
+                    <span class="bp">self</span><span class="o">.</span><span class="n">fft_cache</span><span class="p">[</span><span class="n">key</span><span class="p">]</span> <span class="o">=</span> <span class="kc">None</span>
+            <span class="k">else</span><span class="p">:</span>
+                <span class="kn">from</span> <span class="nn">mpi4py_fft</span> <span class="kn">import</span> <span class="n">PFFT</span>
+
+                <span class="n">_fft</span> <span class="o">=</span> <span class="n">PFFT</span><span class="p">(</span>
+                    <span class="n">comm</span><span class="o">=</span><span class="bp">self</span><span class="o">.</span><span class="n">comm</span><span class="p">,</span>
+                    <span class="n">shape</span><span class="o">=</span><span class="n">shape</span><span class="p">,</span>
+                    <span class="n">axes</span><span class="o">=</span><span class="nb">sorted</span><span class="p">(</span><span class="n">axes</span><span class="p">),</span>
+                    <span class="n">dtype</span><span class="o">=</span><span class="s1">&#39;D&#39;</span><span class="p">,</span>
+                    <span class="n">collapse</span><span class="o">=</span><span class="kc">False</span><span class="p">,</span>
+                    <span class="n">backend</span><span class="o">=</span><span class="bp">self</span><span class="o">.</span><span class="n">fft_backend</span><span class="p">,</span>
+                    <span class="n">comm_backend</span><span class="o">=</span><span class="bp">self</span><span class="o">.</span><span class="n">fft_comm_backend</span><span class="p">,</span>
+                    <span class="n">padding</span><span class="o">=</span><span class="n">padding</span><span class="p">,</span>
+                <span class="p">)</span>
+                <span class="k">if</span> <span class="n">direction</span> <span class="o">==</span> <span class="s1">&#39;forward&#39;</span><span class="p">:</span>
+                    <span class="bp">self</span><span class="o">.</span><span class="n">fft_cache</span><span class="p">[</span><span class="n">key</span><span class="p">]</span> <span class="o">=</span> <span class="n">_fft</span><span class="o">.</span><span class="n">forward</span>
+                <span class="k">elif</span> <span class="n">direction</span> <span class="o">==</span> <span class="s1">&#39;backward&#39;</span><span class="p">:</span>
+                    <span class="bp">self</span><span class="o">.</span><span class="n">fft_cache</span><span class="p">[</span><span class="n">key</span><span class="p">]</span> <span class="o">=</span> <span class="n">_fft</span><span class="o">.</span><span class="n">backward</span>
+                <span class="k">elif</span> <span class="n">direction</span> <span class="o">==</span> <span class="s1">&#39;object&#39;</span><span class="p">:</span>
+                    <span class="bp">self</span><span class="o">.</span><span class="n">fft_cache</span><span class="p">[</span><span class="n">key</span><span class="p">]</span> <span class="o">=</span> <span class="n">_fft</span>
+
+        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">fft_cache</span><span class="p">[</span><span class="n">key</span><span class="p">]</span></div>
+
+
+<div class="viewcode-block" id="SpectralHelper.setup_fft">
+<a class="viewcode-back" href="../../pySDC/helpers.spectral_helper.html#helpers.spectral_helper.SpectralHelper.setup_fft">[docs]</a>
+    <span class="k">def</span> <span class="nf">setup_fft</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">real_spectral_coefficients</span><span class="o">=</span><span class="kc">False</span><span class="p">):</span>
+<span class="w">        </span><span class="sd">&quot;&quot;&quot;</span>
+<span class="sd">        This function must be called after all axes have been setup in order to prepare the local shapes of the data.</span>
+<span class="sd">        This must also be called before setting up any BCs.</span>
+
+<span class="sd">        Args:</span>
+<span class="sd">            real_spectral_coefficients (bool): Allow only real coefficients in spectral space</span>
+<span class="sd">        &quot;&quot;&quot;</span>
+        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">components</span><span class="p">)</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
+            <span class="bp">self</span><span class="o">.</span><span class="n">add_component</span><span class="p">(</span><span class="s1">&#39;u&#39;</span><span class="p">)</span>
+
+        <span class="bp">self</span><span class="o">.</span><span class="n">global_shape</span> <span class="o">=</span> <span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">components</span><span class="p">),)</span> <span class="o">+</span> <span class="nb">tuple</span><span class="p">(</span><span class="n">me</span><span class="o">.</span><span class="n">N</span> <span class="k">for</span> <span class="n">me</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">axes</span><span class="p">)</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">local_slice</span> <span class="o">=</span> <span class="p">[</span><span class="nb">slice</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">me</span><span class="o">.</span><span class="n">N</span><span class="p">)</span> <span class="k">for</span> <span class="n">me</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">axes</span><span class="p">]</span>
+
+        <span class="n">axes</span> <span class="o">=</span> <span class="nb">tuple</span><span class="p">(</span><span class="n">i</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">axes</span><span class="p">)))</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">fft_obj</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">get_fft</span><span class="p">(</span><span class="n">axes</span><span class="o">=</span><span class="n">axes</span><span class="p">,</span> <span class="n">direction</span><span class="o">=</span><span class="s1">&#39;object&#39;</span><span class="p">)</span>
+        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">fft_obj</span> <span class="ow">is</span> <span class="ow">not</span> <span class="kc">None</span><span class="p">:</span>
+            <span class="bp">self</span><span class="o">.</span><span class="n">local_slice</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">fft_obj</span><span class="o">.</span><span class="n">local_slice</span><span class="p">(</span><span class="kc">False</span><span class="p">)</span>
+
+        <span class="bp">self</span><span class="o">.</span><span class="n">init</span> <span class="o">=</span> <span class="p">(</span>
+            <span class="n">np</span><span class="o">.</span><span class="n">empty</span><span class="p">(</span><span class="n">shape</span><span class="o">=</span><span class="bp">self</span><span class="o">.</span><span class="n">global_shape</span><span class="p">)[</span>
+                <span class="p">(</span>
+                    <span class="o">...</span><span class="p">,</span>
+                    <span class="o">*</span><span class="bp">self</span><span class="o">.</span><span class="n">local_slice</span><span class="p">,</span>
+                <span class="p">)</span>
+            <span class="p">]</span><span class="o">.</span><span class="n">shape</span><span class="p">,</span>
+            <span class="bp">self</span><span class="o">.</span><span class="n">comm</span><span class="p">,</span>
+            <span class="n">np</span><span class="o">.</span><span class="n">dtype</span><span class="p">(</span><span class="s1">&#39;float&#39;</span><span class="p">),</span>
+        <span class="p">)</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">init_forward</span> <span class="o">=</span> <span class="p">(</span>
+            <span class="n">np</span><span class="o">.</span><span class="n">empty</span><span class="p">(</span><span class="n">shape</span><span class="o">=</span><span class="bp">self</span><span class="o">.</span><span class="n">global_shape</span><span class="p">)[</span>
+                <span class="p">(</span>
+                    <span class="o">...</span><span class="p">,</span>
+                    <span class="o">*</span><span class="bp">self</span><span class="o">.</span><span class="n">local_slice</span><span class="p">,</span>
+                <span class="p">)</span>
+            <span class="p">]</span><span class="o">.</span><span class="n">shape</span><span class="p">,</span>
+            <span class="bp">self</span><span class="o">.</span><span class="n">comm</span><span class="p">,</span>
+            <span class="n">np</span><span class="o">.</span><span class="n">dtype</span><span class="p">(</span><span class="s1">&#39;float&#39;</span><span class="p">)</span> <span class="k">if</span> <span class="n">real_spectral_coefficients</span> <span class="k">else</span> <span class="n">np</span><span class="o">.</span><span class="n">dtype</span><span class="p">(</span><span class="s1">&#39;complex128&#39;</span><span class="p">),</span>
+        <span class="p">)</span>
+
+        <span class="bp">self</span><span class="o">.</span><span class="n">BC_mat</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">get_empty_operator_matrix</span><span class="p">()</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">BC_rhs_mask</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">xp</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span>
+            <span class="n">shape</span><span class="o">=</span><span class="bp">self</span><span class="o">.</span><span class="n">init</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span>
+            <span class="n">dtype</span><span class="o">=</span><span class="nb">bool</span><span class="p">,</span>
+        <span class="p">)</span></div>
+
+
+    <span class="k">def</span> <span class="nf">_transform_fft</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">axes</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
+<span class="w">        </span><span class="sd">&quot;&quot;&quot;</span>
+<span class="sd">        FFT along `axes`</span>
+
+<span class="sd">        Args:</span>
+<span class="sd">            u: The solution</span>
+<span class="sd">            axes (tuple): Axes you want to transform over</span>
+
+<span class="sd">        Returns:</span>
+<span class="sd">            transformed solution</span>
+<span class="sd">        &quot;&quot;&quot;</span>
+        <span class="c1"># TODO: clean up and try putting more of this in the 1D bases</span>
+        <span class="n">fft</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">get_fft</span><span class="p">(</span><span class="n">axes</span><span class="p">,</span> <span class="s1">&#39;forward&#39;</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
+        <span class="k">return</span> <span class="n">fft</span><span class="p">(</span><span class="n">u</span><span class="p">,</span> <span class="n">axes</span><span class="o">=</span><span class="n">axes</span><span class="p">)</span>
+
+    <span class="k">def</span> <span class="nf">_transform_dct</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">axes</span><span class="p">,</span> <span class="n">padding</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
+<span class="w">        </span><span class="sd">&#39;&#39;&#39;</span>
+<span class="sd">        DCT along `axes`.</span>
+<span class="sd">        This will only return real values!</span>
+<span class="sd">        When padding the solution, we cannot just use the mpi4py-fft implementation, because of the unusual ordering of</span>
+<span class="sd">        wavenumbers in FFTs.</span>
+
+<span class="sd">        Args:</span>
+<span class="sd">            u: The solution</span>
+<span class="sd">            axes (tuple): Axes you want to transform over</span>
+
+<span class="sd">        Returns:</span>
+<span class="sd">            transformed solution</span>
+<span class="sd">        &#39;&#39;&#39;</span>
+        <span class="c1"># TODO: clean up and try putting more of this in the 1D bases</span>
+        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">debug</span><span class="p">:</span>
+            <span class="k">assert</span> <span class="bp">self</span><span class="o">.</span><span class="n">xp</span><span class="o">.</span><span class="n">allclose</span><span class="p">(</span><span class="n">u</span><span class="o">.</span><span class="n">imag</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="s1">&#39;This function can only handle real input.&#39;</span>
+
+        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">axes</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mi">1</span><span class="p">:</span>
+            <span class="n">v</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_transform_dct</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_transform_dct</span><span class="p">(</span><span class="n">u</span><span class="p">,</span> <span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">:],</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">),</span> <span class="p">(</span><span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">],),</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
+        <span class="k">else</span><span class="p">:</span>
+            <span class="n">v</span> <span class="o">=</span> <span class="n">u</span><span class="o">.</span><span class="n">copy</span><span class="p">()</span><span class="o">.</span><span class="n">astype</span><span class="p">(</span><span class="nb">complex</span><span class="p">)</span>
+            <span class="n">axis</span> <span class="o">=</span> <span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
+            <span class="n">base</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">axes</span><span class="p">[</span><span class="n">axis</span><span class="p">]</span>
+
+            <span class="n">shuffle</span> <span class="o">=</span> <span class="p">[</span><span class="nb">slice</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">s</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span> <span class="k">for</span> <span class="n">s</span> <span class="ow">in</span> <span class="n">u</span><span class="o">.</span><span class="n">shape</span><span class="p">]</span>
+            <span class="n">shuffle</span><span class="p">[</span><span class="n">axis</span><span class="p">]</span> <span class="o">=</span> <span class="n">base</span><span class="o">.</span><span class="n">get_fft_shuffle</span><span class="p">(</span><span class="kc">True</span><span class="p">,</span> <span class="n">N</span><span class="o">=</span><span class="n">v</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="n">axis</span><span class="p">])</span>
+            <span class="n">v</span> <span class="o">=</span> <span class="n">v</span><span class="p">[(</span><span class="o">*</span><span class="n">shuffle</span><span class="p">,)]</span>
+
+            <span class="k">if</span> <span class="n">padding</span> <span class="ow">is</span> <span class="ow">not</span> <span class="kc">None</span><span class="p">:</span>
+                <span class="n">shape</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">v</span><span class="o">.</span><span class="n">shape</span><span class="p">)</span>
+                <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">comm</span><span class="p">:</span>
+                    <span class="n">shape</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">comm</span><span class="o">.</span><span class="n">allreduce</span><span class="p">(</span><span class="n">v</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
+                <span class="n">fft</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">get_fft</span><span class="p">(</span><span class="n">axes</span><span class="p">,</span> <span class="s1">&#39;forward&#39;</span><span class="p">,</span> <span class="n">shape</span><span class="o">=</span><span class="n">shape</span><span class="p">)</span>
+            <span class="k">else</span><span class="p">:</span>
+                <span class="n">fft</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">get_fft</span><span class="p">(</span><span class="n">axes</span><span class="p">,</span> <span class="s1">&#39;forward&#39;</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
+
+            <span class="n">v</span> <span class="o">=</span> <span class="n">fft</span><span class="p">(</span><span class="n">v</span><span class="p">,</span> <span class="n">axes</span><span class="o">=</span><span class="n">axes</span><span class="p">)</span>
+
+            <span class="n">expansion</span> <span class="o">=</span> <span class="p">[</span><span class="n">np</span><span class="o">.</span><span class="n">newaxis</span> <span class="k">for</span> <span class="n">_</span> <span class="ow">in</span> <span class="n">u</span><span class="o">.</span><span class="n">shape</span><span class="p">]</span>
+            <span class="n">expansion</span><span class="p">[</span><span class="n">axis</span><span class="p">]</span> <span class="o">=</span> <span class="nb">slice</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">v</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="n">axis</span><span class="p">],</span> <span class="mi">1</span><span class="p">)</span>
+
+            <span class="k">if</span> <span class="n">padding</span> <span class="ow">is</span> <span class="ow">not</span> <span class="kc">None</span><span class="p">:</span>
+                <span class="n">shift</span> <span class="o">=</span> <span class="n">base</span><span class="o">.</span><span class="n">get_fft_shift</span><span class="p">(</span><span class="kc">True</span><span class="p">,</span> <span class="n">v</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="n">axis</span><span class="p">])</span>
+
+                <span class="k">if</span> <span class="n">padding</span><span class="p">[</span><span class="n">axis</span><span class="p">]</span> <span class="o">!=</span> <span class="mi">1</span><span class="p">:</span>
+                    <span class="n">N</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">ceil</span><span class="p">(</span><span class="n">v</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="n">axis</span><span class="p">]</span> <span class="o">/</span> <span class="n">padding</span><span class="p">[</span><span class="n">axis</span><span class="p">]))</span>
+                    <span class="n">_expansion</span> <span class="o">=</span> <span class="p">[</span><span class="nb">slice</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span> <span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="n">v</span><span class="o">.</span><span class="n">shape</span><span class="p">]</span>
+                    <span class="n">_expansion</span><span class="p">[</span><span class="n">axis</span><span class="p">]</span> <span class="o">=</span> <span class="nb">slice</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">N</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
+                    <span class="n">v</span> <span class="o">=</span> <span class="n">v</span><span class="p">[(</span><span class="o">*</span><span class="n">_expansion</span><span class="p">,)]</span>
+            <span class="k">else</span><span class="p">:</span>
+                <span class="n">shift</span> <span class="o">=</span> <span class="n">base</span><span class="o">.</span><span class="n">fft_utils</span><span class="p">[</span><span class="s1">&#39;fwd&#39;</span><span class="p">][</span><span class="s1">&#39;shift&#39;</span><span class="p">]</span>
+
+            <span class="n">v</span> <span class="o">*=</span> <span class="n">shift</span><span class="p">[(</span><span class="o">*</span><span class="n">expansion</span><span class="p">,)]</span>
+
+        <span class="k">return</span> <span class="n">v</span><span class="o">.</span><span class="n">real</span>
+
+<div class="viewcode-block" id="SpectralHelper.transform_single_component">
+<a class="viewcode-back" href="../../pySDC/helpers.spectral_helper.html#helpers.spectral_helper.SpectralHelper.transform_single_component">[docs]</a>
+    <span class="k">def</span> <span class="nf">transform_single_component</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">axes</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">padding</span><span class="o">=</span><span class="kc">None</span><span class="p">):</span>
+<span class="w">        </span><span class="sd">&quot;&quot;&quot;</span>
+<span class="sd">        Transform a single component of the solution</span>
+
+<span class="sd">        Args:</span>
+<span class="sd">            u data to transform:</span>
+<span class="sd">            axes (tuple): Axes over which to transform</span>
+<span class="sd">            padding (list): Padding factor for transform. E.g. a padding factor of 2 will discard the upper half of modes after transforming</span>
+
+<span class="sd">        Returns:</span>
+<span class="sd">            Transformed data</span>
+<span class="sd">        &quot;&quot;&quot;</span>
+        <span class="c1"># TODO: clean up and try putting more of this in the 1D bases</span>
+        <span class="n">trfs</span> <span class="o">=</span> <span class="p">{</span>
+            <span class="n">ChebychevHelper</span><span class="p">:</span> <span class="bp">self</span><span class="o">.</span><span class="n">_transform_dct</span><span class="p">,</span>
+            <span class="n">UltrasphericalHelper</span><span class="p">:</span> <span class="bp">self</span><span class="o">.</span><span class="n">_transform_dct</span><span class="p">,</span>
+            <span class="n">FFTHelper</span><span class="p">:</span> <span class="bp">self</span><span class="o">.</span><span class="n">_transform_fft</span><span class="p">,</span>
+        <span class="p">}</span>
+
+        <span class="n">axes</span> <span class="o">=</span> <span class="nb">tuple</span><span class="p">(</span><span class="o">-</span><span class="n">i</span> <span class="o">-</span> <span class="mi">1</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">ndim</span><span class="p">)[::</span><span class="o">-</span><span class="mi">1</span><span class="p">])</span> <span class="k">if</span> <span class="n">axes</span> <span class="ow">is</span> <span class="kc">None</span> <span class="k">else</span> <span class="n">axes</span>
+        <span class="n">padding</span> <span class="o">=</span> <span class="p">(</span>
+            <span class="p">[</span>
+                <span class="mi">1</span><span class="p">,</span>
+            <span class="p">]</span>
+            <span class="o">*</span> <span class="bp">self</span><span class="o">.</span><span class="n">ndim</span>
+            <span class="k">if</span> <span class="n">padding</span> <span class="ow">is</span> <span class="kc">None</span>
+            <span class="k">else</span> <span class="n">padding</span>
+        <span class="p">)</span>  <span class="c1"># You know, sometimes I feel very strongly about Black still. This atrocious formatting is readable by Sauron only.</span>
+
+        <span class="n">result</span> <span class="o">=</span> <span class="n">u</span><span class="o">.</span><span class="n">copy</span><span class="p">()</span><span class="o">.</span><span class="n">astype</span><span class="p">(</span><span class="nb">complex</span><span class="p">)</span>
+        <span class="n">alignment</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">ndim</span> <span class="o">-</span> <span class="mi">1</span>
+
+        <span class="n">axes_collapsed</span> <span class="o">=</span> <span class="p">[</span><span class="nb">tuple</span><span class="p">(</span><span class="nb">sorted</span><span class="p">(</span><span class="n">me</span> <span class="k">for</span> <span class="n">me</span> <span class="ow">in</span> <span class="n">axes</span> <span class="k">if</span> <span class="nb">type</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">axes</span><span class="p">[</span><span class="n">me</span><span class="p">])</span> <span class="o">==</span> <span class="n">base</span><span class="p">))</span> <span class="k">for</span> <span class="n">base</span> <span class="ow">in</span> <span class="n">trfs</span><span class="o">.</span><span class="n">keys</span><span class="p">()]</span>
+        <span class="n">bases</span> <span class="o">=</span> <span class="p">[</span><span class="nb">list</span><span class="p">(</span><span class="n">trfs</span><span class="o">.</span><span class="n">keys</span><span class="p">())[</span><span class="n">i</span><span class="p">]</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">axes_collapsed</span><span class="p">))</span> <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">axes_collapsed</span><span class="p">[</span><span class="n">i</span><span class="p">])</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">]</span>
+        <span class="n">axes_collapsed</span> <span class="o">=</span> <span class="p">[</span><span class="n">me</span> <span class="k">for</span> <span class="n">me</span> <span class="ow">in</span> <span class="n">axes_collapsed</span> <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">me</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">]</span>
+        <span class="n">shape</span> <span class="o">=</span> <span class="p">[</span><span class="nb">max</span><span class="p">(</span><span class="n">u</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="bp">self</span><span class="o">.</span><span class="n">global_shape</span><span class="p">[</span><span class="mi">1</span> <span class="o">+</span> <span class="n">i</span><span class="p">])</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">ndim</span><span class="p">)]</span>
+
+        <span class="n">fft</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">get_fft</span><span class="p">(</span><span class="n">axes</span><span class="o">=</span><span class="n">axes</span><span class="p">,</span> <span class="n">padding</span><span class="o">=</span><span class="n">padding</span><span class="p">,</span> <span class="n">direction</span><span class="o">=</span><span class="s1">&#39;object&#39;</span><span class="p">)</span>
+        <span class="k">if</span> <span class="n">fft</span> <span class="ow">is</span> <span class="ow">not</span> <span class="kc">None</span><span class="p">:</span>
+            <span class="n">shape</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">fft</span><span class="o">.</span><span class="n">global_shape</span><span class="p">(</span><span class="kc">False</span><span class="p">))</span>
+
+        <span class="k">for</span> <span class="n">trf</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">axes_collapsed</span><span class="p">)):</span>
+            <span class="n">_axes</span> <span class="o">=</span> <span class="n">axes_collapsed</span><span class="p">[</span><span class="n">trf</span><span class="p">]</span>
+            <span class="n">base</span> <span class="o">=</span> <span class="n">bases</span><span class="p">[</span><span class="n">trf</span><span class="p">]</span>
+
+            <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">_axes</span><span class="p">)</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
+                <span class="k">continue</span>
+
+            <span class="k">for</span> <span class="n">_ax</span> <span class="ow">in</span> <span class="n">_axes</span><span class="p">:</span>
+                <span class="n">shape</span><span class="p">[</span><span class="n">_ax</span><span class="p">]</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">global_shape</span><span class="p">[</span><span class="mi">1</span> <span class="o">+</span> <span class="bp">self</span><span class="o">.</span><span class="n">ndim</span> <span class="o">+</span> <span class="n">_ax</span><span class="p">]</span>
+
+            <span class="n">fft</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">get_fft</span><span class="p">(</span><span class="n">_axes</span><span class="p">,</span> <span class="s1">&#39;object&#39;</span><span class="p">,</span> <span class="n">padding</span><span class="o">=</span><span class="n">padding</span><span class="p">,</span> <span class="n">shape</span><span class="o">=</span><span class="n">shape</span><span class="p">)</span>
+
+            <span class="n">_in</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">get_aligned</span><span class="p">(</span>
+                <span class="n">result</span><span class="p">,</span> <span class="n">axis_in</span><span class="o">=</span><span class="n">alignment</span><span class="p">,</span> <span class="n">axis_out</span><span class="o">=</span><span class="bp">self</span><span class="o">.</span><span class="n">ndim</span> <span class="o">+</span> <span class="n">_axes</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">],</span> <span class="n">forward</span><span class="o">=</span><span class="kc">False</span><span class="p">,</span> <span class="n">fft</span><span class="o">=</span><span class="n">fft</span><span class="p">,</span> <span class="n">shape</span><span class="o">=</span><span class="n">shape</span>
+            <span class="p">)</span>
+
+            <span class="n">alignment</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">ndim</span> <span class="o">+</span> <span class="n">_axes</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
+
+            <span class="n">_out</span> <span class="o">=</span> <span class="n">trfs</span><span class="p">[</span><span class="n">base</span><span class="p">](</span><span class="n">_in</span><span class="p">,</span> <span class="n">axes</span><span class="o">=</span><span class="n">_axes</span><span class="p">,</span> <span class="n">padding</span><span class="o">=</span><span class="n">padding</span><span class="p">,</span> <span class="n">shape</span><span class="o">=</span><span class="n">shape</span><span class="p">)</span>
+
+            <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">comm</span> <span class="ow">is</span> <span class="ow">not</span> <span class="kc">None</span><span class="p">:</span>
+                <span class="n">_out</span> <span class="o">*=</span> <span class="n">np</span><span class="o">.</span><span class="n">prod</span><span class="p">([</span><span class="bp">self</span><span class="o">.</span><span class="n">axes</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">.</span><span class="n">N</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">_axes</span><span class="p">])</span>
+
+            <span class="n">axes_next_base</span> <span class="o">=</span> <span class="n">axes_collapsed</span><span class="p">[(</span><span class="n">trf</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> <span class="o">%</span> <span class="nb">len</span><span class="p">(</span><span class="n">axes_collapsed</span><span class="p">)]</span>
+            <span class="n">alignment</span> <span class="o">=</span> <span class="n">alignment</span> <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">axes_next_base</span><span class="p">)</span> <span class="o">==</span> <span class="mi">0</span> <span class="k">else</span> <span class="bp">self</span><span class="o">.</span><span class="n">ndim</span> <span class="o">+</span> <span class="n">axes_next_base</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
+            <span class="n">result</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">get_aligned</span><span class="p">(</span>
+                <span class="n">_out</span><span class="p">,</span> <span class="n">axis_in</span><span class="o">=</span><span class="bp">self</span><span class="o">.</span><span class="n">ndim</span> <span class="o">+</span> <span class="n">_axes</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">axis_out</span><span class="o">=</span><span class="n">alignment</span><span class="p">,</span> <span class="n">fft</span><span class="o">=</span><span class="n">fft</span><span class="p">,</span> <span class="n">forward</span><span class="o">=</span><span class="kc">True</span><span class="p">,</span> <span class="n">shape</span><span class="o">=</span><span class="n">shape</span>
+            <span class="p">)</span>
+
+        <span class="n">fft</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">get_fft</span><span class="p">(</span><span class="n">axes</span><span class="o">=</span><span class="n">axes</span><span class="p">,</span> <span class="n">padding</span><span class="o">=</span><span class="n">padding</span><span class="p">)</span>
+        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">get_aligned</span><span class="p">(</span><span class="n">result</span><span class="p">,</span> <span class="n">axis_in</span><span class="o">=</span><span class="n">alignment</span><span class="p">,</span> <span class="n">axis_out</span><span class="o">=</span><span class="bp">self</span><span class="o">.</span><span class="n">ndim</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">fft</span><span class="o">=</span><span class="n">fft</span><span class="p">,</span> <span class="n">forward</span><span class="o">=</span><span class="kc">True</span><span class="p">,</span> <span class="n">shape</span><span class="o">=</span><span class="n">shape</span><span class="p">)</span></div>
+
+
+<div class="viewcode-block" id="SpectralHelper.transform">
+<a class="viewcode-back" href="../../pySDC/helpers.spectral_helper.html#helpers.spectral_helper.SpectralHelper.transform">[docs]</a>
+    <span class="k">def</span> <span class="nf">transform</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">axes</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">padding</span><span class="o">=</span><span class="kc">None</span><span class="p">):</span>
+<span class="w">        </span><span class="sd">&quot;&quot;&quot;</span>
+<span class="sd">        Transform all components from physical space to spectral space</span>
+
+<span class="sd">        Args:</span>
+<span class="sd">            u data to transform:</span>
+<span class="sd">            axes (tuple): Axes over which to transform</span>
+<span class="sd">            padding (list): Padding factor for transform. E.g. a padding factor of 2 will discard the upper half of modes after transforming</span>
+
+<span class="sd">        Returns:</span>
+<span class="sd">            Transformed data</span>
+<span class="sd">        &quot;&quot;&quot;</span>
+        <span class="n">axes</span> <span class="o">=</span> <span class="nb">tuple</span><span class="p">(</span><span class="o">-</span><span class="n">i</span> <span class="o">-</span> <span class="mi">1</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">ndim</span><span class="p">)[::</span><span class="o">-</span><span class="mi">1</span><span class="p">])</span> <span class="k">if</span> <span class="n">axes</span> <span class="ow">is</span> <span class="kc">None</span> <span class="k">else</span> <span class="n">axes</span>
+        <span class="n">padding</span> <span class="o">=</span> <span class="p">(</span>
+            <span class="p">[</span>
+                <span class="mi">1</span><span class="p">,</span>
+            <span class="p">]</span>
+            <span class="o">*</span> <span class="bp">self</span><span class="o">.</span><span class="n">ndim</span>
+            <span class="k">if</span> <span class="n">padding</span> <span class="ow">is</span> <span class="kc">None</span>
+            <span class="k">else</span> <span class="n">padding</span>
+        <span class="p">)</span>
+
+        <span class="n">result</span> <span class="o">=</span> <span class="p">[</span>
+            <span class="kc">None</span><span class="p">,</span>
+        <span class="p">]</span> <span class="o">*</span> <span class="bp">self</span><span class="o">.</span><span class="n">ncomponents</span>
+        <span class="k">for</span> <span class="n">comp</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">components</span><span class="p">:</span>
+            <span class="n">i</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">index</span><span class="p">(</span><span class="n">comp</span><span class="p">)</span>
+
+            <span class="n">result</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">transform_single_component</span><span class="p">(</span><span class="n">u</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">axes</span><span class="o">=</span><span class="n">axes</span><span class="p">,</span> <span class="n">padding</span><span class="o">=</span><span class="n">padding</span><span class="p">)</span>
+
+        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">xp</span><span class="o">.</span><span class="n">stack</span><span class="p">(</span><span class="n">result</span><span class="p">)</span></div>
+
+
+    <span class="k">def</span> <span class="nf">_transform_ifft</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">axes</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
+        <span class="c1"># TODO: clean up and try putting more of this in the 1D bases</span>
+        <span class="n">ifft</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">get_fft</span><span class="p">(</span><span class="n">axes</span><span class="p">,</span> <span class="s1">&#39;backward&#39;</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
+        <span class="k">return</span> <span class="n">ifft</span><span class="p">(</span><span class="n">u</span><span class="p">,</span> <span class="n">axes</span><span class="o">=</span><span class="n">axes</span><span class="p">)</span>
+
+    <span class="k">def</span> <span class="nf">_transform_idct</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">axes</span><span class="p">,</span> <span class="n">padding</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
+<span class="w">        </span><span class="sd">&#39;&#39;&#39;</span>
+<span class="sd">        This will only ever return real values!</span>
+<span class="sd">        &#39;&#39;&#39;</span>
+        <span class="c1"># TODO: clean up and try putting more of this in the 1D bases</span>
+        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">debug</span><span class="p">:</span>
+            <span class="k">assert</span> <span class="bp">self</span><span class="o">.</span><span class="n">xp</span><span class="o">.</span><span class="n">allclose</span><span class="p">(</span><span class="n">u</span><span class="o">.</span><span class="n">imag</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="s1">&#39;This function can only handle real input.&#39;</span>
+
+        <span class="n">v</span> <span class="o">=</span> <span class="n">u</span><span class="o">.</span><span class="n">copy</span><span class="p">()</span><span class="o">.</span><span class="n">astype</span><span class="p">(</span><span class="nb">complex</span><span class="p">)</span>
+
+        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">axes</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mi">1</span><span class="p">:</span>
+            <span class="n">v</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_transform_idct</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_transform_idct</span><span class="p">(</span><span class="n">u</span><span class="p">,</span> <span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">:]),</span> <span class="p">(</span><span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">],))</span>
+        <span class="k">else</span><span class="p">:</span>
+            <span class="n">axis</span> <span class="o">=</span> <span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
+            <span class="n">base</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">axes</span><span class="p">[</span><span class="n">axis</span><span class="p">]</span>
+
+            <span class="k">if</span> <span class="n">padding</span> <span class="ow">is</span> <span class="ow">not</span> <span class="kc">None</span><span class="p">:</span>
+                <span class="k">if</span> <span class="n">padding</span><span class="p">[</span><span class="n">axis</span><span class="p">]</span> <span class="o">!=</span> <span class="mi">1</span><span class="p">:</span>
+                    <span class="n">N_pad</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">ceil</span><span class="p">(</span><span class="n">v</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="n">axis</span><span class="p">]</span> <span class="o">*</span> <span class="n">padding</span><span class="p">[</span><span class="n">axis</span><span class="p">]))</span>
+                    <span class="n">_pad</span> <span class="o">=</span> <span class="p">[[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span> <span class="k">for</span> <span class="n">_</span> <span class="ow">in</span> <span class="n">v</span><span class="o">.</span><span class="n">shape</span><span class="p">]</span>
+                    <span class="n">_pad</span><span class="p">[</span><span class="n">axis</span><span class="p">]</span> <span class="o">=</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="n">N_pad</span> <span class="o">-</span> <span class="n">base</span><span class="o">.</span><span class="n">N</span><span class="p">]</span>
+                    <span class="n">v</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">xp</span><span class="o">.</span><span class="n">pad</span><span class="p">(</span><span class="n">v</span><span class="p">,</span> <span class="n">_pad</span><span class="p">,</span> <span class="s1">&#39;constant&#39;</span><span class="p">)</span>
+
+                    <span class="n">shift</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">xp</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="mi">1</span><span class="n">j</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">pi</span> <span class="o">*</span> <span class="bp">self</span><span class="o">.</span><span class="n">xp</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="n">N_pad</span><span class="p">)</span> <span class="o">/</span> <span class="p">(</span><span class="mi">2</span> <span class="o">*</span> <span class="n">N_pad</span><span class="p">))</span> <span class="o">*</span> <span class="n">base</span><span class="o">.</span><span class="n">N</span>
+                <span class="k">else</span><span class="p">:</span>
+                    <span class="n">shift</span> <span class="o">=</span> <span class="n">base</span><span class="o">.</span><span class="n">fft_utils</span><span class="p">[</span><span class="s1">&#39;bck&#39;</span><span class="p">][</span><span class="s1">&#39;shift&#39;</span><span class="p">]</span>
+            <span class="k">else</span><span class="p">:</span>
+                <span class="n">shift</span> <span class="o">=</span> <span class="n">base</span><span class="o">.</span><span class="n">fft_utils</span><span class="p">[</span><span class="s1">&#39;bck&#39;</span><span class="p">][</span><span class="s1">&#39;shift&#39;</span><span class="p">]</span>
+
+            <span class="n">expansion</span> <span class="o">=</span> <span class="p">[</span><span class="n">np</span><span class="o">.</span><span class="n">newaxis</span> <span class="k">for</span> <span class="n">_</span> <span class="ow">in</span> <span class="n">u</span><span class="o">.</span><span class="n">shape</span><span class="p">]</span>
+            <span class="n">expansion</span><span class="p">[</span><span class="n">axis</span><span class="p">]</span> <span class="o">=</span> <span class="nb">slice</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">v</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="n">axis</span><span class="p">],</span> <span class="mi">1</span><span class="p">)</span>
+
+            <span class="n">v</span> <span class="o">*=</span> <span class="n">shift</span><span class="p">[(</span><span class="o">*</span><span class="n">expansion</span><span class="p">,)]</span>
+
+            <span class="k">if</span> <span class="n">padding</span> <span class="ow">is</span> <span class="ow">not</span> <span class="kc">None</span><span class="p">:</span>
+                <span class="k">if</span> <span class="n">padding</span><span class="p">[</span><span class="n">axis</span><span class="p">]</span> <span class="o">!=</span> <span class="mi">1</span><span class="p">:</span>
+                    <span class="n">shape</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">v</span><span class="o">.</span><span class="n">shape</span><span class="p">)</span>
+                    <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">comm</span><span class="p">:</span>
+                        <span class="n">shape</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">comm</span><span class="o">.</span><span class="n">allreduce</span><span class="p">(</span><span class="n">v</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
+                    <span class="n">ifft</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">get_fft</span><span class="p">(</span><span class="n">axes</span><span class="p">,</span> <span class="s1">&#39;backward&#39;</span><span class="p">,</span> <span class="n">shape</span><span class="o">=</span><span class="n">shape</span><span class="p">)</span>
+                <span class="k">else</span><span class="p">:</span>
+                    <span class="n">ifft</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">get_fft</span><span class="p">(</span><span class="n">axes</span><span class="p">,</span> <span class="s1">&#39;backward&#39;</span><span class="p">,</span> <span class="n">padding</span><span class="o">=</span><span class="n">padding</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
+            <span class="k">else</span><span class="p">:</span>
+                <span class="n">ifft</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">get_fft</span><span class="p">(</span><span class="n">axes</span><span class="p">,</span> <span class="s1">&#39;backward&#39;</span><span class="p">,</span> <span class="n">padding</span><span class="o">=</span><span class="n">padding</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
+            <span class="n">v</span> <span class="o">=</span> <span class="n">ifft</span><span class="p">(</span><span class="n">v</span><span class="p">,</span> <span class="n">axes</span><span class="o">=</span><span class="n">axes</span><span class="p">)</span>
+
+            <span class="n">shuffle</span> <span class="o">=</span> <span class="p">[</span><span class="nb">slice</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">s</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span> <span class="k">for</span> <span class="n">s</span> <span class="ow">in</span> <span class="n">v</span><span class="o">.</span><span class="n">shape</span><span class="p">]</span>
+            <span class="n">shuffle</span><span class="p">[</span><span class="n">axis</span><span class="p">]</span> <span class="o">=</span> <span class="n">base</span><span class="o">.</span><span class="n">get_fft_shuffle</span><span class="p">(</span><span class="kc">False</span><span class="p">,</span> <span class="n">N</span><span class="o">=</span><span class="n">v</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="n">axis</span><span class="p">])</span>
+            <span class="n">v</span> <span class="o">=</span> <span class="n">v</span><span class="p">[(</span><span class="o">*</span><span class="n">shuffle</span><span class="p">,)]</span>
+
+        <span class="k">return</span> <span class="n">v</span><span class="o">.</span><span class="n">real</span>
+
+<div class="viewcode-block" id="SpectralHelper.itransform_single_component">
+<a class="viewcode-back" href="../../pySDC/helpers.spectral_helper.html#helpers.spectral_helper.SpectralHelper.itransform_single_component">[docs]</a>
+    <span class="k">def</span> <span class="nf">itransform_single_component</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">axes</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">padding</span><span class="o">=</span><span class="kc">None</span><span class="p">):</span>
+<span class="w">        </span><span class="sd">&quot;&quot;&quot;</span>
+<span class="sd">        Inverse transform over single component of the solution</span>
+
+<span class="sd">        Args:</span>
+<span class="sd">            u data to transform:</span>
+<span class="sd">            axes (tuple): Axes over which to transform</span>
+<span class="sd">            padding (list): Padding factor for transform. E.g. a padding factor of 2 will add as many zeros as there were modes before before transforming</span>
+
+<span class="sd">        Returns:</span>
+<span class="sd">            Transformed data</span>
+<span class="sd">        &quot;&quot;&quot;</span>
+        <span class="c1"># TODO: clean up and try putting more of this in the 1D bases</span>
+        <span class="n">trfs</span> <span class="o">=</span> <span class="p">{</span>
+            <span class="n">FFTHelper</span><span class="p">:</span> <span class="bp">self</span><span class="o">.</span><span class="n">_transform_ifft</span><span class="p">,</span>
+            <span class="n">ChebychevHelper</span><span class="p">:</span> <span class="bp">self</span><span class="o">.</span><span class="n">_transform_idct</span><span class="p">,</span>
+            <span class="n">UltrasphericalHelper</span><span class="p">:</span> <span class="bp">self</span><span class="o">.</span><span class="n">_transform_idct</span><span class="p">,</span>
+        <span class="p">}</span>
+
+        <span class="n">axes</span> <span class="o">=</span> <span class="nb">tuple</span><span class="p">(</span><span class="o">-</span><span class="n">i</span> <span class="o">-</span> <span class="mi">1</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">ndim</span><span class="p">)[::</span><span class="o">-</span><span class="mi">1</span><span class="p">])</span> <span class="k">if</span> <span class="n">axes</span> <span class="ow">is</span> <span class="kc">None</span> <span class="k">else</span> <span class="n">axes</span>
+        <span class="n">padding</span> <span class="o">=</span> <span class="p">(</span>
+            <span class="p">[</span>
+                <span class="mi">1</span><span class="p">,</span>
+            <span class="p">]</span>
+            <span class="o">*</span> <span class="bp">self</span><span class="o">.</span><span class="n">ndim</span>
+            <span class="k">if</span> <span class="n">padding</span> <span class="ow">is</span> <span class="kc">None</span>
+            <span class="k">else</span> <span class="n">padding</span>
+        <span class="p">)</span>
+
+        <span class="n">result</span> <span class="o">=</span> <span class="n">u</span><span class="o">.</span><span class="n">copy</span><span class="p">()</span><span class="o">.</span><span class="n">astype</span><span class="p">(</span><span class="nb">complex</span><span class="p">)</span>
+        <span class="n">alignment</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">ndim</span> <span class="o">-</span> <span class="mi">1</span>
+
+        <span class="n">axes_collapsed</span> <span class="o">=</span> <span class="p">[</span><span class="nb">tuple</span><span class="p">(</span><span class="nb">sorted</span><span class="p">(</span><span class="n">me</span> <span class="k">for</span> <span class="n">me</span> <span class="ow">in</span> <span class="n">axes</span> <span class="k">if</span> <span class="nb">type</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">axes</span><span class="p">[</span><span class="n">me</span><span class="p">])</span> <span class="o">==</span> <span class="n">base</span><span class="p">))</span> <span class="k">for</span> <span class="n">base</span> <span class="ow">in</span> <span class="n">trfs</span><span class="o">.</span><span class="n">keys</span><span class="p">()]</span>
+        <span class="n">bases</span> <span class="o">=</span> <span class="p">[</span><span class="nb">list</span><span class="p">(</span><span class="n">trfs</span><span class="o">.</span><span class="n">keys</span><span class="p">())[</span><span class="n">i</span><span class="p">]</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">axes_collapsed</span><span class="p">))</span> <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">axes_collapsed</span><span class="p">[</span><span class="n">i</span><span class="p">])</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">]</span>
+        <span class="n">axes_collapsed</span> <span class="o">=</span> <span class="p">[</span><span class="n">me</span> <span class="k">for</span> <span class="n">me</span> <span class="ow">in</span> <span class="n">axes_collapsed</span> <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">me</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">]</span>
+        <span class="n">shape</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">global_shape</span><span class="p">[</span><span class="mi">1</span><span class="p">:])</span>
+
+        <span class="k">for</span> <span class="n">trf</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">axes_collapsed</span><span class="p">)):</span>
+            <span class="n">_axes</span> <span class="o">=</span> <span class="n">axes_collapsed</span><span class="p">[</span><span class="n">trf</span><span class="p">]</span>
+            <span class="n">base</span> <span class="o">=</span> <span class="n">bases</span><span class="p">[</span><span class="n">trf</span><span class="p">]</span>
+
+            <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">_axes</span><span class="p">)</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
+                <span class="k">continue</span>
+
+            <span class="n">fft</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">get_fft</span><span class="p">(</span><span class="n">_axes</span><span class="p">,</span> <span class="s1">&#39;object&#39;</span><span class="p">,</span> <span class="n">padding</span><span class="o">=</span><span class="n">padding</span><span class="p">,</span> <span class="n">shape</span><span class="o">=</span><span class="n">shape</span><span class="p">)</span>
+
+            <span class="n">_in</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">get_aligned</span><span class="p">(</span>
+                <span class="n">result</span><span class="p">,</span> <span class="n">axis_in</span><span class="o">=</span><span class="n">alignment</span><span class="p">,</span> <span class="n">axis_out</span><span class="o">=</span><span class="bp">self</span><span class="o">.</span><span class="n">ndim</span> <span class="o">+</span> <span class="n">_axes</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">forward</span><span class="o">=</span><span class="kc">True</span><span class="p">,</span> <span class="n">fft</span><span class="o">=</span><span class="n">fft</span><span class="p">,</span> <span class="n">shape</span><span class="o">=</span><span class="n">shape</span>
+            <span class="p">)</span>
+            <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">comm</span> <span class="ow">is</span> <span class="ow">not</span> <span class="kc">None</span><span class="p">:</span>
+                <span class="n">_in</span> <span class="o">/=</span> <span class="n">np</span><span class="o">.</span><span class="n">prod</span><span class="p">([</span><span class="bp">self</span><span class="o">.</span><span class="n">axes</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">.</span><span class="n">N</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">_axes</span><span class="p">])</span>
+
+            <span class="n">alignment</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">ndim</span> <span class="o">+</span> <span class="n">_axes</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
+
+            <span class="n">_out</span> <span class="o">=</span> <span class="n">trfs</span><span class="p">[</span><span class="n">base</span><span class="p">](</span><span class="n">_in</span><span class="p">,</span> <span class="n">axes</span><span class="o">=</span><span class="n">_axes</span><span class="p">,</span> <span class="n">padding</span><span class="o">=</span><span class="n">padding</span><span class="p">,</span> <span class="n">shape</span><span class="o">=</span><span class="n">shape</span><span class="p">)</span>
+
+            <span class="k">for</span> <span class="n">_ax</span> <span class="ow">in</span> <span class="n">_axes</span><span class="p">:</span>
+                <span class="k">if</span> <span class="n">fft</span><span class="p">:</span>
+                    <span class="n">shape</span><span class="p">[</span><span class="n">_ax</span><span class="p">]</span> <span class="o">=</span> <span class="n">fft</span><span class="o">.</span><span class="n">_input_shape</span><span class="p">[</span><span class="n">_ax</span><span class="p">]</span>
+                <span class="k">else</span><span class="p">:</span>
+                    <span class="n">shape</span><span class="p">[</span><span class="n">_ax</span><span class="p">]</span> <span class="o">=</span> <span class="n">_out</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="n">_ax</span><span class="p">]</span>
+
+            <span class="n">axes_next_base</span> <span class="o">=</span> <span class="n">axes_collapsed</span><span class="p">[(</span><span class="n">trf</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> <span class="o">%</span> <span class="nb">len</span><span class="p">(</span><span class="n">axes_collapsed</span><span class="p">)]</span>
+            <span class="n">alignment</span> <span class="o">=</span> <span class="n">alignment</span> <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">axes_next_base</span><span class="p">)</span> <span class="o">==</span> <span class="mi">0</span> <span class="k">else</span> <span class="bp">self</span><span class="o">.</span><span class="n">ndim</span> <span class="o">+</span> <span class="n">axes_next_base</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
+            <span class="n">result</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">get_aligned</span><span class="p">(</span>
+                <span class="n">_out</span><span class="p">,</span> <span class="n">axis_in</span><span class="o">=</span><span class="bp">self</span><span class="o">.</span><span class="n">ndim</span> <span class="o">+</span> <span class="n">_axes</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">],</span> <span class="n">axis_out</span><span class="o">=</span><span class="n">alignment</span><span class="p">,</span> <span class="n">fft</span><span class="o">=</span><span class="n">fft</span><span class="p">,</span> <span class="n">forward</span><span class="o">=</span><span class="kc">False</span><span class="p">,</span> <span class="n">shape</span><span class="o">=</span><span class="n">shape</span>
+            <span class="p">)</span>
+
+        <span class="n">fft</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">get_fft</span><span class="p">(</span><span class="n">axes</span><span class="o">=</span><span class="n">axes</span><span class="p">,</span> <span class="n">padding</span><span class="o">=</span><span class="n">padding</span><span class="p">)</span>
+        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">get_aligned</span><span class="p">(</span><span class="n">result</span><span class="p">,</span> <span class="n">axis_in</span><span class="o">=</span><span class="n">alignment</span><span class="p">,</span> <span class="n">axis_out</span><span class="o">=</span><span class="bp">self</span><span class="o">.</span><span class="n">ndim</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">fft</span><span class="o">=</span><span class="n">fft</span><span class="p">,</span> <span class="n">shape</span><span class="o">=</span><span class="n">shape</span><span class="p">)</span></div>
+
+
+<div class="viewcode-block" id="SpectralHelper.get_aligned">
+<a class="viewcode-back" href="../../pySDC/helpers.spectral_helper.html#helpers.spectral_helper.SpectralHelper.get_aligned">[docs]</a>
+    <span class="k">def</span> <span class="nf">get_aligned</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">axis_in</span><span class="p">,</span> <span class="n">axis_out</span><span class="p">,</span> <span class="n">fft</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">forward</span><span class="o">=</span><span class="kc">False</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
+<span class="w">        </span><span class="sd">&quot;&quot;&quot;</span>
+<span class="sd">        Realign the data along the axis when using distributed FFTs. `kwargs` will be used to get the correct PFFT</span>
+<span class="sd">        object from `mpi4py-fft`, which has suitable transfer classes for the shape of data. Hence, they should include</span>
+<span class="sd">        shape especially, if applicable.</span>
+
+<span class="sd">        Args:</span>
+<span class="sd">            u: The solution</span>
+<span class="sd">            axis_in (int): Current alignment</span>
+<span class="sd">            axis_out (int): New alignment</span>
+<span class="sd">            fft (mpi4py_fft.PFFT), optional: parallel FFT object</span>
+<span class="sd">            forward (bool): Whether the input is in spectral space or not</span>
+
+<span class="sd">        Returns:</span>
+<span class="sd">            solution aligned on `axis_in`</span>
+<span class="sd">        &quot;&quot;&quot;</span>
+        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">comm</span> <span class="ow">is</span> <span class="kc">None</span> <span class="ow">or</span> <span class="n">axis_in</span> <span class="o">==</span> <span class="n">axis_out</span><span class="p">:</span>
+            <span class="k">return</span> <span class="n">u</span><span class="o">.</span><span class="n">copy</span><span class="p">()</span>
+        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">comm</span><span class="o">.</span><span class="n">size</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
+            <span class="k">return</span> <span class="n">u</span><span class="o">.</span><span class="n">copy</span><span class="p">()</span>
+
+        <span class="n">fft</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">get_fft</span><span class="p">(</span><span class="o">**</span><span class="n">kwargs</span><span class="p">)</span> <span class="k">if</span> <span class="n">fft</span> <span class="ow">is</span> <span class="kc">None</span> <span class="k">else</span> <span class="n">fft</span>
+
+        <span class="n">global_fft</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">get_fft</span><span class="p">(</span><span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
+        <span class="n">axisA</span> <span class="o">=</span> <span class="p">[</span><span class="n">me</span><span class="o">.</span><span class="n">axisA</span> <span class="k">for</span> <span class="n">me</span> <span class="ow">in</span> <span class="n">global_fft</span><span class="o">.</span><span class="n">transfer</span><span class="p">]</span>
+        <span class="n">axisB</span> <span class="o">=</span> <span class="p">[</span><span class="n">me</span><span class="o">.</span><span class="n">axisB</span> <span class="k">for</span> <span class="n">me</span> <span class="ow">in</span> <span class="n">global_fft</span><span class="o">.</span><span class="n">transfer</span><span class="p">]</span>
+
+        <span class="n">current_axis</span> <span class="o">=</span> <span class="n">axis_in</span>
+
+        <span class="k">if</span> <span class="n">axis_in</span> <span class="ow">in</span> <span class="n">axisA</span> <span class="ow">and</span> <span class="n">axis_out</span> <span class="ow">in</span> <span class="n">axisB</span><span class="p">:</span>
+            <span class="k">while</span> <span class="n">current_axis</span> <span class="o">!=</span> <span class="n">axis_out</span><span class="p">:</span>
+                <span class="n">transfer</span> <span class="o">=</span> <span class="n">global_fft</span><span class="o">.</span><span class="n">transfer</span><span class="p">[</span><span class="n">axisA</span><span class="o">.</span><span class="n">index</span><span class="p">(</span><span class="n">current_axis</span><span class="p">)]</span>
+
+                <span class="n">arrayB</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">xp</span><span class="o">.</span><span class="n">empty</span><span class="p">(</span><span class="n">shape</span><span class="o">=</span><span class="n">transfer</span><span class="o">.</span><span class="n">subshapeB</span><span class="p">,</span> <span class="n">dtype</span><span class="o">=</span><span class="n">transfer</span><span class="o">.</span><span class="n">dtype</span><span class="p">)</span>
+                <span class="n">arrayA</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">xp</span><span class="o">.</span><span class="n">empty</span><span class="p">(</span><span class="n">shape</span><span class="o">=</span><span class="n">transfer</span><span class="o">.</span><span class="n">subshapeA</span><span class="p">,</span> <span class="n">dtype</span><span class="o">=</span><span class="n">transfer</span><span class="o">.</span><span class="n">dtype</span><span class="p">)</span>
+                <span class="n">arrayA</span><span class="p">[:]</span> <span class="o">=</span> <span class="n">u</span><span class="p">[:]</span>
+
+                <span class="n">transfer</span><span class="o">.</span><span class="n">forward</span><span class="p">(</span><span class="n">arrayA</span><span class="o">=</span><span class="n">arrayA</span><span class="p">,</span> <span class="n">arrayB</span><span class="o">=</span><span class="n">arrayB</span><span class="p">)</span>
+
+                <span class="n">current_axis</span> <span class="o">=</span> <span class="n">transfer</span><span class="o">.</span><span class="n">axisB</span>
+                <span class="n">u</span> <span class="o">=</span> <span class="n">arrayB</span>
+
+            <span class="k">return</span> <span class="n">u</span>
+        <span class="k">elif</span> <span class="n">axis_in</span> <span class="ow">in</span> <span class="n">axisB</span> <span class="ow">and</span> <span class="n">axis_out</span> <span class="ow">in</span> <span class="n">axisA</span><span class="p">:</span>
+            <span class="k">while</span> <span class="n">current_axis</span> <span class="o">!=</span> <span class="n">axis_out</span><span class="p">:</span>
+                <span class="n">transfer</span> <span class="o">=</span> <span class="n">global_fft</span><span class="o">.</span><span class="n">transfer</span><span class="p">[</span><span class="n">axisB</span><span class="o">.</span><span class="n">index</span><span class="p">(</span><span class="n">current_axis</span><span class="p">)]</span>
+
+                <span class="n">arrayB</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">xp</span><span class="o">.</span><span class="n">empty</span><span class="p">(</span><span class="n">shape</span><span class="o">=</span><span class="n">transfer</span><span class="o">.</span><span class="n">subshapeB</span><span class="p">,</span> <span class="n">dtype</span><span class="o">=</span><span class="n">transfer</span><span class="o">.</span><span class="n">dtype</span><span class="p">)</span>
+                <span class="n">arrayA</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">xp</span><span class="o">.</span><span class="n">empty</span><span class="p">(</span><span class="n">shape</span><span class="o">=</span><span class="n">transfer</span><span class="o">.</span><span class="n">subshapeA</span><span class="p">,</span> <span class="n">dtype</span><span class="o">=</span><span class="n">transfer</span><span class="o">.</span><span class="n">dtype</span><span class="p">)</span>
+                <span class="n">arrayB</span><span class="p">[:]</span> <span class="o">=</span> <span class="n">u</span><span class="p">[:]</span>
+
+                <span class="n">transfer</span><span class="o">.</span><span class="n">backward</span><span class="p">(</span><span class="n">arrayA</span><span class="o">=</span><span class="n">arrayA</span><span class="p">,</span> <span class="n">arrayB</span><span class="o">=</span><span class="n">arrayB</span><span class="p">)</span>
+
+                <span class="n">current_axis</span> <span class="o">=</span> <span class="n">transfer</span><span class="o">.</span><span class="n">axisA</span>
+                <span class="n">u</span> <span class="o">=</span> <span class="n">arrayA</span>
+
+            <span class="k">return</span> <span class="n">u</span>
+        <span class="k">else</span><span class="p">:</span>  <span class="c1"># go the potentially slower route of not reusing transfer classes</span>
+            <span class="kn">from</span> <span class="nn">mpi4py_fft</span> <span class="kn">import</span> <span class="n">newDistArray</span>
+
+            <span class="n">_in</span> <span class="o">=</span> <span class="n">newDistArray</span><span class="p">(</span><span class="n">fft</span><span class="p">,</span> <span class="n">forward</span><span class="p">)</span><span class="o">.</span><span class="n">redistribute</span><span class="p">(</span><span class="n">axis_in</span><span class="p">)</span>
+            <span class="n">_in</span><span class="p">[</span><span class="o">...</span><span class="p">]</span> <span class="o">=</span> <span class="n">u</span>
+
+            <span class="k">return</span> <span class="n">_in</span><span class="o">.</span><span class="n">redistribute</span><span class="p">(</span><span class="n">axis_out</span><span class="p">)</span></div>
+
+
+<div class="viewcode-block" id="SpectralHelper.itransform">
+<a class="viewcode-back" href="../../pySDC/helpers.spectral_helper.html#helpers.spectral_helper.SpectralHelper.itransform">[docs]</a>
+    <span class="k">def</span> <span class="nf">itransform</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">axes</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">padding</span><span class="o">=</span><span class="kc">None</span><span class="p">):</span>
+<span class="w">        </span><span class="sd">&quot;&quot;&quot;</span>
+<span class="sd">        Inverse transform over all components of the solution</span>
+
+<span class="sd">        Args:</span>
+<span class="sd">            u data to transform:</span>
+<span class="sd">            axes (tuple): Axes over which to transform</span>
+<span class="sd">            padding (list): Padding factor for transform. E.g. a padding factor of 2 will add as many zeros as there were modes before before transforming</span>
+
+<span class="sd">        Returns:</span>
+<span class="sd">            Transformed data</span>
+<span class="sd">        &quot;&quot;&quot;</span>
+        <span class="n">axes</span> <span class="o">=</span> <span class="nb">tuple</span><span class="p">(</span><span class="o">-</span><span class="n">i</span> <span class="o">-</span> <span class="mi">1</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">ndim</span><span class="p">)[::</span><span class="o">-</span><span class="mi">1</span><span class="p">])</span> <span class="k">if</span> <span class="n">axes</span> <span class="ow">is</span> <span class="kc">None</span> <span class="k">else</span> <span class="n">axes</span>
+        <span class="n">padding</span> <span class="o">=</span> <span class="p">(</span>
+            <span class="p">[</span>
+                <span class="mi">1</span><span class="p">,</span>
+            <span class="p">]</span>
+            <span class="o">*</span> <span class="bp">self</span><span class="o">.</span><span class="n">ndim</span>
+            <span class="k">if</span> <span class="n">padding</span> <span class="ow">is</span> <span class="kc">None</span>
+            <span class="k">else</span> <span class="n">padding</span>
+        <span class="p">)</span>
+
+        <span class="n">result</span> <span class="o">=</span> <span class="p">[</span>
+            <span class="kc">None</span><span class="p">,</span>
+        <span class="p">]</span> <span class="o">*</span> <span class="bp">self</span><span class="o">.</span><span class="n">ncomponents</span>
+        <span class="k">for</span> <span class="n">comp</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">components</span><span class="p">:</span>
+            <span class="n">i</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">index</span><span class="p">(</span><span class="n">comp</span><span class="p">)</span>
+
+            <span class="n">result</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">itransform_single_component</span><span class="p">(</span><span class="n">u</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">axes</span><span class="o">=</span><span class="n">axes</span><span class="p">,</span> <span class="n">padding</span><span class="o">=</span><span class="n">padding</span><span class="p">)</span>
+
+        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">xp</span><span class="o">.</span><span class="n">stack</span><span class="p">(</span><span class="n">result</span><span class="p">)</span></div>
+
+
+<div class="viewcode-block" id="SpectralHelper.get_local_slice_of_1D_matrix">
+<a class="viewcode-back" href="../../pySDC/helpers.spectral_helper.html#helpers.spectral_helper.SpectralHelper.get_local_slice_of_1D_matrix">[docs]</a>
+    <span class="k">def</span> <span class="nf">get_local_slice_of_1D_matrix</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">M</span><span class="p">,</span> <span class="n">axis</span><span class="p">):</span>
+<span class="w">        </span><span class="sd">&quot;&quot;&quot;</span>
+<span class="sd">        Get the local version of a 1D matrix. When using distributed FFTs, each rank will carry only a subset of modes,</span>
+<span class="sd">        which you can sort out via the `SpectralHelper.local_slice` attribute. When constructing a 1D matrix, you can</span>
+<span class="sd">        use this method to get the part corresponding to the modes carried by this rank.</span>
+
+<span class="sd">        Args:</span>
+<span class="sd">            M (sparse matrix): Global 1D matrix you want to get the local version of</span>
+<span class="sd">            axis (int): Direction in which you want the local version. You will get the global matrix in other directions. This means slab decomposition only.</span>
+
+<span class="sd">        Returns:</span>
+<span class="sd">            sparse local matrix</span>
+<span class="sd">        &quot;&quot;&quot;</span>
+        <span class="k">return</span> <span class="n">M</span><span class="o">.</span><span class="n">tocsc</span><span class="p">()[</span><span class="bp">self</span><span class="o">.</span><span class="n">local_slice</span><span class="p">[</span><span class="n">axis</span><span class="p">],</span> <span class="bp">self</span><span class="o">.</span><span class="n">local_slice</span><span class="p">[</span><span class="n">axis</span><span class="p">]]</span></div>
+
+
+<div class="viewcode-block" id="SpectralHelper.get_filter_matrix">
+<a class="viewcode-back" href="../../pySDC/helpers.spectral_helper.html#helpers.spectral_helper.SpectralHelper.get_filter_matrix">[docs]</a>
+    <span class="k">def</span> <span class="nf">get_filter_matrix</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">axis</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
+<span class="w">        </span><span class="sd">&quot;&quot;&quot;</span>
+<span class="sd">        Get bandpass filter along `axis`. See the documentation `get_filter_matrix` in the 1D bases for what kwargs are</span>
+<span class="sd">        admissible.</span>
+
+<span class="sd">        Returns:</span>
+<span class="sd">            sparse bandpass matrix</span>
+<span class="sd">        &quot;&quot;&quot;</span>
+        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">ndim</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
+            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">get_filter_matrix</span><span class="p">(</span><span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
+
+        <span class="n">mats</span> <span class="o">=</span> <span class="p">[</span><span class="n">base</span><span class="o">.</span><span class="n">get_Id</span><span class="p">()</span> <span class="k">for</span> <span class="n">base</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">axes</span><span class="p">]</span>
+        <span class="n">mats</span><span class="p">[</span><span class="n">axis</span><span class="p">]</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">axes</span><span class="p">[</span><span class="n">axis</span><span class="p">]</span><span class="o">.</span><span class="n">get_filter_matrix</span><span class="p">(</span><span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
+        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">sparse_lib</span><span class="o">.</span><span class="n">kron</span><span class="p">(</span><span class="o">*</span><span class="n">mats</span><span class="p">)</span></div>
+
+
+<div class="viewcode-block" id="SpectralHelper.get_differentiation_matrix">
+<a class="viewcode-back" href="../../pySDC/helpers.spectral_helper.html#helpers.spectral_helper.SpectralHelper.get_differentiation_matrix">[docs]</a>
+    <span class="k">def</span> <span class="nf">get_differentiation_matrix</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">axes</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
+<span class="w">        </span><span class="sd">&quot;&quot;&quot;</span>
+<span class="sd">        Get differentiation matrix along specified axis. `kwargs` are forwarded to the 1D base implementation.</span>
+
+<span class="sd">        Args:</span>
+<span class="sd">            axes (tuple): Axes along which to differentiate.</span>
+
+<span class="sd">        Returns:</span>
+<span class="sd">            sparse differentiation matrix</span>
+<span class="sd">        &quot;&quot;&quot;</span>
+        <span class="n">sp</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">sparse_lib</span>
+        <span class="n">ndim</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">ndim</span>
+
+        <span class="k">if</span> <span class="n">ndim</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
+            <span class="n">D</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">get_differentiation_matrix</span><span class="p">(</span><span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
+        <span class="k">elif</span> <span class="n">ndim</span> <span class="o">==</span> <span class="mi">2</span><span class="p">:</span>
+            <span class="k">for</span> <span class="n">axis</span> <span class="ow">in</span> <span class="n">axes</span><span class="p">:</span>
+                <span class="n">axis2</span> <span class="o">=</span> <span class="p">(</span><span class="n">axis</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> <span class="o">%</span> <span class="n">ndim</span>
+                <span class="n">D1D</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">axes</span><span class="p">[</span><span class="n">axis</span><span class="p">]</span><span class="o">.</span><span class="n">get_differentiation_matrix</span><span class="p">(</span><span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
+
+                <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">axes</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mi">1</span><span class="p">:</span>
+                    <span class="n">I1D</span> <span class="o">=</span> <span class="n">sp</span><span class="o">.</span><span class="n">eye</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">axes</span><span class="p">[</span><span class="n">axis2</span><span class="p">]</span><span class="o">.</span><span class="n">N</span><span class="p">)</span>
+                <span class="k">else</span><span class="p">:</span>
+                    <span class="n">I1D</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">axes</span><span class="p">[</span><span class="n">axis2</span><span class="p">]</span><span class="o">.</span><span class="n">get_Id</span><span class="p">()</span>
+
+                <span class="n">mats</span> <span class="o">=</span> <span class="p">[</span><span class="kc">None</span><span class="p">]</span> <span class="o">*</span> <span class="n">ndim</span>
+                <span class="n">mats</span><span class="p">[</span><span class="n">axis</span><span class="p">]</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">get_local_slice_of_1D_matrix</span><span class="p">(</span><span class="n">D1D</span><span class="p">,</span> <span class="n">axis</span><span class="p">)</span>
+                <span class="n">mats</span><span class="p">[</span><span class="n">axis2</span><span class="p">]</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">get_local_slice_of_1D_matrix</span><span class="p">(</span><span class="n">I1D</span><span class="p">,</span> <span class="n">axis2</span><span class="p">)</span>
+
+                <span class="k">if</span> <span class="n">axis</span> <span class="o">==</span> <span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">]:</span>
+                    <span class="n">D</span> <span class="o">=</span> <span class="n">sp</span><span class="o">.</span><span class="n">kron</span><span class="p">(</span><span class="o">*</span><span class="n">mats</span><span class="p">)</span>
+                <span class="k">else</span><span class="p">:</span>
+                    <span class="n">D</span> <span class="o">=</span> <span class="n">D</span> <span class="o">@</span> <span class="n">sp</span><span class="o">.</span><span class="n">kron</span><span class="p">(</span><span class="o">*</span><span class="n">mats</span><span class="p">)</span>
+        <span class="k">else</span><span class="p">:</span>
+            <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span><span class="sa">f</span><span class="s1">&#39;Differentiation matrix not implemented for </span><span class="si">{</span><span class="n">ndim</span><span class="si">}</span><span class="s1"> dimension!&#39;</span><span class="p">)</span>
+
+        <span class="k">return</span> <span class="n">D</span></div>
+
+
+<div class="viewcode-block" id="SpectralHelper.get_integration_matrix">
+<a class="viewcode-back" href="../../pySDC/helpers.spectral_helper.html#helpers.spectral_helper.SpectralHelper.get_integration_matrix">[docs]</a>
+    <span class="k">def</span> <span class="nf">get_integration_matrix</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">axes</span><span class="p">):</span>
+<span class="w">        </span><span class="sd">&quot;&quot;&quot;</span>
+<span class="sd">        Get integration matrix to integrate along specified axis.</span>
+
+<span class="sd">        Args:</span>
+<span class="sd">            axes (tuple): Axes along which to integrate over.</span>
+
+<span class="sd">        Returns:</span>
+<span class="sd">            sparse integration matrix</span>
+<span class="sd">        &quot;&quot;&quot;</span>
+        <span class="n">sp</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">sparse_lib</span>
+        <span class="n">ndim</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">axes</span><span class="p">)</span>
+
+        <span class="k">if</span> <span class="n">ndim</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
+            <span class="n">S</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">get_integration_matrix</span><span class="p">()</span>
+        <span class="k">elif</span> <span class="n">ndim</span> <span class="o">==</span> <span class="mi">2</span><span class="p">:</span>
+            <span class="k">for</span> <span class="n">axis</span> <span class="ow">in</span> <span class="n">axes</span><span class="p">:</span>
+                <span class="n">axis2</span> <span class="o">=</span> <span class="p">(</span><span class="n">axis</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> <span class="o">%</span> <span class="n">ndim</span>
+                <span class="n">S1D</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">axes</span><span class="p">[</span><span class="n">axis</span><span class="p">]</span><span class="o">.</span><span class="n">get_integration_matrix</span><span class="p">()</span>
+
+                <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">axes</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mi">1</span><span class="p">:</span>
+                    <span class="n">I1D</span> <span class="o">=</span> <span class="n">sp</span><span class="o">.</span><span class="n">eye</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">axes</span><span class="p">[</span><span class="n">axis2</span><span class="p">]</span><span class="o">.</span><span class="n">N</span><span class="p">)</span>
+                <span class="k">else</span><span class="p">:</span>
+                    <span class="n">I1D</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">axes</span><span class="p">[</span><span class="n">axis2</span><span class="p">]</span><span class="o">.</span><span class="n">get_Id</span><span class="p">()</span>
+
+                <span class="n">mats</span> <span class="o">=</span> <span class="p">[</span><span class="kc">None</span><span class="p">]</span> <span class="o">*</span> <span class="n">ndim</span>
+                <span class="n">mats</span><span class="p">[</span><span class="n">axis</span><span class="p">]</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">get_local_slice_of_1D_matrix</span><span class="p">(</span><span class="n">S1D</span><span class="p">,</span> <span class="n">axis</span><span class="p">)</span>
+                <span class="n">mats</span><span class="p">[</span><span class="n">axis2</span><span class="p">]</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">get_local_slice_of_1D_matrix</span><span class="p">(</span><span class="n">I1D</span><span class="p">,</span> <span class="n">axis2</span><span class="p">)</span>
+
+                <span class="k">if</span> <span class="n">axis</span> <span class="o">==</span> <span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">]:</span>
+                    <span class="n">S</span> <span class="o">=</span> <span class="n">sp</span><span class="o">.</span><span class="n">kron</span><span class="p">(</span><span class="o">*</span><span class="n">mats</span><span class="p">)</span>
+                <span class="k">else</span><span class="p">:</span>
+                    <span class="n">S</span> <span class="o">=</span> <span class="n">S</span> <span class="o">@</span> <span class="n">sp</span><span class="o">.</span><span class="n">kron</span><span class="p">(</span><span class="o">*</span><span class="n">mats</span><span class="p">)</span>
+        <span class="k">else</span><span class="p">:</span>
+            <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span><span class="sa">f</span><span class="s1">&#39;Integration matrix not implemented for </span><span class="si">{</span><span class="n">ndim</span><span class="si">}</span><span class="s1"> dimension!&#39;</span><span class="p">)</span>
+
+        <span class="k">return</span> <span class="n">S</span></div>
+
+
+<div class="viewcode-block" id="SpectralHelper.get_Id">
+<a class="viewcode-back" href="../../pySDC/helpers.spectral_helper.html#helpers.spectral_helper.SpectralHelper.get_Id">[docs]</a>
+    <span class="k">def</span> <span class="nf">get_Id</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
+<span class="w">        </span><span class="sd">&quot;&quot;&quot;</span>
+<span class="sd">        Get identity matrix</span>
+
+<span class="sd">        Returns:</span>
+<span class="sd">            sparse identity matrix</span>
+<span class="sd">        &quot;&quot;&quot;</span>
+        <span class="n">sp</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">sparse_lib</span>
+        <span class="n">ndim</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">ndim</span>
+        <span class="n">I</span> <span class="o">=</span> <span class="n">sp</span><span class="o">.</span><span class="n">eye</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">prod</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">init</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">1</span><span class="p">:]),</span> <span class="n">dtype</span><span class="o">=</span><span class="nb">complex</span><span class="p">)</span>
+
+        <span class="k">if</span> <span class="n">ndim</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
+            <span class="n">I</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">get_Id</span><span class="p">()</span>
+        <span class="k">elif</span> <span class="n">ndim</span> <span class="o">==</span> <span class="mi">2</span><span class="p">:</span>
+            <span class="k">for</span> <span class="n">axis</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">ndim</span><span class="p">):</span>
+                <span class="n">axis2</span> <span class="o">=</span> <span class="p">(</span><span class="n">axis</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> <span class="o">%</span> <span class="n">ndim</span>
+                <span class="n">I1D</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">axes</span><span class="p">[</span><span class="n">axis</span><span class="p">]</span><span class="o">.</span><span class="n">get_Id</span><span class="p">()</span>
+
+                <span class="n">I1D2</span> <span class="o">=</span> <span class="n">sp</span><span class="o">.</span><span class="n">eye</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">axes</span><span class="p">[</span><span class="n">axis2</span><span class="p">]</span><span class="o">.</span><span class="n">N</span><span class="p">)</span>
+
+                <span class="n">mats</span> <span class="o">=</span> <span class="p">[</span><span class="kc">None</span><span class="p">]</span> <span class="o">*</span> <span class="n">ndim</span>
+                <span class="n">mats</span><span class="p">[</span><span class="n">axis</span><span class="p">]</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">get_local_slice_of_1D_matrix</span><span class="p">(</span><span class="n">I1D</span><span class="p">,</span> <span class="n">axis</span><span class="p">)</span>
+                <span class="n">mats</span><span class="p">[</span><span class="n">axis2</span><span class="p">]</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">get_local_slice_of_1D_matrix</span><span class="p">(</span><span class="n">I1D2</span><span class="p">,</span> <span class="n">axis2</span><span class="p">)</span>
+
+                <span class="n">I</span> <span class="o">=</span> <span class="n">I</span> <span class="o">@</span> <span class="n">sp</span><span class="o">.</span><span class="n">kron</span><span class="p">(</span><span class="o">*</span><span class="n">mats</span><span class="p">)</span>
+        <span class="k">else</span><span class="p">:</span>
+            <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span><span class="sa">f</span><span class="s1">&#39;Identity matrix not implemented for </span><span class="si">{</span><span class="n">ndim</span><span class="si">}</span><span class="s1"> dimension!&#39;</span><span class="p">)</span>
+
+        <span class="k">return</span> <span class="n">I</span></div>
+
+
+<div class="viewcode-block" id="SpectralHelper.get_Dirichlet_recombination_matrix">
+<a class="viewcode-back" href="../../pySDC/helpers.spectral_helper.html#helpers.spectral_helper.SpectralHelper.get_Dirichlet_recombination_matrix">[docs]</a>
+    <span class="k">def</span> <span class="nf">get_Dirichlet_recombination_matrix</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">axis</span><span class="o">=-</span><span class="mi">1</span><span class="p">):</span>
+<span class="w">        </span><span class="sd">&quot;&quot;&quot;</span>
+<span class="sd">        Get Dirichlet recombination matrix along axis. Not that it only makes sense in directions discretized with variations of Chebychev bases.</span>
+
+<span class="sd">        Args:</span>
+<span class="sd">            axis (int): Axis you discretized with Chebychev</span>
+
+<span class="sd">        Returns:</span>
+<span class="sd">            sparse matrix</span>
+<span class="sd">        &quot;&quot;&quot;</span>
+        <span class="n">sp</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">sparse_lib</span>
+        <span class="n">ndim</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">axes</span><span class="p">)</span>
+
+        <span class="k">if</span> <span class="n">ndim</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
+            <span class="n">C</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">get_Dirichlet_recombination_matrix</span><span class="p">()</span>
+        <span class="k">elif</span> <span class="n">ndim</span> <span class="o">==</span> <span class="mi">2</span><span class="p">:</span>
+            <span class="n">axis2</span> <span class="o">=</span> <span class="p">(</span><span class="n">axis</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> <span class="o">%</span> <span class="n">ndim</span>
+            <span class="n">C1D</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">axes</span><span class="p">[</span><span class="n">axis</span><span class="p">]</span><span class="o">.</span><span class="n">get_Dirichlet_recombination_matrix</span><span class="p">()</span>
+
+            <span class="n">I1D</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">axes</span><span class="p">[</span><span class="n">axis2</span><span class="p">]</span><span class="o">.</span><span class="n">get_Id</span><span class="p">()</span>
+
+            <span class="n">mats</span> <span class="o">=</span> <span class="p">[</span><span class="kc">None</span><span class="p">]</span> <span class="o">*</span> <span class="n">ndim</span>
+            <span class="n">mats</span><span class="p">[</span><span class="n">axis</span><span class="p">]</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">get_local_slice_of_1D_matrix</span><span class="p">(</span><span class="n">C1D</span><span class="p">,</span> <span class="n">axis</span><span class="p">)</span>
+            <span class="n">mats</span><span class="p">[</span><span class="n">axis2</span><span class="p">]</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">get_local_slice_of_1D_matrix</span><span class="p">(</span><span class="n">I1D</span><span class="p">,</span> <span class="n">axis2</span><span class="p">)</span>
+
+            <span class="n">C</span> <span class="o">=</span> <span class="n">sp</span><span class="o">.</span><span class="n">kron</span><span class="p">(</span><span class="o">*</span><span class="n">mats</span><span class="p">)</span>
+        <span class="k">else</span><span class="p">:</span>
+            <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span><span class="sa">f</span><span class="s1">&#39;Basis change matrix not implemented for </span><span class="si">{</span><span class="n">ndim</span><span class="si">}</span><span class="s1"> dimension!&#39;</span><span class="p">)</span>
+
+        <span class="k">return</span> <span class="n">C</span></div>
+
+
+<div class="viewcode-block" id="SpectralHelper.get_basis_change_matrix">
+<a class="viewcode-back" href="../../pySDC/helpers.spectral_helper.html#helpers.spectral_helper.SpectralHelper.get_basis_change_matrix">[docs]</a>
+    <span class="k">def</span> <span class="nf">get_basis_change_matrix</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">axes</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
+<span class="w">        </span><span class="sd">&quot;&quot;&quot;</span>
+<span class="sd">        Some spectral bases do a change between bases while differentiating. This method returns matrices that changes the basis to whatever you want.</span>
+<span class="sd">        Refer to the methods of the same name of the 1D bases to learn what parameters you need to pass here as `kwargs`.</span>
+
+<span class="sd">        Args:</span>
+<span class="sd">            axes (tuple): Axes along which to change basis.</span>
+
+<span class="sd">        Returns:</span>
+<span class="sd">            sparse basis change matrix</span>
+<span class="sd">        &quot;&quot;&quot;</span>
+        <span class="n">axes</span> <span class="o">=</span> <span class="nb">tuple</span><span class="p">(</span><span class="o">-</span><span class="n">i</span> <span class="o">-</span> <span class="mi">1</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">ndim</span><span class="p">))</span> <span class="k">if</span> <span class="n">axes</span> <span class="ow">is</span> <span class="kc">None</span> <span class="k">else</span> <span class="n">axes</span>
+
+        <span class="n">sp</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">sparse_lib</span>
+        <span class="n">ndim</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">axes</span><span class="p">)</span>
+
+        <span class="k">if</span> <span class="n">ndim</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
+            <span class="n">C</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">get_basis_change_matrix</span><span class="p">(</span><span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
+        <span class="k">elif</span> <span class="n">ndim</span> <span class="o">==</span> <span class="mi">2</span><span class="p">:</span>
+            <span class="k">for</span> <span class="n">axis</span> <span class="ow">in</span> <span class="n">axes</span><span class="p">:</span>
+                <span class="n">axis2</span> <span class="o">=</span> <span class="p">(</span><span class="n">axis</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> <span class="o">%</span> <span class="n">ndim</span>
+                <span class="n">C1D</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">axes</span><span class="p">[</span><span class="n">axis</span><span class="p">]</span><span class="o">.</span><span class="n">get_basis_change_matrix</span><span class="p">(</span><span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
+
+                <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">axes</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mi">1</span><span class="p">:</span>
+                    <span class="n">I1D</span> <span class="o">=</span> <span class="n">sp</span><span class="o">.</span><span class="n">eye</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">axes</span><span class="p">[</span><span class="n">axis2</span><span class="p">]</span><span class="o">.</span><span class="n">N</span><span class="p">)</span>
+                <span class="k">else</span><span class="p">:</span>
+                    <span class="n">I1D</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">axes</span><span class="p">[</span><span class="n">axis2</span><span class="p">]</span><span class="o">.</span><span class="n">get_Id</span><span class="p">()</span>
+
+                <span class="n">mats</span> <span class="o">=</span> <span class="p">[</span><span class="kc">None</span><span class="p">]</span> <span class="o">*</span> <span class="n">ndim</span>
+                <span class="n">mats</span><span class="p">[</span><span class="n">axis</span><span class="p">]</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">get_local_slice_of_1D_matrix</span><span class="p">(</span><span class="n">C1D</span><span class="p">,</span> <span class="n">axis</span><span class="p">)</span>
+                <span class="n">mats</span><span class="p">[</span><span class="n">axis2</span><span class="p">]</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">get_local_slice_of_1D_matrix</span><span class="p">(</span><span class="n">I1D</span><span class="p">,</span> <span class="n">axis2</span><span class="p">)</span>
+
+                <span class="k">if</span> <span class="n">axis</span> <span class="o">==</span> <span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">]:</span>
+                    <span class="n">C</span> <span class="o">=</span> <span class="n">sp</span><span class="o">.</span><span class="n">kron</span><span class="p">(</span><span class="o">*</span><span class="n">mats</span><span class="p">)</span>
+                <span class="k">else</span><span class="p">:</span>
+                    <span class="n">C</span> <span class="o">=</span> <span class="n">C</span> <span class="o">@</span> <span class="n">sp</span><span class="o">.</span><span class="n">kron</span><span class="p">(</span><span class="o">*</span><span class="n">mats</span><span class="p">)</span>
+        <span class="k">else</span><span class="p">:</span>
+            <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span><span class="sa">f</span><span class="s1">&#39;Basis change matrix not implemented for </span><span class="si">{</span><span class="n">ndim</span><span class="si">}</span><span class="s1"> dimension!&#39;</span><span class="p">)</span>
+
+        <span class="k">return</span> <span class="n">C</span></div>
+</div>
+
+</pre></div>
+
+            <div class="clearer"></div>
+          </div>
+        </div>
+      </div>
+      <div class="sphinxsidebar" role="navigation" aria-label="Main">
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+          <a href="../../genindex.html" title="General Index"
+             >index</a></li>
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+          <a href="../../py-modindex.html" title="Python Module Index"
+             >modules</a> |</li>
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+          <li class="nav-item nav-item-1"><a href="../index.html" >Module code</a> &#187;</li>
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+      Created using <a href="https://www.sphinx-doc.org/">Sphinx</a> 8.0.2.
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diff --git a/_modules/implementations/convergence_controller_classes/interpolate_between_restarts.html b/_modules/implementations/convergence_controller_classes/interpolate_between_restarts.html
index 92d1dbdce6..37316d747b 100644
--- a/_modules/implementations/convergence_controller_classes/interpolate_between_restarts.html
+++ b/_modules/implementations/convergence_controller_classes/interpolate_between_restarts.html
@@ -105,7 +105,7 @@ <h1>Source code for implementations.convergence_controller_classes.interpolate_b
 
                 <span class="k">for</span> <span class="n">m</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">level</span><span class="o">.</span><span class="n">u</span><span class="p">)):</span>
                     <span class="n">level</span><span class="o">.</span><span class="n">u</span><span class="p">[</span><span class="n">m</span><span class="p">][:]</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">status</span><span class="o">.</span><span class="n">u_inter</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">m</span><span class="p">]</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="n">level</span><span class="o">.</span><span class="n">prob</span><span class="o">.</span><span class="n">init</span><span class="p">[</span><span class="mi">0</span><span class="p">])[:]</span>
-                    <span class="n">level</span><span class="o">.</span><span class="n">f</span><span class="p">[</span><span class="n">m</span><span class="p">][:]</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">status</span><span class="o">.</span><span class="n">f_inter</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">m</span><span class="p">]</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="n">level</span><span class="o">.</span><span class="n">prob</span><span class="o">.</span><span class="n">init</span><span class="p">[</span><span class="mi">0</span><span class="p">])[:]</span>
+                    <span class="n">level</span><span class="o">.</span><span class="n">f</span><span class="p">[</span><span class="n">m</span><span class="p">][:]</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">status</span><span class="o">.</span><span class="n">f_inter</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">m</span><span class="p">]</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="n">level</span><span class="o">.</span><span class="n">f</span><span class="p">[</span><span class="n">m</span><span class="p">]</span><span class="o">.</span><span class="n">shape</span><span class="p">)[:]</span>
 
             <span class="c1"># reset the status variables</span>
             <span class="bp">self</span><span class="o">.</span><span class="n">status</span><span class="o">.</span><span class="n">perform_interpolation</span> <span class="o">=</span> <span class="kc">False</span>
diff --git a/_modules/implementations/problem_classes/Burgers.html b/_modules/implementations/problem_classes/Burgers.html
new file mode 100644
index 0000000000..fb628c58b1
--- /dev/null
+++ b/_modules/implementations/problem_classes/Burgers.html
@@ -0,0 +1,451 @@
+<!DOCTYPE html>
+
+<html lang="en" data-content_root="../../../">
+  <head>
+    <meta charset="utf-8" />
+    <meta name="viewport" content="width=device-width, initial-scale=1.0" />
+    <title>implementations.problem_classes.Burgers &#8212; pySDC 5.5.1 documentation</title>
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+             >modules</a> |</li>
+        <li class="nav-item nav-item-0"><a href="../../../index.html">pySDC 5.5.1 documentation</a> &#187;</li>
+          <li class="nav-item nav-item-1"><a href="../../index.html" accesskey="U">Module code</a> &#187;</li>
+        <li class="nav-item nav-item-this"><a href="">implementations.problem_classes.Burgers</a></li> 
+      </ul>
+    </div>  
+
+    <div class="document">
+      <div class="documentwrapper">
+        <div class="bodywrapper">
+          <div class="body" role="main">
+            
+  <h1>Source code for implementations.problem_classes.Burgers</h1><div class="highlight"><pre>
+<span></span><span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
+
+<span class="kn">from</span> <span class="nn">pySDC.implementations.datatype_classes.mesh</span> <span class="kn">import</span> <span class="n">mesh</span><span class="p">,</span> <span class="n">imex_mesh</span>
+<span class="kn">from</span> <span class="nn">pySDC.implementations.problem_classes.generic_spectral</span> <span class="kn">import</span> <span class="n">GenericSpectralLinear</span>
+
+
+<div class="viewcode-block" id="Burgers1D">
+<a class="viewcode-back" href="../../../pySDC/implementations.problem_classes.Burgers.html#implementations.problem_classes.Burgers.Burgers1D">[docs]</a>
+<span class="k">class</span> <span class="nc">Burgers1D</span><span class="p">(</span><span class="n">GenericSpectralLinear</span><span class="p">):</span>
+<span class="w">    </span><span class="sd">&quot;&quot;&quot;</span>
+<span class="sd">    See https://en.wikipedia.org/wiki/Burgers&#39;_equation for the equation that is solved.</span>
+<span class="sd">    Discretization is done with a Chebychev method, which requires a first order derivative formulation.</span>
+<span class="sd">    Feel free to do a more efficient implementation using an ultraspherical method to avoid the first order business.</span>
+
+<span class="sd">    Parameters:</span>
+<span class="sd">       N (int): Spatial resolution</span>
+<span class="sd">       epsilon (float): viscosity</span>
+<span class="sd">       BCl (float): Value at left boundary</span>
+<span class="sd">       BCr (float): Value at right boundary</span>
+<span class="sd">       f (int): Frequency of the initial conditions</span>
+<span class="sd">       mode (str): &#39;T2U&#39; or &#39;T2T&#39;. Use &#39;T2U&#39; to get sparse differentiation matrices</span>
+<span class="sd">    &quot;&quot;&quot;</span>
+
+    <span class="n">dtype_u</span> <span class="o">=</span> <span class="n">mesh</span>
+    <span class="n">dtype_f</span> <span class="o">=</span> <span class="n">imex_mesh</span>
+
+    <span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">N</span><span class="o">=</span><span class="mi">64</span><span class="p">,</span> <span class="n">epsilon</span><span class="o">=</span><span class="mf">0.1</span><span class="p">,</span> <span class="n">BCl</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">BCr</span><span class="o">=-</span><span class="mi">1</span><span class="p">,</span> <span class="n">f</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span> <span class="n">mode</span><span class="o">=</span><span class="s1">&#39;T2U&#39;</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
+<span class="w">        </span><span class="sd">&quot;&quot;&quot;</span>
+<span class="sd">        Constructor. `kwargs` are forwarded to parent class constructor.</span>
+
+<span class="sd">        Args:</span>
+<span class="sd">            N (int): Spatial resolution</span>
+<span class="sd">            epsilon (float): viscosity</span>
+<span class="sd">            BCl (float): Value at left boundary</span>
+<span class="sd">            BCr (float): Value at right boundary</span>
+<span class="sd">            f (int): Frequency of the initial conditions</span>
+<span class="sd">            mode (str): &#39;T2U&#39; or &#39;T2T&#39;. Use &#39;T2U&#39; to get sparse differentiation matrices</span>
+<span class="sd">        &quot;&quot;&quot;</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">_makeAttributeAndRegister</span><span class="p">(</span><span class="s1">&#39;N&#39;</span><span class="p">,</span> <span class="s1">&#39;epsilon&#39;</span><span class="p">,</span> <span class="s1">&#39;BCl&#39;</span><span class="p">,</span> <span class="s1">&#39;BCr&#39;</span><span class="p">,</span> <span class="s1">&#39;f&#39;</span><span class="p">,</span> <span class="s1">&#39;mode&#39;</span><span class="p">,</span> <span class="n">localVars</span><span class="o">=</span><span class="nb">locals</span><span class="p">(),</span> <span class="n">readOnly</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
+
+        <span class="n">bases</span> <span class="o">=</span> <span class="p">[{</span><span class="s1">&#39;base&#39;</span><span class="p">:</span> <span class="s1">&#39;cheby&#39;</span><span class="p">,</span> <span class="s1">&#39;N&#39;</span><span class="p">:</span> <span class="n">N</span><span class="p">}]</span>
+        <span class="n">components</span> <span class="o">=</span> <span class="p">[</span><span class="s1">&#39;u&#39;</span><span class="p">,</span> <span class="s1">&#39;ux&#39;</span><span class="p">]</span>
+
+        <span class="nb">super</span><span class="p">()</span><span class="o">.</span><span class="fm">__init__</span><span class="p">(</span><span class="n">bases</span><span class="o">=</span><span class="n">bases</span><span class="p">,</span> <span class="n">components</span><span class="o">=</span><span class="n">components</span><span class="p">,</span> <span class="n">spectral_space</span><span class="o">=</span><span class="kc">False</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
+
+        <span class="bp">self</span><span class="o">.</span><span class="n">x</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">get_grid</span><span class="p">()[</span><span class="mi">0</span><span class="p">]</span>
+
+        <span class="c1"># prepare matrices</span>
+        <span class="n">Dx</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">get_differentiation_matrix</span><span class="p">(</span><span class="n">axes</span><span class="o">=</span><span class="p">(</span><span class="mi">0</span><span class="p">,))</span>
+        <span class="n">I</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">get_Id</span><span class="p">()</span>
+
+        <span class="n">T2U</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">get_basis_change_matrix</span><span class="p">(</span><span class="n">conv</span><span class="o">=</span><span class="n">mode</span><span class="p">)</span>
+
+        <span class="bp">self</span><span class="o">.</span><span class="n">Dx</span> <span class="o">=</span> <span class="n">Dx</span>
+
+        <span class="c1"># construct linear operator</span>
+        <span class="n">L_lhs</span> <span class="o">=</span> <span class="p">{</span><span class="s1">&#39;u&#39;</span><span class="p">:</span> <span class="p">{</span><span class="s1">&#39;ux&#39;</span><span class="p">:</span> <span class="o">-</span><span class="n">epsilon</span> <span class="o">*</span> <span class="p">(</span><span class="n">T2U</span> <span class="o">@</span> <span class="n">Dx</span><span class="p">)},</span> <span class="s1">&#39;ux&#39;</span><span class="p">:</span> <span class="p">{</span><span class="s1">&#39;u&#39;</span><span class="p">:</span> <span class="o">-</span><span class="n">T2U</span> <span class="o">@</span> <span class="n">Dx</span><span class="p">,</span> <span class="s1">&#39;ux&#39;</span><span class="p">:</span> <span class="n">T2U</span> <span class="o">@</span> <span class="n">I</span><span class="p">}}</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">setup_L</span><span class="p">(</span><span class="n">L_lhs</span><span class="p">)</span>
+
+        <span class="c1"># construct mass matrix</span>
+        <span class="n">M_lhs</span> <span class="o">=</span> <span class="p">{</span><span class="s1">&#39;u&#39;</span><span class="p">:</span> <span class="p">{</span><span class="s1">&#39;u&#39;</span><span class="p">:</span> <span class="n">T2U</span> <span class="o">@</span> <span class="n">I</span><span class="p">}}</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">setup_M</span><span class="p">(</span><span class="n">M_lhs</span><span class="p">)</span>
+
+        <span class="c1"># boundary conditions</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">add_BC</span><span class="p">(</span><span class="n">component</span><span class="o">=</span><span class="s1">&#39;u&#39;</span><span class="p">,</span> <span class="n">equation</span><span class="o">=</span><span class="s1">&#39;u&#39;</span><span class="p">,</span> <span class="n">axis</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span> <span class="n">x</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">v</span><span class="o">=</span><span class="n">BCr</span><span class="p">,</span> <span class="n">kind</span><span class="o">=</span><span class="s1">&#39;Dirichlet&#39;</span><span class="p">)</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">add_BC</span><span class="p">(</span><span class="n">component</span><span class="o">=</span><span class="s1">&#39;u&#39;</span><span class="p">,</span> <span class="n">equation</span><span class="o">=</span><span class="s1">&#39;ux&#39;</span><span class="p">,</span> <span class="n">axis</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span> <span class="n">x</span><span class="o">=-</span><span class="mi">1</span><span class="p">,</span> <span class="n">v</span><span class="o">=</span><span class="n">BCl</span><span class="p">,</span> <span class="n">kind</span><span class="o">=</span><span class="s1">&#39;Dirichlet&#39;</span><span class="p">)</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">setup_BCs</span><span class="p">()</span>
+
+<div class="viewcode-block" id="Burgers1D.u_exact">
+<a class="viewcode-back" href="../../../pySDC/implementations.problem_classes.Burgers.html#implementations.problem_classes.Burgers.Burgers1D.u_exact">[docs]</a>
+    <span class="k">def</span> <span class="nf">u_exact</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">t</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
+        <span class="n">me</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">u_init</span>
+
+        <span class="c1"># x = (self.x + 1) / 2</span>
+        <span class="c1"># g = 4 * (1 + np.exp(-(4 * x + t)/self.epsilon/32))</span>
+        <span class="c1"># g_x = 4 * np.exp(-(4 * x + t)/self.epsilon/32) * (-4/self.epsilon/32)</span>
+
+        <span class="c1"># me[0] = 3./4. - 1./g</span>
+        <span class="c1"># me[1] = 1/g**2 * g_x</span>
+
+        <span class="c1"># return me</span>
+
+        <span class="k">if</span> <span class="n">t</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
+            <span class="n">me</span><span class="p">[</span><span class="bp">self</span><span class="o">.</span><span class="n">index</span><span class="p">(</span><span class="s1">&#39;u&#39;</span><span class="p">)][:]</span> <span class="o">=</span> <span class="p">((</span><span class="bp">self</span><span class="o">.</span><span class="n">BCr</span> <span class="o">+</span> <span class="bp">self</span><span class="o">.</span><span class="n">BCl</span><span class="p">)</span> <span class="o">/</span> <span class="mi">2</span> <span class="o">+</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">BCr</span> <span class="o">-</span> <span class="bp">self</span><span class="o">.</span><span class="n">BCl</span><span class="p">)</span> <span class="o">/</span> <span class="mi">2</span> <span class="o">*</span> <span class="bp">self</span><span class="o">.</span><span class="n">x</span><span class="p">)</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">cos</span><span class="p">(</span>
+                <span class="bp">self</span><span class="o">.</span><span class="n">x</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">pi</span> <span class="o">*</span> <span class="bp">self</span><span class="o">.</span><span class="n">f</span>
+            <span class="p">)</span>
+            <span class="n">me</span><span class="p">[</span><span class="bp">self</span><span class="o">.</span><span class="n">index</span><span class="p">(</span><span class="s1">&#39;ux&#39;</span><span class="p">)][:]</span> <span class="o">=</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">BCr</span> <span class="o">-</span> <span class="bp">self</span><span class="o">.</span><span class="n">BCl</span><span class="p">)</span> <span class="o">/</span> <span class="mi">2</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">cos</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">x</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">pi</span> <span class="o">*</span> <span class="bp">self</span><span class="o">.</span><span class="n">f</span><span class="p">)</span> <span class="o">+</span> <span class="p">(</span>
+                <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">BCr</span> <span class="o">+</span> <span class="bp">self</span><span class="o">.</span><span class="n">BCl</span><span class="p">)</span> <span class="o">/</span> <span class="mi">2</span> <span class="o">+</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">BCr</span> <span class="o">-</span> <span class="bp">self</span><span class="o">.</span><span class="n">BCl</span><span class="p">)</span> <span class="o">/</span> <span class="mi">2</span> <span class="o">*</span> <span class="bp">self</span><span class="o">.</span><span class="n">x</span>
+            <span class="p">)</span> <span class="o">*</span> <span class="bp">self</span><span class="o">.</span><span class="n">f</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">pi</span> <span class="o">*</span> <span class="o">-</span><span class="n">np</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">x</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">pi</span> <span class="o">*</span> <span class="bp">self</span><span class="o">.</span><span class="n">f</span><span class="p">)</span>
+        <span class="k">elif</span> <span class="n">t</span> <span class="o">==</span> <span class="n">np</span><span class="o">.</span><span class="n">inf</span> <span class="ow">and</span> <span class="bp">self</span><span class="o">.</span><span class="n">f</span> <span class="o">==</span> <span class="mi">0</span> <span class="ow">and</span> <span class="bp">self</span><span class="o">.</span><span class="n">BCl</span> <span class="o">==</span> <span class="o">-</span><span class="bp">self</span><span class="o">.</span><span class="n">BCr</span><span class="p">:</span>
+            <span class="n">me</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">BCl</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">exp</span><span class="p">((</span><span class="bp">self</span><span class="o">.</span><span class="n">BCr</span> <span class="o">-</span> <span class="bp">self</span><span class="o">.</span><span class="n">BCl</span><span class="p">)</span> <span class="o">/</span> <span class="p">(</span><span class="mi">2</span> <span class="o">*</span> <span class="bp">self</span><span class="o">.</span><span class="n">epsilon</span><span class="p">)</span> <span class="o">*</span> <span class="bp">self</span><span class="o">.</span><span class="n">x</span><span class="p">)</span> <span class="o">+</span> <span class="bp">self</span><span class="o">.</span><span class="n">BCr</span><span class="p">)</span> <span class="o">/</span> <span class="p">(</span>
+                <span class="n">np</span><span class="o">.</span><span class="n">exp</span><span class="p">((</span><span class="bp">self</span><span class="o">.</span><span class="n">BCr</span> <span class="o">-</span> <span class="bp">self</span><span class="o">.</span><span class="n">BCl</span><span class="p">)</span> <span class="o">/</span> <span class="p">(</span><span class="mi">2</span> <span class="o">*</span> <span class="bp">self</span><span class="o">.</span><span class="n">epsilon</span><span class="p">)</span> <span class="o">*</span> <span class="bp">self</span><span class="o">.</span><span class="n">x</span><span class="p">)</span> <span class="o">+</span> <span class="mi">1</span>
+            <span class="p">)</span>
+        <span class="k">else</span><span class="p">:</span>
+            <span class="k">raise</span> <span class="ne">NotImplementedError</span>
+
+        <span class="k">return</span> <span class="n">me</span></div>
+
+
+<div class="viewcode-block" id="Burgers1D.eval_f">
+<a class="viewcode-back" href="../../../pySDC/implementations.problem_classes.Burgers.html#implementations.problem_classes.Burgers.Burgers1D.eval_f">[docs]</a>
+    <span class="k">def</span> <span class="nf">eval_f</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
+        <span class="n">f</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">f_init</span>
+        <span class="n">iu</span><span class="p">,</span> <span class="n">iux</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">index</span><span class="p">(</span><span class="s1">&#39;u&#39;</span><span class="p">),</span> <span class="bp">self</span><span class="o">.</span><span class="n">index</span><span class="p">(</span><span class="s1">&#39;ux&#39;</span><span class="p">)</span>
+
+        <span class="n">u_hat</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">transform</span><span class="p">(</span><span class="n">u</span><span class="p">)</span>
+
+        <span class="n">Dx_u_hat</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">u_init_forward</span>
+        <span class="n">Dx_u_hat</span><span class="p">[</span><span class="n">iux</span><span class="p">]</span> <span class="o">=</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">Dx</span> <span class="o">@</span> <span class="n">u_hat</span><span class="p">[</span><span class="n">iux</span><span class="p">]</span><span class="o">.</span><span class="n">flatten</span><span class="p">())</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="n">u_hat</span><span class="p">[</span><span class="n">iu</span><span class="p">]</span><span class="o">.</span><span class="n">shape</span><span class="p">)</span>
+
+        <span class="n">f</span><span class="o">.</span><span class="n">impl</span><span class="p">[</span><span class="n">iu</span><span class="p">]</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">epsilon</span> <span class="o">*</span> <span class="bp">self</span><span class="o">.</span><span class="n">itransform</span><span class="p">(</span><span class="n">Dx_u_hat</span><span class="p">)[</span><span class="n">iux</span><span class="p">]</span><span class="o">.</span><span class="n">real</span>
+        <span class="n">f</span><span class="o">.</span><span class="n">expl</span><span class="p">[</span><span class="n">iu</span><span class="p">]</span> <span class="o">=</span> <span class="o">-</span><span class="n">u</span><span class="p">[</span><span class="n">iu</span><span class="p">]</span> <span class="o">*</span> <span class="n">u</span><span class="p">[</span><span class="n">iux</span><span class="p">]</span>
+        <span class="k">return</span> <span class="n">f</span></div>
+
+
+<div class="viewcode-block" id="Burgers1D.get_fig">
+<a class="viewcode-back" href="../../../pySDC/implementations.problem_classes.Burgers.html#implementations.problem_classes.Burgers.Burgers1D.get_fig">[docs]</a>
+    <span class="k">def</span> <span class="nf">get_fig</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>  <span class="c1"># pragma: no cover</span>
+<span class="w">        </span><span class="sd">&quot;&quot;&quot;</span>
+<span class="sd">        Get a figure suitable to plot the solution of this problem.</span>
+
+<span class="sd">        Returns</span>
+<span class="sd">        -------</span>
+<span class="sd">        self.fig : matplotlib.pyplot.figure.Figure</span>
+<span class="sd">        &quot;&quot;&quot;</span>
+        <span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
+
+        <span class="n">plt</span><span class="o">.</span><span class="n">rcParams</span><span class="p">[</span><span class="s1">&#39;figure.constrained_layout.use&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="kc">True</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">fig</span><span class="p">,</span> <span class="n">axs</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">()</span>
+        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">fig</span></div>
+
+
+<div class="viewcode-block" id="Burgers1D.plot">
+<a class="viewcode-back" href="../../../pySDC/implementations.problem_classes.Burgers.html#implementations.problem_classes.Burgers.Burgers1D.plot">[docs]</a>
+    <span class="k">def</span> <span class="nf">plot</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">t</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">fig</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">comp</span><span class="o">=</span><span class="s1">&#39;u&#39;</span><span class="p">):</span>  <span class="c1"># pragma: no cover</span>
+<span class="w">        </span><span class="sa">r</span><span class="sd">&quot;&quot;&quot;</span>
+<span class="sd">        Plot the solution.</span>
+
+<span class="sd">        Parameters</span>
+<span class="sd">        ----------</span>
+<span class="sd">        u : dtype_u</span>
+<span class="sd">            Solution to be plotted</span>
+<span class="sd">        t : float</span>
+<span class="sd">            Time to display at the top of the figure</span>
+<span class="sd">        fig : matplotlib.pyplot.figure.Figure, optional</span>
+<span class="sd">            Figure with the same structure as a figure generated by `self.get_fig`. If none is supplied, a new figure will be generated.</span>
+
+<span class="sd">        Returns</span>
+<span class="sd">        -------</span>
+<span class="sd">        None</span>
+<span class="sd">        &quot;&quot;&quot;</span>
+        <span class="n">fig</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">get_fig</span><span class="p">()</span> <span class="k">if</span> <span class="n">fig</span> <span class="ow">is</span> <span class="kc">None</span> <span class="k">else</span> <span class="n">fig</span>
+        <span class="n">ax</span> <span class="o">=</span> <span class="n">fig</span><span class="o">.</span><span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
+
+        <span class="n">ax</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">x</span><span class="p">,</span> <span class="n">u</span><span class="p">[</span><span class="bp">self</span><span class="o">.</span><span class="n">index</span><span class="p">(</span><span class="n">comp</span><span class="p">)])</span>
+
+        <span class="k">if</span> <span class="n">t</span> <span class="ow">is</span> <span class="ow">not</span> <span class="kc">None</span><span class="p">:</span>
+            <span class="n">fig</span><span class="o">.</span><span class="n">suptitle</span><span class="p">(</span><span class="sa">f</span><span class="s1">&#39;t = </span><span class="si">{</span><span class="n">t</span><span class="si">:</span><span class="s1">.2e</span><span class="si">}</span><span class="s1">&#39;</span><span class="p">)</span>
+
+        <span class="n">ax</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="sa">r</span><span class="s1">&#39;$x$&#39;</span><span class="p">)</span>
+        <span class="n">ax</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="sa">r</span><span class="s1">&#39;$u$&#39;</span><span class="p">)</span></div>
+</div>
+
+
+
+<div class="viewcode-block" id="Burgers2D">
+<a class="viewcode-back" href="../../../pySDC/implementations.problem_classes.Burgers.html#implementations.problem_classes.Burgers.Burgers2D">[docs]</a>
+<span class="k">class</span> <span class="nc">Burgers2D</span><span class="p">(</span><span class="n">GenericSpectralLinear</span><span class="p">):</span>
+<span class="w">    </span><span class="sd">&quot;&quot;&quot;</span>
+<span class="sd">    See https://en.wikipedia.org/wiki/Burgers&#39;_equation for the equation that is solved.</span>
+<span class="sd">    This implementation is discretized with FFTs in x and Chebychev in z.</span>
+
+<span class="sd">    Parameters:</span>
+<span class="sd">       nx (int): Spatial resolution in x direction</span>
+<span class="sd">       nz (int): Spatial resolution in z direction</span>
+<span class="sd">       epsilon (float): viscosity</span>
+<span class="sd">       BCl (float): Value at left boundary</span>
+<span class="sd">       BCr (float): Value at right boundary</span>
+<span class="sd">       fux (int): Frequency of the initial conditions in x-direction</span>
+<span class="sd">       fuz (int): Frequency of the initial conditions in z-direction</span>
+<span class="sd">       mode (str): &#39;T2U&#39; or &#39;T2T&#39;. Use &#39;T2U&#39; to get sparse differentiation matrices</span>
+<span class="sd">    &quot;&quot;&quot;</span>
+
+    <span class="n">dtype_u</span> <span class="o">=</span> <span class="n">mesh</span>
+    <span class="n">dtype_f</span> <span class="o">=</span> <span class="n">imex_mesh</span>
+
+    <span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">nx</span><span class="o">=</span><span class="mi">64</span><span class="p">,</span> <span class="n">nz</span><span class="o">=</span><span class="mi">64</span><span class="p">,</span> <span class="n">epsilon</span><span class="o">=</span><span class="mf">0.1</span><span class="p">,</span> <span class="n">fux</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span> <span class="n">fuz</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">mode</span><span class="o">=</span><span class="s1">&#39;T2U&#39;</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
+<span class="w">        </span><span class="sd">&quot;&quot;&quot;</span>
+<span class="sd">         Constructor. `kwargs` are forwarded to parent class constructor.</span>
+
+<span class="sd">         Args:</span>
+<span class="sd">        nx (int): Spatial resolution in x direction</span>
+<span class="sd">        nz (int): Spatial resolution in z direction</span>
+<span class="sd">        epsilon (float): viscosity</span>
+<span class="sd">        fux (int): Frequency of the initial conditions in x-direction</span>
+<span class="sd">        fuz (int): Frequency of the initial conditions in z-direction</span>
+<span class="sd">        mode (str): &#39;T2U&#39; or &#39;T2T&#39;. Use &#39;T2U&#39; to get sparse differentiation matrices</span>
+<span class="sd">        &quot;&quot;&quot;</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">_makeAttributeAndRegister</span><span class="p">(</span><span class="s1">&#39;nx&#39;</span><span class="p">,</span> <span class="s1">&#39;nz&#39;</span><span class="p">,</span> <span class="s1">&#39;epsilon&#39;</span><span class="p">,</span> <span class="s1">&#39;fux&#39;</span><span class="p">,</span> <span class="s1">&#39;fuz&#39;</span><span class="p">,</span> <span class="s1">&#39;mode&#39;</span><span class="p">,</span> <span class="n">localVars</span><span class="o">=</span><span class="nb">locals</span><span class="p">(),</span> <span class="n">readOnly</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
+
+        <span class="n">bases</span> <span class="o">=</span> <span class="p">[</span>
+            <span class="p">{</span><span class="s1">&#39;base&#39;</span><span class="p">:</span> <span class="s1">&#39;fft&#39;</span><span class="p">,</span> <span class="s1">&#39;N&#39;</span><span class="p">:</span> <span class="n">nx</span><span class="p">},</span>
+            <span class="p">{</span><span class="s1">&#39;base&#39;</span><span class="p">:</span> <span class="s1">&#39;cheby&#39;</span><span class="p">,</span> <span class="s1">&#39;N&#39;</span><span class="p">:</span> <span class="n">nz</span><span class="p">},</span>
+        <span class="p">]</span>
+        <span class="n">components</span> <span class="o">=</span> <span class="p">[</span><span class="s1">&#39;u&#39;</span><span class="p">,</span> <span class="s1">&#39;v&#39;</span><span class="p">,</span> <span class="s1">&#39;ux&#39;</span><span class="p">,</span> <span class="s1">&#39;uz&#39;</span><span class="p">,</span> <span class="s1">&#39;vx&#39;</span><span class="p">,</span> <span class="s1">&#39;vz&#39;</span><span class="p">]</span>
+        <span class="nb">super</span><span class="p">()</span><span class="o">.</span><span class="fm">__init__</span><span class="p">(</span><span class="n">bases</span><span class="o">=</span><span class="n">bases</span><span class="p">,</span> <span class="n">components</span><span class="o">=</span><span class="n">components</span><span class="p">,</span> <span class="n">spectral_space</span><span class="o">=</span><span class="kc">False</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
+
+        <span class="bp">self</span><span class="o">.</span><span class="n">Z</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">X</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">get_grid</span><span class="p">()</span>
+
+        <span class="c1"># prepare matrices</span>
+        <span class="n">Dx</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">get_differentiation_matrix</span><span class="p">(</span><span class="n">axes</span><span class="o">=</span><span class="p">(</span><span class="mi">0</span><span class="p">,))</span>
+        <span class="n">Dz</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">get_differentiation_matrix</span><span class="p">(</span><span class="n">axes</span><span class="o">=</span><span class="p">(</span><span class="mi">1</span><span class="p">,))</span>
+        <span class="n">I</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">get_Id</span><span class="p">()</span>
+
+        <span class="n">T2U</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">get_basis_change_matrix</span><span class="p">(</span><span class="n">axes</span><span class="o">=</span><span class="p">(</span><span class="mi">1</span><span class="p">,),</span> <span class="n">conv</span><span class="o">=</span><span class="n">mode</span><span class="p">)</span>
+
+        <span class="bp">self</span><span class="o">.</span><span class="n">Dx</span> <span class="o">=</span> <span class="n">Dx</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">Dz</span> <span class="o">=</span> <span class="n">Dz</span>
+
+        <span class="c1"># construct linear operator</span>
+        <span class="n">L_lhs</span> <span class="o">=</span> <span class="p">{</span>
+            <span class="s1">&#39;u&#39;</span><span class="p">:</span> <span class="p">{</span><span class="s1">&#39;ux&#39;</span><span class="p">:</span> <span class="o">-</span><span class="n">epsilon</span> <span class="o">*</span> <span class="n">T2U</span> <span class="o">@</span> <span class="n">Dx</span><span class="p">,</span> <span class="s1">&#39;uz&#39;</span><span class="p">:</span> <span class="o">-</span><span class="n">epsilon</span> <span class="o">*</span> <span class="n">T2U</span> <span class="o">@</span> <span class="n">Dz</span><span class="p">},</span>
+            <span class="s1">&#39;v&#39;</span><span class="p">:</span> <span class="p">{</span><span class="s1">&#39;vx&#39;</span><span class="p">:</span> <span class="o">-</span><span class="n">epsilon</span> <span class="o">*</span> <span class="n">T2U</span> <span class="o">@</span> <span class="n">Dx</span><span class="p">,</span> <span class="s1">&#39;vz&#39;</span><span class="p">:</span> <span class="o">-</span><span class="n">epsilon</span> <span class="o">*</span> <span class="n">T2U</span> <span class="o">@</span> <span class="n">Dz</span><span class="p">},</span>
+            <span class="s1">&#39;ux&#39;</span><span class="p">:</span> <span class="p">{</span><span class="s1">&#39;u&#39;</span><span class="p">:</span> <span class="o">-</span><span class="n">T2U</span> <span class="o">@</span> <span class="n">Dx</span><span class="p">,</span> <span class="s1">&#39;ux&#39;</span><span class="p">:</span> <span class="n">T2U</span> <span class="o">@</span> <span class="n">I</span><span class="p">},</span>
+            <span class="s1">&#39;uz&#39;</span><span class="p">:</span> <span class="p">{</span><span class="s1">&#39;u&#39;</span><span class="p">:</span> <span class="o">-</span><span class="n">T2U</span> <span class="o">@</span> <span class="n">Dz</span><span class="p">,</span> <span class="s1">&#39;uz&#39;</span><span class="p">:</span> <span class="n">T2U</span> <span class="o">@</span> <span class="n">I</span><span class="p">},</span>
+            <span class="s1">&#39;vx&#39;</span><span class="p">:</span> <span class="p">{</span><span class="s1">&#39;v&#39;</span><span class="p">:</span> <span class="o">-</span><span class="n">T2U</span> <span class="o">@</span> <span class="n">Dx</span><span class="p">,</span> <span class="s1">&#39;vx&#39;</span><span class="p">:</span> <span class="n">T2U</span> <span class="o">@</span> <span class="n">I</span><span class="p">},</span>
+            <span class="s1">&#39;vz&#39;</span><span class="p">:</span> <span class="p">{</span><span class="s1">&#39;v&#39;</span><span class="p">:</span> <span class="o">-</span><span class="n">T2U</span> <span class="o">@</span> <span class="n">Dz</span><span class="p">,</span> <span class="s1">&#39;vz&#39;</span><span class="p">:</span> <span class="n">T2U</span> <span class="o">@</span> <span class="n">I</span><span class="p">},</span>
+        <span class="p">}</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">setup_L</span><span class="p">(</span><span class="n">L_lhs</span><span class="p">)</span>
+
+        <span class="c1"># construct mass matrix</span>
+        <span class="n">M_lhs</span> <span class="o">=</span> <span class="p">{</span>
+            <span class="s1">&#39;u&#39;</span><span class="p">:</span> <span class="p">{</span><span class="s1">&#39;u&#39;</span><span class="p">:</span> <span class="n">T2U</span> <span class="o">@</span> <span class="n">I</span><span class="p">},</span>
+            <span class="s1">&#39;v&#39;</span><span class="p">:</span> <span class="p">{</span><span class="s1">&#39;v&#39;</span><span class="p">:</span> <span class="n">T2U</span> <span class="o">@</span> <span class="n">I</span><span class="p">},</span>
+        <span class="p">}</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">setup_M</span><span class="p">(</span><span class="n">M_lhs</span><span class="p">)</span>
+
+        <span class="c1"># boundary conditions</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">BCtop</span> <span class="o">=</span> <span class="mi">1</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">BCbottom</span> <span class="o">=</span> <span class="o">-</span><span class="bp">self</span><span class="o">.</span><span class="n">BCtop</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">BCtopu</span> <span class="o">=</span> <span class="mi">0</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">add_BC</span><span class="p">(</span><span class="n">component</span><span class="o">=</span><span class="s1">&#39;v&#39;</span><span class="p">,</span> <span class="n">equation</span><span class="o">=</span><span class="s1">&#39;v&#39;</span><span class="p">,</span> <span class="n">axis</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">v</span><span class="o">=</span><span class="bp">self</span><span class="o">.</span><span class="n">BCtop</span><span class="p">,</span> <span class="n">x</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">kind</span><span class="o">=</span><span class="s1">&#39;Dirichlet&#39;</span><span class="p">)</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">add_BC</span><span class="p">(</span><span class="n">component</span><span class="o">=</span><span class="s1">&#39;v&#39;</span><span class="p">,</span> <span class="n">equation</span><span class="o">=</span><span class="s1">&#39;vz&#39;</span><span class="p">,</span> <span class="n">axis</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">v</span><span class="o">=</span><span class="bp">self</span><span class="o">.</span><span class="n">BCbottom</span><span class="p">,</span> <span class="n">x</span><span class="o">=-</span><span class="mi">1</span><span class="p">,</span> <span class="n">kind</span><span class="o">=</span><span class="s1">&#39;Dirichlet&#39;</span><span class="p">)</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">add_BC</span><span class="p">(</span><span class="n">component</span><span class="o">=</span><span class="s1">&#39;u&#39;</span><span class="p">,</span> <span class="n">equation</span><span class="o">=</span><span class="s1">&#39;uz&#39;</span><span class="p">,</span> <span class="n">axis</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">v</span><span class="o">=</span><span class="bp">self</span><span class="o">.</span><span class="n">BCtopu</span><span class="p">,</span> <span class="n">x</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">kind</span><span class="o">=</span><span class="s1">&#39;Dirichlet&#39;</span><span class="p">)</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">add_BC</span><span class="p">(</span><span class="n">component</span><span class="o">=</span><span class="s1">&#39;u&#39;</span><span class="p">,</span> <span class="n">equation</span><span class="o">=</span><span class="s1">&#39;u&#39;</span><span class="p">,</span> <span class="n">axis</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">v</span><span class="o">=</span><span class="bp">self</span><span class="o">.</span><span class="n">BCtopu</span><span class="p">,</span> <span class="n">x</span><span class="o">=-</span><span class="mi">1</span><span class="p">,</span> <span class="n">kind</span><span class="o">=</span><span class="s1">&#39;Dirichlet&#39;</span><span class="p">)</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">setup_BCs</span><span class="p">()</span>
+
+<div class="viewcode-block" id="Burgers2D.u_exact">
+<a class="viewcode-back" href="../../../pySDC/implementations.problem_classes.Burgers.html#implementations.problem_classes.Burgers.Burgers2D.u_exact">[docs]</a>
+    <span class="k">def</span> <span class="nf">u_exact</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">t</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="n">noise_level</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
+        <span class="n">me</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">u_init</span>
+
+        <span class="n">iu</span><span class="p">,</span> <span class="n">iv</span><span class="p">,</span> <span class="n">iux</span><span class="p">,</span> <span class="n">iuz</span><span class="p">,</span> <span class="n">ivx</span><span class="p">,</span> <span class="n">ivz</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">index</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">components</span><span class="p">)</span>
+        <span class="k">if</span> <span class="n">t</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
+            <span class="n">me</span><span class="p">[</span><span class="n">iu</span><span class="p">]</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">xp</span><span class="o">.</span><span class="n">cos</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">X</span> <span class="o">*</span> <span class="bp">self</span><span class="o">.</span><span class="n">fux</span><span class="p">)</span> <span class="o">*</span> <span class="bp">self</span><span class="o">.</span><span class="n">xp</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">Z</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">pi</span> <span class="o">*</span> <span class="bp">self</span><span class="o">.</span><span class="n">fuz</span><span class="p">)</span> <span class="o">+</span> <span class="bp">self</span><span class="o">.</span><span class="n">BCtopu</span>
+            <span class="n">me</span><span class="p">[</span><span class="n">iux</span><span class="p">]</span> <span class="o">=</span> <span class="o">-</span><span class="bp">self</span><span class="o">.</span><span class="n">xp</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">X</span> <span class="o">*</span> <span class="bp">self</span><span class="o">.</span><span class="n">fux</span><span class="p">)</span> <span class="o">*</span> <span class="bp">self</span><span class="o">.</span><span class="n">fux</span> <span class="o">*</span> <span class="bp">self</span><span class="o">.</span><span class="n">xp</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">Z</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">pi</span> <span class="o">*</span> <span class="bp">self</span><span class="o">.</span><span class="n">fuz</span><span class="p">)</span>
+            <span class="n">me</span><span class="p">[</span><span class="n">iuz</span><span class="p">]</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">xp</span><span class="o">.</span><span class="n">cos</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">X</span> <span class="o">*</span> <span class="bp">self</span><span class="o">.</span><span class="n">fux</span><span class="p">)</span> <span class="o">*</span> <span class="bp">self</span><span class="o">.</span><span class="n">xp</span><span class="o">.</span><span class="n">cos</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">Z</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">pi</span> <span class="o">*</span> <span class="bp">self</span><span class="o">.</span><span class="n">fuz</span><span class="p">)</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">pi</span> <span class="o">*</span> <span class="bp">self</span><span class="o">.</span><span class="n">fuz</span>
+
+            <span class="n">me</span><span class="p">[</span><span class="n">iv</span><span class="p">]</span> <span class="o">=</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">BCtop</span> <span class="o">+</span> <span class="bp">self</span><span class="o">.</span><span class="n">BCbottom</span><span class="p">)</span> <span class="o">/</span> <span class="mi">2</span> <span class="o">+</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">BCtop</span> <span class="o">-</span> <span class="bp">self</span><span class="o">.</span><span class="n">BCbottom</span><span class="p">)</span> <span class="o">/</span> <span class="mi">2</span> <span class="o">*</span> <span class="bp">self</span><span class="o">.</span><span class="n">Z</span>
+            <span class="n">me</span><span class="p">[</span><span class="n">ivz</span><span class="p">][:]</span> <span class="o">=</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">BCtop</span> <span class="o">-</span> <span class="bp">self</span><span class="o">.</span><span class="n">BCbottom</span><span class="p">)</span> <span class="o">/</span> <span class="mi">2</span>
+
+            <span class="c1"># add noise</span>
+            <span class="n">rng</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">xp</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">default_rng</span><span class="p">(</span><span class="n">seed</span><span class="o">=</span><span class="mi">99</span><span class="p">)</span>
+            <span class="n">me</span><span class="p">[</span><span class="n">iv</span><span class="p">]</span><span class="o">.</span><span class="n">real</span> <span class="o">+=</span> <span class="n">rng</span><span class="o">.</span><span class="n">normal</span><span class="p">(</span><span class="n">size</span><span class="o">=</span><span class="n">me</span><span class="p">[</span><span class="n">iv</span><span class="p">]</span><span class="o">.</span><span class="n">shape</span><span class="p">)</span> <span class="o">*</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">Z</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span> <span class="o">*</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">Z</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> <span class="o">*</span> <span class="n">noise_level</span>
+
+        <span class="k">else</span><span class="p">:</span>
+            <span class="k">raise</span> <span class="ne">NotImplementedError</span>
+
+        <span class="k">return</span> <span class="n">me</span></div>
+
+
+<div class="viewcode-block" id="Burgers2D.eval_f">
+<a class="viewcode-back" href="../../../pySDC/implementations.problem_classes.Burgers.html#implementations.problem_classes.Burgers.Burgers2D.eval_f">[docs]</a>
+    <span class="k">def</span> <span class="nf">eval_f</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
+        <span class="n">f</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">f_init</span>
+        <span class="n">iu</span><span class="p">,</span> <span class="n">iv</span><span class="p">,</span> <span class="n">iux</span><span class="p">,</span> <span class="n">iuz</span><span class="p">,</span> <span class="n">ivx</span><span class="p">,</span> <span class="n">ivz</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">index</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">components</span><span class="p">)</span>
+
+        <span class="n">u_hat</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">transform</span><span class="p">(</span><span class="n">u</span><span class="p">)</span>
+        <span class="n">f_hat</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">u_init_forward</span>
+        <span class="n">f_hat</span><span class="p">[</span><span class="n">iu</span><span class="p">]</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">epsilon</span> <span class="o">*</span> <span class="p">((</span><span class="bp">self</span><span class="o">.</span><span class="n">Dx</span> <span class="o">@</span> <span class="n">u_hat</span><span class="p">[</span><span class="n">iux</span><span class="p">]</span><span class="o">.</span><span class="n">flatten</span><span class="p">()</span> <span class="o">+</span> <span class="bp">self</span><span class="o">.</span><span class="n">Dz</span> <span class="o">@</span> <span class="n">u_hat</span><span class="p">[</span><span class="n">iuz</span><span class="p">]</span><span class="o">.</span><span class="n">flatten</span><span class="p">()))</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span>
+            <span class="n">u_hat</span><span class="p">[</span><span class="n">iux</span><span class="p">]</span><span class="o">.</span><span class="n">shape</span>
+        <span class="p">)</span>
+        <span class="n">f_hat</span><span class="p">[</span><span class="n">iv</span><span class="p">]</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">epsilon</span> <span class="o">*</span> <span class="p">((</span><span class="bp">self</span><span class="o">.</span><span class="n">Dx</span> <span class="o">@</span> <span class="n">u_hat</span><span class="p">[</span><span class="n">ivx</span><span class="p">]</span><span class="o">.</span><span class="n">flatten</span><span class="p">()</span> <span class="o">+</span> <span class="bp">self</span><span class="o">.</span><span class="n">Dz</span> <span class="o">@</span> <span class="n">u_hat</span><span class="p">[</span><span class="n">ivz</span><span class="p">]</span><span class="o">.</span><span class="n">flatten</span><span class="p">()))</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span>
+            <span class="n">u_hat</span><span class="p">[</span><span class="n">iux</span><span class="p">]</span><span class="o">.</span><span class="n">shape</span>
+        <span class="p">)</span>
+        <span class="n">f</span><span class="o">.</span><span class="n">impl</span><span class="p">[</span><span class="o">...</span><span class="p">]</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">itransform</span><span class="p">(</span><span class="n">f_hat</span><span class="p">)</span><span class="o">.</span><span class="n">real</span>
+
+        <span class="n">f</span><span class="o">.</span><span class="n">expl</span><span class="p">[</span><span class="n">iu</span><span class="p">]</span> <span class="o">=</span> <span class="o">-</span><span class="p">(</span><span class="n">u</span><span class="p">[</span><span class="n">iu</span><span class="p">]</span> <span class="o">*</span> <span class="n">u</span><span class="p">[</span><span class="n">iux</span><span class="p">]</span> <span class="o">+</span> <span class="n">u</span><span class="p">[</span><span class="n">iv</span><span class="p">]</span> <span class="o">*</span> <span class="n">u</span><span class="p">[</span><span class="n">iuz</span><span class="p">])</span>
+        <span class="n">f</span><span class="o">.</span><span class="n">expl</span><span class="p">[</span><span class="n">iv</span><span class="p">]</span> <span class="o">=</span> <span class="o">-</span><span class="p">(</span><span class="n">u</span><span class="p">[</span><span class="n">iu</span><span class="p">]</span> <span class="o">*</span> <span class="n">u</span><span class="p">[</span><span class="n">ivx</span><span class="p">]</span> <span class="o">+</span> <span class="n">u</span><span class="p">[</span><span class="n">iv</span><span class="p">]</span> <span class="o">*</span> <span class="n">u</span><span class="p">[</span><span class="n">ivz</span><span class="p">])</span>
+        <span class="k">return</span> <span class="n">f</span></div>
+
+
+<div class="viewcode-block" id="Burgers2D.compute_vorticity">
+<a class="viewcode-back" href="../../../pySDC/implementations.problem_classes.Burgers.html#implementations.problem_classes.Burgers.Burgers2D.compute_vorticity">[docs]</a>
+    <span class="k">def</span> <span class="nf">compute_vorticity</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">u</span><span class="p">):</span>
+        <span class="n">me</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">u_init_forward</span>
+
+        <span class="n">u_hat</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">transform</span><span class="p">(</span><span class="n">u</span><span class="p">)</span>
+        <span class="n">iu</span><span class="p">,</span> <span class="n">iv</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">index</span><span class="p">([</span><span class="s1">&#39;u&#39;</span><span class="p">,</span> <span class="s1">&#39;v&#39;</span><span class="p">])</span>
+
+        <span class="n">me</span><span class="p">[</span><span class="n">iu</span><span class="p">]</span> <span class="o">=</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">Dx</span> <span class="o">@</span> <span class="n">u_hat</span><span class="p">[</span><span class="n">iv</span><span class="p">]</span><span class="o">.</span><span class="n">flatten</span><span class="p">()</span> <span class="o">+</span> <span class="bp">self</span><span class="o">.</span><span class="n">Dz</span> <span class="o">@</span> <span class="n">u_hat</span><span class="p">[</span><span class="n">iu</span><span class="p">]</span><span class="o">.</span><span class="n">flatten</span><span class="p">())</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="n">u</span><span class="p">[</span><span class="n">iu</span><span class="p">]</span><span class="o">.</span><span class="n">shape</span><span class="p">)</span>
+        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">itransform</span><span class="p">(</span><span class="n">me</span><span class="p">)[</span><span class="n">iu</span><span class="p">]</span><span class="o">.</span><span class="n">real</span></div>
+
+
+<div class="viewcode-block" id="Burgers2D.get_fig">
+<a class="viewcode-back" href="../../../pySDC/implementations.problem_classes.Burgers.html#implementations.problem_classes.Burgers.Burgers2D.get_fig">[docs]</a>
+    <span class="k">def</span> <span class="nf">get_fig</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>  <span class="c1"># pragma: no cover</span>
+<span class="w">        </span><span class="sd">&quot;&quot;&quot;</span>
+<span class="sd">        Get a figure suitable to plot the solution of this problem</span>
+
+<span class="sd">        Returns</span>
+<span class="sd">        -------</span>
+<span class="sd">        self.fig : matplotlib.pyplot.figure.Figure</span>
+<span class="sd">        &quot;&quot;&quot;</span>
+        <span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
+        <span class="kn">from</span> <span class="nn">mpl_toolkits.axes_grid1</span> <span class="kn">import</span> <span class="n">make_axes_locatable</span>
+
+        <span class="n">plt</span><span class="o">.</span><span class="n">rcParams</span><span class="p">[</span><span class="s1">&#39;figure.constrained_layout.use&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="kc">True</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">fig</span><span class="p">,</span> <span class="n">axs</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">sharex</span><span class="o">=</span><span class="kc">True</span><span class="p">,</span> <span class="n">sharey</span><span class="o">=</span><span class="kc">True</span><span class="p">,</span> <span class="n">figsize</span><span class="o">=</span><span class="p">((</span><span class="mi">8</span><span class="p">,</span> <span class="mi">7</span><span class="p">)))</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">cax</span> <span class="o">=</span> <span class="p">[]</span>
+        <span class="n">divider</span> <span class="o">=</span> <span class="n">make_axes_locatable</span><span class="p">(</span><span class="n">axs</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">cax</span> <span class="o">+=</span> <span class="p">[</span><span class="n">divider</span><span class="o">.</span><span class="n">append_axes</span><span class="p">(</span><span class="s1">&#39;right&#39;</span><span class="p">,</span> <span class="n">size</span><span class="o">=</span><span class="s1">&#39;3%&#39;</span><span class="p">,</span> <span class="n">pad</span><span class="o">=</span><span class="mf">0.03</span><span class="p">)]</span>
+        <span class="n">divider2</span> <span class="o">=</span> <span class="n">make_axes_locatable</span><span class="p">(</span><span class="n">axs</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">cax</span> <span class="o">+=</span> <span class="p">[</span><span class="n">divider2</span><span class="o">.</span><span class="n">append_axes</span><span class="p">(</span><span class="s1">&#39;right&#39;</span><span class="p">,</span> <span class="n">size</span><span class="o">=</span><span class="s1">&#39;3%&#39;</span><span class="p">,</span> <span class="n">pad</span><span class="o">=</span><span class="mf">0.03</span><span class="p">)]</span>
+        <span class="n">divider3</span> <span class="o">=</span> <span class="n">make_axes_locatable</span><span class="p">(</span><span class="n">axs</span><span class="p">[</span><span class="mi">2</span><span class="p">])</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">cax</span> <span class="o">+=</span> <span class="p">[</span><span class="n">divider3</span><span class="o">.</span><span class="n">append_axes</span><span class="p">(</span><span class="s1">&#39;right&#39;</span><span class="p">,</span> <span class="n">size</span><span class="o">=</span><span class="s1">&#39;3%&#39;</span><span class="p">,</span> <span class="n">pad</span><span class="o">=</span><span class="mf">0.03</span><span class="p">)]</span>
+        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">fig</span></div>
+
+
+<div class="viewcode-block" id="Burgers2D.plot">
+<a class="viewcode-back" href="../../../pySDC/implementations.problem_classes.Burgers.html#implementations.problem_classes.Burgers.Burgers2D.plot">[docs]</a>
+    <span class="k">def</span> <span class="nf">plot</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">t</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">fig</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">vmin</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">vmax</span><span class="o">=</span><span class="kc">None</span><span class="p">):</span>  <span class="c1"># pragma: no cover</span>
+<span class="w">        </span><span class="sa">r</span><span class="sd">&quot;&quot;&quot;</span>
+<span class="sd">        Plot the solution. Please supply a figure with the same structure as returned by ``self.get_fig``.</span>
+
+<span class="sd">        Parameters</span>
+<span class="sd">        ----------</span>
+<span class="sd">        u : dtype_u</span>
+<span class="sd">            Solution to be plotted</span>
+<span class="sd">        t : float</span>
+<span class="sd">            Time to display at the top of the figure</span>
+<span class="sd">        fig : matplotlib.pyplot.figure.Figure</span>
+<span class="sd">            Figure with the correct structure</span>
+
+<span class="sd">        Returns</span>
+<span class="sd">        -------</span>
+<span class="sd">        None</span>
+<span class="sd">        &quot;&quot;&quot;</span>
+        <span class="n">fig</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">get_fig</span><span class="p">()</span> <span class="k">if</span> <span class="n">fig</span> <span class="ow">is</span> <span class="kc">None</span> <span class="k">else</span> <span class="n">fig</span>
+        <span class="n">axs</span> <span class="o">=</span> <span class="n">fig</span><span class="o">.</span><span class="n">axes</span>
+
+        <span class="n">iu</span><span class="p">,</span> <span class="n">iv</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">index</span><span class="p">([</span><span class="s1">&#39;u&#39;</span><span class="p">,</span> <span class="s1">&#39;v&#39;</span><span class="p">])</span>
+
+        <span class="n">imU</span> <span class="o">=</span> <span class="n">axs</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">pcolormesh</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">X</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">Z</span><span class="p">,</span> <span class="n">u</span><span class="p">[</span><span class="n">iu</span><span class="p">]</span><span class="o">.</span><span class="n">real</span><span class="p">,</span> <span class="n">vmin</span><span class="o">=</span><span class="n">vmin</span><span class="p">,</span> <span class="n">vmax</span><span class="o">=</span><span class="n">vmax</span><span class="p">)</span>
+        <span class="n">imV</span> <span class="o">=</span> <span class="n">axs</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">pcolormesh</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">X</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">Z</span><span class="p">,</span> <span class="n">u</span><span class="p">[</span><span class="n">iv</span><span class="p">]</span><span class="o">.</span><span class="n">real</span><span class="p">,</span> <span class="n">vmin</span><span class="o">=</span><span class="n">vmin</span><span class="p">,</span> <span class="n">vmax</span><span class="o">=</span><span class="n">vmax</span><span class="p">)</span>
+        <span class="n">imVort</span> <span class="o">=</span> <span class="n">axs</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span><span class="o">.</span><span class="n">pcolormesh</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">X</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">Z</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">compute_vorticity</span><span class="p">(</span><span class="n">u</span><span class="p">)</span><span class="o">.</span><span class="n">real</span><span class="p">)</span>
+
+        <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">label</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">],</span> <span class="p">[</span><span class="sa">r</span><span class="s1">&#39;$u$&#39;</span><span class="p">,</span> <span class="s1">&#39;$v$&#39;</span><span class="p">,</span> <span class="s1">&#39;vorticity&#39;</span><span class="p">]):</span>
+            <span class="n">axs</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">.</span><span class="n">set_aspect</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
+            <span class="n">axs</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="n">label</span><span class="p">)</span>
+
+        <span class="k">if</span> <span class="n">t</span> <span class="ow">is</span> <span class="ow">not</span> <span class="kc">None</span><span class="p">:</span>
+            <span class="n">fig</span><span class="o">.</span><span class="n">suptitle</span><span class="p">(</span><span class="sa">f</span><span class="s1">&#39;t = </span><span class="si">{</span><span class="n">t</span><span class="si">:</span><span class="s1">.2e</span><span class="si">}</span><span class="s1">&#39;</span><span class="p">)</span>
+        <span class="n">axs</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="sa">r</span><span class="s1">&#39;$x$&#39;</span><span class="p">)</span>
+        <span class="n">axs</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="sa">r</span><span class="s1">&#39;$z$&#39;</span><span class="p">)</span>
+        <span class="n">fig</span><span class="o">.</span><span class="n">colorbar</span><span class="p">(</span><span class="n">imU</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">cax</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
+        <span class="n">fig</span><span class="o">.</span><span class="n">colorbar</span><span class="p">(</span><span class="n">imV</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">cax</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
+        <span class="n">fig</span><span class="o">.</span><span class="n">colorbar</span><span class="p">(</span><span class="n">imVort</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">cax</span><span class="p">[</span><span class="mi">2</span><span class="p">])</span></div>
+</div>
+
+</pre></div>
+
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diff --git a/_modules/implementations/problem_classes/HeatEquation_Chebychev.html b/_modules/implementations/problem_classes/HeatEquation_Chebychev.html
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+        <li class="nav-item nav-item-this"><a href="">implementations.problem_classes.HeatEquation_Chebychev</a></li> 
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+            
+  <h1>Source code for implementations.problem_classes.HeatEquation_Chebychev</h1><div class="highlight"><pre>
+<span></span><span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
+<span class="kn">from</span> <span class="nn">scipy</span> <span class="kn">import</span> <span class="n">sparse</span> <span class="k">as</span> <span class="n">sp</span>
+
+<span class="kn">from</span> <span class="nn">pySDC.core.problem</span> <span class="kn">import</span> <span class="n">Problem</span>
+<span class="kn">from</span> <span class="nn">pySDC.implementations.datatype_classes.mesh</span> <span class="kn">import</span> <span class="n">mesh</span>
+<span class="kn">from</span> <span class="nn">pySDC.implementations.problem_classes.generic_spectral</span> <span class="kn">import</span> <span class="n">GenericSpectralLinear</span>
+
+
+<div class="viewcode-block" id="Heat1DChebychev">
+<a class="viewcode-back" href="../../../pySDC/implementations.problem_classes.HeatEquation_Chebychev.html#implementations.problem_classes.HeatEquation_Chebychev.Heat1DChebychev">[docs]</a>
+<span class="k">class</span> <span class="nc">Heat1DChebychev</span><span class="p">(</span><span class="n">GenericSpectralLinear</span><span class="p">):</span>
+<span class="w">    </span><span class="sd">&quot;&quot;&quot;</span>
+<span class="sd">    1D Heat equation with Dirichlet Boundary conditions discretized on (-1, 1) using a Chebychev spectral method.</span>
+<span class="sd">    &quot;&quot;&quot;</span>
+
+    <span class="n">dtype_u</span> <span class="o">=</span> <span class="n">mesh</span>
+    <span class="n">dtype_f</span> <span class="o">=</span> <span class="n">mesh</span>
+
+    <span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">nvars</span><span class="o">=</span><span class="mi">128</span><span class="p">,</span> <span class="n">a</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span> <span class="n">b</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span> <span class="n">f</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">nu</span><span class="o">=</span><span class="mf">1.0</span><span class="p">,</span> <span class="n">mode</span><span class="o">=</span><span class="s1">&#39;T2U&#39;</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
+<span class="w">        </span><span class="sd">&quot;&quot;&quot;</span>
+<span class="sd">        Constructor. `kwargs` are forwarded to parent class constructor.</span>
+
+<span class="sd">        Args:</span>
+<span class="sd">            nvars (int): Resolution</span>
+<span class="sd">            a (float): Left BC value</span>
+<span class="sd">            b (float): Right BC value</span>
+<span class="sd">            f (int): Frequency of the solution</span>
+<span class="sd">            nu (float): Diffusion parameter</span>
+<span class="sd">            mode (&#39;T2T&#39; or &#39;T2U&#39;): Mode for Chebychev method.</span>
+
+<span class="sd">        &quot;&quot;&quot;</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">_makeAttributeAndRegister</span><span class="p">(</span><span class="s1">&#39;nvars&#39;</span><span class="p">,</span> <span class="s1">&#39;a&#39;</span><span class="p">,</span> <span class="s1">&#39;b&#39;</span><span class="p">,</span> <span class="s1">&#39;f&#39;</span><span class="p">,</span> <span class="s1">&#39;nu&#39;</span><span class="p">,</span> <span class="s1">&#39;mode&#39;</span><span class="p">,</span> <span class="n">localVars</span><span class="o">=</span><span class="nb">locals</span><span class="p">(),</span> <span class="n">readOnly</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
+
+        <span class="n">bases</span> <span class="o">=</span> <span class="p">[{</span><span class="s1">&#39;base&#39;</span><span class="p">:</span> <span class="s1">&#39;chebychev&#39;</span><span class="p">,</span> <span class="s1">&#39;N&#39;</span><span class="p">:</span> <span class="n">nvars</span><span class="p">}]</span>
+        <span class="n">components</span> <span class="o">=</span> <span class="p">[</span><span class="s1">&#39;u&#39;</span><span class="p">,</span> <span class="s1">&#39;ux&#39;</span><span class="p">]</span>
+
+        <span class="nb">super</span><span class="p">()</span><span class="o">.</span><span class="fm">__init__</span><span class="p">(</span><span class="n">bases</span><span class="p">,</span> <span class="n">components</span><span class="p">,</span> <span class="n">real_spectral_coefficients</span><span class="o">=</span><span class="kc">True</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
+
+        <span class="bp">self</span><span class="o">.</span><span class="n">x</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">get_grid</span><span class="p">()[</span><span class="mi">0</span><span class="p">]</span>
+
+        <span class="n">I</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">get_Id</span><span class="p">()</span>
+        <span class="n">Dx</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">get_differentiation_matrix</span><span class="p">(</span><span class="n">axes</span><span class="o">=</span><span class="p">(</span><span class="mi">0</span><span class="p">,))</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">Dx</span> <span class="o">=</span> <span class="n">Dx</span>
+
+        <span class="bp">self</span><span class="o">.</span><span class="n">T2U</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">get_basis_change_matrix</span><span class="p">(</span><span class="n">conv</span><span class="o">=</span><span class="n">mode</span><span class="p">)</span>
+
+        <span class="n">L_lhs</span> <span class="o">=</span> <span class="p">{</span>
+            <span class="s1">&#39;ux&#39;</span><span class="p">:</span> <span class="p">{</span><span class="s1">&#39;u&#39;</span><span class="p">:</span> <span class="o">-</span><span class="bp">self</span><span class="o">.</span><span class="n">T2U</span> <span class="o">@</span> <span class="n">Dx</span><span class="p">,</span> <span class="s1">&#39;ux&#39;</span><span class="p">:</span> <span class="bp">self</span><span class="o">.</span><span class="n">T2U</span> <span class="o">@</span> <span class="n">I</span><span class="p">},</span>
+            <span class="s1">&#39;u&#39;</span><span class="p">:</span> <span class="p">{</span><span class="s1">&#39;ux&#39;</span><span class="p">:</span> <span class="o">-</span><span class="n">nu</span> <span class="o">*</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">T2U</span> <span class="o">@</span> <span class="n">Dx</span><span class="p">)},</span>
+        <span class="p">}</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">setup_L</span><span class="p">(</span><span class="n">L_lhs</span><span class="p">)</span>
+
+        <span class="n">M_lhs</span> <span class="o">=</span> <span class="p">{</span><span class="s1">&#39;u&#39;</span><span class="p">:</span> <span class="p">{</span><span class="s1">&#39;u&#39;</span><span class="p">:</span> <span class="bp">self</span><span class="o">.</span><span class="n">T2U</span> <span class="o">@</span> <span class="n">I</span><span class="p">}}</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">setup_M</span><span class="p">(</span><span class="n">M_lhs</span><span class="p">)</span>
+
+        <span class="bp">self</span><span class="o">.</span><span class="n">add_BC</span><span class="p">(</span><span class="n">component</span><span class="o">=</span><span class="s1">&#39;u&#39;</span><span class="p">,</span> <span class="n">equation</span><span class="o">=</span><span class="s1">&#39;u&#39;</span><span class="p">,</span> <span class="n">axis</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span> <span class="n">x</span><span class="o">=-</span><span class="mi">1</span><span class="p">,</span> <span class="n">v</span><span class="o">=</span><span class="n">a</span><span class="p">,</span> <span class="n">kind</span><span class="o">=</span><span class="s2">&quot;Dirichlet&quot;</span><span class="p">)</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">add_BC</span><span class="p">(</span><span class="n">component</span><span class="o">=</span><span class="s1">&#39;u&#39;</span><span class="p">,</span> <span class="n">equation</span><span class="o">=</span><span class="s1">&#39;ux&#39;</span><span class="p">,</span> <span class="n">axis</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span> <span class="n">x</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">v</span><span class="o">=</span><span class="n">b</span><span class="p">,</span> <span class="n">kind</span><span class="o">=</span><span class="s2">&quot;Dirichlet&quot;</span><span class="p">)</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">setup_BCs</span><span class="p">()</span>
+
+<div class="viewcode-block" id="Heat1DChebychev.eval_f">
+<a class="viewcode-back" href="../../../pySDC/implementations.problem_classes.HeatEquation_Chebychev.html#implementations.problem_classes.HeatEquation_Chebychev.Heat1DChebychev.eval_f">[docs]</a>
+    <span class="k">def</span> <span class="nf">eval_f</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
+        <span class="n">f</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">f_init</span>
+        <span class="n">iu</span><span class="p">,</span> <span class="n">iux</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">index</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">components</span><span class="p">)</span>
+
+        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">spectral_space</span><span class="p">:</span>
+            <span class="n">u_hat</span> <span class="o">=</span> <span class="n">u</span><span class="o">.</span><span class="n">copy</span><span class="p">()</span>
+        <span class="k">else</span><span class="p">:</span>
+            <span class="n">u_hat</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">transform</span><span class="p">(</span><span class="n">u</span><span class="p">)</span>
+
+        <span class="n">u_hat</span><span class="p">[</span><span class="n">iu</span><span class="p">]</span> <span class="o">=</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">nu</span> <span class="o">*</span> <span class="bp">self</span><span class="o">.</span><span class="n">Dx</span> <span class="o">@</span> <span class="n">u_hat</span><span class="p">[</span><span class="n">iux</span><span class="p">]</span><span class="o">.</span><span class="n">flatten</span><span class="p">())</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="n">u_hat</span><span class="p">[</span><span class="n">iu</span><span class="p">]</span><span class="o">.</span><span class="n">shape</span><span class="p">)</span>
+
+        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">spectral_space</span><span class="p">:</span>
+            <span class="n">me</span> <span class="o">=</span> <span class="n">u_hat</span>
+        <span class="k">else</span><span class="p">:</span>
+            <span class="n">me</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">itransform</span><span class="p">(</span><span class="n">u_hat</span><span class="p">)</span><span class="o">.</span><span class="n">real</span>
+
+        <span class="n">f</span><span class="p">[</span><span class="n">iu</span><span class="p">]</span> <span class="o">=</span> <span class="n">me</span><span class="p">[</span><span class="n">iu</span><span class="p">]</span>
+        <span class="k">return</span> <span class="n">f</span></div>
+
+
+<div class="viewcode-block" id="Heat1DChebychev.u_exact">
+<a class="viewcode-back" href="../../../pySDC/implementations.problem_classes.HeatEquation_Chebychev.html#implementations.problem_classes.HeatEquation_Chebychev.Heat1DChebychev.u_exact">[docs]</a>
+    <span class="k">def</span> <span class="nf">u_exact</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">t</span><span class="p">,</span> <span class="n">noise</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span> <span class="n">seed</span><span class="o">=</span><span class="mi">666</span><span class="p">):</span>
+<span class="w">        </span><span class="sd">&quot;&quot;&quot;</span>
+<span class="sd">        Get exact solution at time `t`</span>
+
+<span class="sd">        Args:</span>
+<span class="sd">            t (float): When you want the exact solution</span>
+<span class="sd">            noise (float): Add noise of this level</span>
+<span class="sd">            seed (int): Random seed for the noise</span>
+
+<span class="sd">        Returns:</span>
+<span class="sd">            Heat1DChebychev.dtype_u: Exact solution</span>
+<span class="sd">        &quot;&quot;&quot;</span>
+        <span class="n">xp</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">xp</span>
+        <span class="n">iu</span><span class="p">,</span> <span class="n">iux</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">index</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">components</span><span class="p">)</span>
+        <span class="n">u</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">spectral</span><span class="o">.</span><span class="n">u_init</span>
+
+        <span class="n">u</span><span class="p">[</span><span class="n">iu</span><span class="p">]</span> <span class="o">=</span> <span class="p">(</span>
+            <span class="n">xp</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">f</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">pi</span> <span class="o">*</span> <span class="bp">self</span><span class="o">.</span><span class="n">x</span><span class="p">)</span> <span class="o">*</span> <span class="n">xp</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="bp">self</span><span class="o">.</span><span class="n">nu</span> <span class="o">*</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">f</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">pi</span><span class="p">)</span> <span class="o">**</span> <span class="mi">2</span> <span class="o">*</span> <span class="n">t</span><span class="p">)</span>
+            <span class="o">+</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">b</span> <span class="o">-</span> <span class="bp">self</span><span class="o">.</span><span class="n">a</span><span class="p">)</span> <span class="o">/</span> <span class="mi">2</span> <span class="o">*</span> <span class="bp">self</span><span class="o">.</span><span class="n">x</span>
+            <span class="o">+</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">b</span> <span class="o">+</span> <span class="bp">self</span><span class="o">.</span><span class="n">a</span><span class="p">)</span> <span class="o">/</span> <span class="mi">2</span>
+        <span class="p">)</span>
+        <span class="n">u</span><span class="p">[</span><span class="n">iux</span><span class="p">]</span> <span class="o">=</span> <span class="p">(</span>
+            <span class="bp">self</span><span class="o">.</span><span class="n">f</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">pi</span> <span class="o">*</span> <span class="n">xp</span><span class="o">.</span><span class="n">cos</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">f</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">pi</span> <span class="o">*</span> <span class="bp">self</span><span class="o">.</span><span class="n">x</span><span class="p">)</span> <span class="o">*</span> <span class="n">xp</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="bp">self</span><span class="o">.</span><span class="n">nu</span> <span class="o">*</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">f</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">pi</span><span class="p">)</span> <span class="o">**</span> <span class="mi">2</span> <span class="o">*</span> <span class="n">t</span><span class="p">)</span>
+            <span class="o">+</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">b</span> <span class="o">-</span> <span class="bp">self</span><span class="o">.</span><span class="n">a</span><span class="p">)</span> <span class="o">/</span> <span class="mi">2</span>
+        <span class="p">)</span>
+
+        <span class="k">if</span> <span class="n">noise</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">:</span>
+            <span class="k">assert</span> <span class="n">t</span> <span class="o">==</span> <span class="mi">0</span>
+            <span class="n">_noise</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">u_init</span>
+            <span class="n">rng</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">xp</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">default_rng</span><span class="p">(</span><span class="n">seed</span><span class="o">=</span><span class="n">seed</span><span class="p">)</span>
+            <span class="n">_noise</span><span class="p">[</span><span class="n">iu</span><span class="p">]</span> <span class="o">=</span> <span class="n">rng</span><span class="o">.</span><span class="n">normal</span><span class="p">(</span><span class="n">size</span><span class="o">=</span><span class="n">u</span><span class="p">[</span><span class="n">iu</span><span class="p">]</span><span class="o">.</span><span class="n">shape</span><span class="p">)</span>
+            <span class="n">noise_hat</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">transform</span><span class="p">(</span><span class="n">_noise</span><span class="p">)</span>
+            <span class="n">low_pass</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">get_filter_matrix</span><span class="p">(</span><span class="n">axis</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span> <span class="n">kmax</span><span class="o">=</span><span class="bp">self</span><span class="o">.</span><span class="n">nvars</span> <span class="o">-</span> <span class="mi">2</span><span class="p">)</span>
+            <span class="n">noise_hat</span><span class="p">[</span><span class="n">iu</span><span class="p">]</span> <span class="o">=</span> <span class="p">(</span><span class="n">low_pass</span> <span class="o">@</span> <span class="n">noise_hat</span><span class="p">[</span><span class="n">iu</span><span class="p">]</span><span class="o">.</span><span class="n">flatten</span><span class="p">())</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="n">noise_hat</span><span class="p">[</span><span class="n">iu</span><span class="p">]</span><span class="o">.</span><span class="n">shape</span><span class="p">)</span>
+            <span class="n">_noise</span><span class="p">[:]</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">itransform</span><span class="p">(</span><span class="n">noise_hat</span><span class="p">)</span>
+            <span class="n">u</span> <span class="o">+=</span> <span class="n">_noise</span> <span class="o">*</span> <span class="n">noise</span> <span class="o">*</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">x</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span> <span class="o">*</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
+
+        <span class="bp">self</span><span class="o">.</span><span class="n">check_BCs</span><span class="p">(</span><span class="n">u</span><span class="p">)</span>
+
+        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">spectral_space</span><span class="p">:</span>
+            <span class="n">u_hat</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">u_init</span>
+            <span class="n">u_hat</span><span class="p">[</span><span class="o">...</span><span class="p">]</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">transform</span><span class="p">(</span><span class="n">u</span><span class="p">)</span>
+            <span class="k">return</span> <span class="n">u_hat</span>
+        <span class="k">else</span><span class="p">:</span>
+            <span class="k">return</span> <span class="n">u</span></div>
+</div>
+
+
+
+<div class="viewcode-block" id="Heat1DUltraspherical">
+<a class="viewcode-back" href="../../../pySDC/implementations.problem_classes.HeatEquation_Chebychev.html#implementations.problem_classes.HeatEquation_Chebychev.Heat1DUltraspherical">[docs]</a>
+<span class="k">class</span> <span class="nc">Heat1DUltraspherical</span><span class="p">(</span><span class="n">GenericSpectralLinear</span><span class="p">):</span>
+<span class="w">    </span><span class="sd">&quot;&quot;&quot;</span>
+<span class="sd">    1D Heat equation with Dirichlet Boundary conditions discretized on (-1, 1) using an ultraspherical spectral method.</span>
+<span class="sd">    &quot;&quot;&quot;</span>
+
+    <span class="n">dtype_u</span> <span class="o">=</span> <span class="n">mesh</span>
+    <span class="n">dtype_f</span> <span class="o">=</span> <span class="n">mesh</span>
+
+    <span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">nvars</span><span class="o">=</span><span class="mi">128</span><span class="p">,</span> <span class="n">a</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span> <span class="n">b</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span> <span class="n">f</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">nu</span><span class="o">=</span><span class="mf">1.0</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
+<span class="w">        </span><span class="sd">&quot;&quot;&quot;</span>
+<span class="sd">        Constructor. `kwargs` are forwarded to parent class constructor.</span>
+
+<span class="sd">        Args:</span>
+<span class="sd">            nvars (int): Resolution</span>
+<span class="sd">            a (float): Left BC value</span>
+<span class="sd">            b (float): Right BC value</span>
+<span class="sd">            f (int): Frequency of the solution</span>
+<span class="sd">            nu (float): Diffusion parameter</span>
+<span class="sd">        &quot;&quot;&quot;</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">_makeAttributeAndRegister</span><span class="p">(</span><span class="s1">&#39;nvars&#39;</span><span class="p">,</span> <span class="s1">&#39;a&#39;</span><span class="p">,</span> <span class="s1">&#39;b&#39;</span><span class="p">,</span> <span class="s1">&#39;f&#39;</span><span class="p">,</span> <span class="s1">&#39;nu&#39;</span><span class="p">,</span> <span class="n">localVars</span><span class="o">=</span><span class="nb">locals</span><span class="p">(),</span> <span class="n">readOnly</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
+
+        <span class="n">bases</span> <span class="o">=</span> <span class="p">[{</span><span class="s1">&#39;base&#39;</span><span class="p">:</span> <span class="s1">&#39;ultraspherical&#39;</span><span class="p">,</span> <span class="s1">&#39;N&#39;</span><span class="p">:</span> <span class="n">nvars</span><span class="p">}]</span>
+        <span class="n">components</span> <span class="o">=</span> <span class="p">[</span><span class="s1">&#39;u&#39;</span><span class="p">]</span>
+
+        <span class="n">GenericSpectralLinear</span><span class="o">.</span><span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">bases</span><span class="p">,</span> <span class="n">components</span><span class="p">,</span> <span class="n">real_spectral_coefficients</span><span class="o">=</span><span class="kc">True</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
+
+        <span class="bp">self</span><span class="o">.</span><span class="n">x</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">get_grid</span><span class="p">()[</span><span class="mi">0</span><span class="p">]</span>
+
+        <span class="n">I</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">get_Id</span><span class="p">()</span>
+        <span class="n">Dxx</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">get_differentiation_matrix</span><span class="p">(</span><span class="n">axes</span><span class="o">=</span><span class="p">(</span><span class="mi">0</span><span class="p">,),</span> <span class="n">p</span><span class="o">=</span><span class="mi">2</span><span class="p">)</span>
+
+        <span class="n">S2</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">get_basis_change_matrix</span><span class="p">(</span><span class="n">p_in</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span> <span class="n">p_out</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>
+        <span class="n">U2</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">get_basis_change_matrix</span><span class="p">(</span><span class="n">p_in</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span> <span class="n">p_out</span><span class="o">=</span><span class="mi">2</span><span class="p">)</span>
+
+        <span class="bp">self</span><span class="o">.</span><span class="n">Dxx</span> <span class="o">=</span> <span class="n">S2</span> <span class="o">@</span> <span class="n">Dxx</span>
+
+        <span class="n">L_lhs</span> <span class="o">=</span> <span class="p">{</span>
+            <span class="s1">&#39;u&#39;</span><span class="p">:</span> <span class="p">{</span><span class="s1">&#39;u&#39;</span><span class="p">:</span> <span class="o">-</span><span class="n">nu</span> <span class="o">*</span> <span class="n">Dxx</span><span class="p">},</span>
+        <span class="p">}</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">setup_L</span><span class="p">(</span><span class="n">L_lhs</span><span class="p">)</span>
+
+        <span class="n">M_lhs</span> <span class="o">=</span> <span class="p">{</span><span class="s1">&#39;u&#39;</span><span class="p">:</span> <span class="p">{</span><span class="s1">&#39;u&#39;</span><span class="p">:</span> <span class="n">U2</span> <span class="o">@</span> <span class="n">I</span><span class="p">}}</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">setup_M</span><span class="p">(</span><span class="n">M_lhs</span><span class="p">)</span>
+
+        <span class="bp">self</span><span class="o">.</span><span class="n">add_BC</span><span class="p">(</span><span class="n">component</span><span class="o">=</span><span class="s1">&#39;u&#39;</span><span class="p">,</span> <span class="n">equation</span><span class="o">=</span><span class="s1">&#39;u&#39;</span><span class="p">,</span> <span class="n">axis</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span> <span class="n">x</span><span class="o">=-</span><span class="mi">1</span><span class="p">,</span> <span class="n">v</span><span class="o">=</span><span class="n">a</span><span class="p">,</span> <span class="n">kind</span><span class="o">=</span><span class="s2">&quot;Dirichlet&quot;</span><span class="p">,</span> <span class="n">line</span><span class="o">=-</span><span class="mi">1</span><span class="p">)</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">add_BC</span><span class="p">(</span><span class="n">component</span><span class="o">=</span><span class="s1">&#39;u&#39;</span><span class="p">,</span> <span class="n">equation</span><span class="o">=</span><span class="s1">&#39;u&#39;</span><span class="p">,</span> <span class="n">axis</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span> <span class="n">x</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">v</span><span class="o">=</span><span class="n">b</span><span class="p">,</span> <span class="n">kind</span><span class="o">=</span><span class="s2">&quot;Dirichlet&quot;</span><span class="p">,</span> <span class="n">line</span><span class="o">=-</span><span class="mi">2</span><span class="p">)</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">setup_BCs</span><span class="p">()</span>
+
+<div class="viewcode-block" id="Heat1DUltraspherical.eval_f">
+<a class="viewcode-back" href="../../../pySDC/implementations.problem_classes.HeatEquation_Chebychev.html#implementations.problem_classes.HeatEquation_Chebychev.Heat1DUltraspherical.eval_f">[docs]</a>
+    <span class="k">def</span> <span class="nf">eval_f</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
+        <span class="n">f</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">f_init</span>
+        <span class="n">iu</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">index</span><span class="p">(</span><span class="s1">&#39;u&#39;</span><span class="p">)</span>
+
+        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">spectral_space</span><span class="p">:</span>
+            <span class="n">u_hat</span> <span class="o">=</span> <span class="n">u</span><span class="o">.</span><span class="n">copy</span><span class="p">()</span>
+        <span class="k">else</span><span class="p">:</span>
+            <span class="n">u_hat</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">transform</span><span class="p">(</span><span class="n">u</span><span class="p">)</span>
+
+        <span class="n">u_hat</span><span class="p">[</span><span class="n">iu</span><span class="p">]</span> <span class="o">=</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">nu</span> <span class="o">*</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">Dxx</span> <span class="o">@</span> <span class="n">u_hat</span><span class="p">[</span><span class="n">iu</span><span class="p">]</span><span class="o">.</span><span class="n">flatten</span><span class="p">()))</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="n">u_hat</span><span class="p">[</span><span class="n">iu</span><span class="p">]</span><span class="o">.</span><span class="n">shape</span><span class="p">)</span>
+
+        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">spectral_space</span><span class="p">:</span>
+            <span class="n">me</span> <span class="o">=</span> <span class="n">u_hat</span>
+        <span class="k">else</span><span class="p">:</span>
+            <span class="n">me</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">itransform</span><span class="p">(</span><span class="n">u_hat</span><span class="p">)</span><span class="o">.</span><span class="n">real</span>
+
+        <span class="n">f</span><span class="p">[</span><span class="n">iu</span><span class="p">][</span><span class="o">...</span><span class="p">]</span> <span class="o">=</span> <span class="n">me</span><span class="p">[</span><span class="n">iu</span><span class="p">]</span>
+        <span class="k">return</span> <span class="n">f</span></div>
+
+
+<div class="viewcode-block" id="Heat1DUltraspherical.u_exact">
+<a class="viewcode-back" href="../../../pySDC/implementations.problem_classes.HeatEquation_Chebychev.html#implementations.problem_classes.HeatEquation_Chebychev.Heat1DUltraspherical.u_exact">[docs]</a>
+    <span class="k">def</span> <span class="nf">u_exact</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">t</span><span class="p">,</span> <span class="n">noise</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span> <span class="n">seed</span><span class="o">=</span><span class="mi">666</span><span class="p">):</span>
+<span class="w">        </span><span class="sd">&quot;&quot;&quot;</span>
+<span class="sd">        Get exact solution at time `t`</span>
+
+<span class="sd">        Args:</span>
+<span class="sd">            t (float): When you want the exact solution</span>
+<span class="sd">            noise (float): Add noise of this level</span>
+<span class="sd">            seed (int): Random seed for the noise</span>
+
+<span class="sd">        Returns:</span>
+<span class="sd">            Heat1DUltraspherical.dtype_u: Exact solution</span>
+<span class="sd">        &quot;&quot;&quot;</span>
+        <span class="n">xp</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">xp</span>
+        <span class="n">iu</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">index</span><span class="p">(</span><span class="s1">&#39;u&#39;</span><span class="p">)</span>
+        <span class="n">u</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">spectral</span><span class="o">.</span><span class="n">u_init</span>
+
+        <span class="n">u</span><span class="p">[</span><span class="n">iu</span><span class="p">]</span> <span class="o">=</span> <span class="p">(</span>
+            <span class="n">xp</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">f</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">pi</span> <span class="o">*</span> <span class="bp">self</span><span class="o">.</span><span class="n">x</span><span class="p">)</span> <span class="o">*</span> <span class="n">xp</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="bp">self</span><span class="o">.</span><span class="n">nu</span> <span class="o">*</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">f</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">pi</span><span class="p">)</span> <span class="o">**</span> <span class="mi">2</span> <span class="o">*</span> <span class="n">t</span><span class="p">)</span>
+            <span class="o">+</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">b</span> <span class="o">-</span> <span class="bp">self</span><span class="o">.</span><span class="n">a</span><span class="p">)</span> <span class="o">/</span> <span class="mi">2</span> <span class="o">*</span> <span class="bp">self</span><span class="o">.</span><span class="n">x</span>
+            <span class="o">+</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">b</span> <span class="o">+</span> <span class="bp">self</span><span class="o">.</span><span class="n">a</span><span class="p">)</span> <span class="o">/</span> <span class="mi">2</span>
+        <span class="p">)</span>
+
+        <span class="k">if</span> <span class="n">noise</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">:</span>
+            <span class="k">assert</span> <span class="n">t</span> <span class="o">==</span> <span class="mi">0</span>
+            <span class="n">_noise</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">u_init</span>
+            <span class="n">rng</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">xp</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">default_rng</span><span class="p">(</span><span class="n">seed</span><span class="o">=</span><span class="n">seed</span><span class="p">)</span>
+            <span class="n">_noise</span><span class="p">[</span><span class="n">iu</span><span class="p">]</span> <span class="o">=</span> <span class="n">rng</span><span class="o">.</span><span class="n">normal</span><span class="p">(</span><span class="n">size</span><span class="o">=</span><span class="n">u</span><span class="p">[</span><span class="n">iu</span><span class="p">]</span><span class="o">.</span><span class="n">shape</span><span class="p">)</span>
+            <span class="n">noise_hat</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">transform</span><span class="p">(</span><span class="n">_noise</span><span class="p">)</span>
+            <span class="n">low_pass</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">get_filter_matrix</span><span class="p">(</span><span class="n">axis</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span> <span class="n">kmax</span><span class="o">=</span><span class="bp">self</span><span class="o">.</span><span class="n">nvars</span> <span class="o">-</span> <span class="mi">2</span><span class="p">)</span>
+            <span class="n">noise_hat</span><span class="p">[</span><span class="n">iu</span><span class="p">]</span> <span class="o">=</span> <span class="p">(</span><span class="n">low_pass</span> <span class="o">@</span> <span class="n">noise_hat</span><span class="p">[</span><span class="n">iu</span><span class="p">]</span><span class="o">.</span><span class="n">flatten</span><span class="p">())</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="n">noise_hat</span><span class="p">[</span><span class="n">iu</span><span class="p">]</span><span class="o">.</span><span class="n">shape</span><span class="p">)</span>
+            <span class="n">_noise</span><span class="p">[:]</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">itransform</span><span class="p">(</span><span class="n">noise_hat</span><span class="p">)</span>
+            <span class="n">u</span> <span class="o">+=</span> <span class="n">_noise</span> <span class="o">*</span> <span class="n">noise</span> <span class="o">*</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">x</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span> <span class="o">*</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
+
+        <span class="bp">self</span><span class="o">.</span><span class="n">check_BCs</span><span class="p">(</span><span class="n">u</span><span class="p">)</span>
+
+        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">spectral_space</span><span class="p">:</span>
+            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">transform</span><span class="p">(</span><span class="n">u</span><span class="p">)</span>
+        <span class="k">else</span><span class="p">:</span>
+            <span class="k">return</span> <span class="n">u</span></div>
+</div>
+
+
+
+<div class="viewcode-block" id="Heat2DChebychev">
+<a class="viewcode-back" href="../../../pySDC/implementations.problem_classes.HeatEquation_Chebychev.html#implementations.problem_classes.HeatEquation_Chebychev.Heat2DChebychev">[docs]</a>
+<span class="k">class</span> <span class="nc">Heat2DChebychev</span><span class="p">(</span><span class="n">GenericSpectralLinear</span><span class="p">):</span>
+<span class="w">    </span><span class="sd">&quot;&quot;&quot;</span>
+<span class="sd">    2D Heat equation with Dirichlet Boundary conditions discretized on (-1, 1)x(-1,1) using spectral methods based on FFT and Chebychev.</span>
+<span class="sd">    &quot;&quot;&quot;</span>
+
+    <span class="n">dtype_u</span> <span class="o">=</span> <span class="n">mesh</span>
+    <span class="n">dtype_f</span> <span class="o">=</span> <span class="n">mesh</span>
+
+    <span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">nx</span><span class="o">=</span><span class="mi">128</span><span class="p">,</span> <span class="n">ny</span><span class="o">=</span><span class="mi">128</span><span class="p">,</span> <span class="n">base_x</span><span class="o">=</span><span class="s1">&#39;fft&#39;</span><span class="p">,</span> <span class="n">base_y</span><span class="o">=</span><span class="s1">&#39;chebychev&#39;</span><span class="p">,</span> <span class="n">a</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span> <span class="n">b</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span> <span class="n">c</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span> <span class="n">fx</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">fy</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">nu</span><span class="o">=</span><span class="mf">1.0</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
+<span class="w">        </span><span class="sd">&quot;&quot;&quot;</span>
+<span class="sd">        Constructor. `kwargs` are forwarded to parent class constructor.</span>
+
+<span class="sd">        Args:</span>
+<span class="sd">            nx (int): Resolution in x-direction</span>
+<span class="sd">            ny (int): Resolution in y-direction</span>
+<span class="sd">            base_x (str): Spectral base in x-direction</span>
+<span class="sd">            base_y (str): Spectral base in y-direction</span>
+<span class="sd">            a (float): BC at y=0 and x=-1</span>
+<span class="sd">            b (float): BC at y=0 and x=+1</span>
+<span class="sd">            c (float): BC at y=1 and x = -1</span>
+<span class="sd">            fx (int): Horizontal frequency of initial conditions</span>
+<span class="sd">            fy (int): Vertical frequency of initial conditions</span>
+<span class="sd">            nu (float): Diffusion parameter</span>
+<span class="sd">        &quot;&quot;&quot;</span>
+        <span class="k">assert</span> <span class="n">nx</span> <span class="o">%</span> <span class="mi">2</span> <span class="o">==</span> <span class="mi">1</span> <span class="ow">or</span> <span class="n">base_x</span> <span class="o">==</span> <span class="s1">&#39;fft&#39;</span>
+        <span class="k">assert</span> <span class="n">ny</span> <span class="o">%</span> <span class="mi">2</span> <span class="o">==</span> <span class="mi">1</span> <span class="ow">or</span> <span class="n">base_y</span> <span class="o">==</span> <span class="s1">&#39;fft&#39;</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">_makeAttributeAndRegister</span><span class="p">(</span>
+            <span class="s1">&#39;nx&#39;</span><span class="p">,</span> <span class="s1">&#39;ny&#39;</span><span class="p">,</span> <span class="s1">&#39;base_x&#39;</span><span class="p">,</span> <span class="s1">&#39;base_y&#39;</span><span class="p">,</span> <span class="s1">&#39;a&#39;</span><span class="p">,</span> <span class="s1">&#39;b&#39;</span><span class="p">,</span> <span class="s1">&#39;c&#39;</span><span class="p">,</span> <span class="s1">&#39;fx&#39;</span><span class="p">,</span> <span class="s1">&#39;fy&#39;</span><span class="p">,</span> <span class="s1">&#39;nu&#39;</span><span class="p">,</span> <span class="n">localVars</span><span class="o">=</span><span class="nb">locals</span><span class="p">(),</span> <span class="n">readOnly</span><span class="o">=</span><span class="kc">True</span>
+        <span class="p">)</span>
+
+        <span class="n">bases</span> <span class="o">=</span> <span class="p">[{</span><span class="s1">&#39;base&#39;</span><span class="p">:</span> <span class="n">base_x</span><span class="p">,</span> <span class="s1">&#39;N&#39;</span><span class="p">:</span> <span class="n">nx</span><span class="p">},</span> <span class="p">{</span><span class="s1">&#39;base&#39;</span><span class="p">:</span> <span class="n">base_y</span><span class="p">,</span> <span class="s1">&#39;N&#39;</span><span class="p">:</span> <span class="n">ny</span><span class="p">}]</span>
+        <span class="n">components</span> <span class="o">=</span> <span class="p">[</span><span class="s1">&#39;u&#39;</span><span class="p">,</span> <span class="s1">&#39;ux&#39;</span><span class="p">,</span> <span class="s1">&#39;uy&#39;</span><span class="p">]</span>
+
+        <span class="nb">super</span><span class="p">()</span><span class="o">.</span><span class="fm">__init__</span><span class="p">(</span><span class="n">bases</span><span class="p">,</span> <span class="n">components</span><span class="p">,</span> <span class="n">Dirichlet_recombination</span><span class="o">=</span><span class="kc">False</span><span class="p">,</span> <span class="n">spectral_space</span><span class="o">=</span><span class="kc">False</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
+
+        <span class="bp">self</span><span class="o">.</span><span class="n">Y</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">X</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">get_grid</span><span class="p">()</span>
+
+        <span class="n">I</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">get_Id</span><span class="p">()</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">Dx</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">get_differentiation_matrix</span><span class="p">(</span><span class="n">axes</span><span class="o">=</span><span class="p">(</span><span class="mi">0</span><span class="p">,))</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">Dy</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">get_differentiation_matrix</span><span class="p">(</span><span class="n">axes</span><span class="o">=</span><span class="p">(</span><span class="mi">1</span><span class="p">,))</span>
+
+        <span class="n">L_lhs</span> <span class="o">=</span> <span class="p">{</span>
+            <span class="s1">&#39;ux&#39;</span><span class="p">:</span> <span class="p">{</span><span class="s1">&#39;u&#39;</span><span class="p">:</span> <span class="o">-</span><span class="bp">self</span><span class="o">.</span><span class="n">Dx</span><span class="p">,</span> <span class="s1">&#39;ux&#39;</span><span class="p">:</span> <span class="n">I</span><span class="p">},</span>
+            <span class="s1">&#39;uy&#39;</span><span class="p">:</span> <span class="p">{</span><span class="s1">&#39;u&#39;</span><span class="p">:</span> <span class="o">-</span><span class="bp">self</span><span class="o">.</span><span class="n">Dy</span><span class="p">,</span> <span class="s1">&#39;uy&#39;</span><span class="p">:</span> <span class="n">I</span><span class="p">},</span>
+            <span class="s1">&#39;u&#39;</span><span class="p">:</span> <span class="p">{</span><span class="s1">&#39;ux&#39;</span><span class="p">:</span> <span class="o">-</span><span class="n">nu</span> <span class="o">*</span> <span class="bp">self</span><span class="o">.</span><span class="n">Dx</span><span class="p">,</span> <span class="s1">&#39;uy&#39;</span><span class="p">:</span> <span class="o">-</span><span class="n">nu</span> <span class="o">*</span> <span class="bp">self</span><span class="o">.</span><span class="n">Dy</span><span class="p">},</span>
+        <span class="p">}</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">setup_L</span><span class="p">(</span><span class="n">L_lhs</span><span class="p">)</span>
+
+        <span class="n">M_lhs</span> <span class="o">=</span> <span class="p">{</span><span class="s1">&#39;u&#39;</span><span class="p">:</span> <span class="p">{</span><span class="s1">&#39;u&#39;</span><span class="p">:</span> <span class="n">I</span><span class="p">}}</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">setup_M</span><span class="p">(</span><span class="n">M_lhs</span><span class="p">)</span>
+
+        <span class="k">for</span> <span class="n">base</span> <span class="ow">in</span> <span class="p">[</span><span class="n">base_x</span><span class="p">,</span> <span class="n">base_y</span><span class="p">]:</span>
+            <span class="k">assert</span> <span class="n">base</span> <span class="ow">in</span> <span class="p">[</span><span class="s1">&#39;chebychev&#39;</span><span class="p">,</span> <span class="s1">&#39;fft&#39;</span><span class="p">]</span>
+
+        <span class="n">alpha</span> <span class="o">=</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">b</span> <span class="o">-</span> <span class="bp">self</span><span class="o">.</span><span class="n">a</span><span class="p">)</span> <span class="o">/</span> <span class="mf">2.0</span>
+        <span class="n">beta</span> <span class="o">=</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">c</span> <span class="o">-</span> <span class="bp">self</span><span class="o">.</span><span class="n">b</span><span class="p">)</span> <span class="o">/</span> <span class="mf">2.0</span>
+        <span class="n">gamma</span> <span class="o">=</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">c</span> <span class="o">+</span> <span class="bp">self</span><span class="o">.</span><span class="n">a</span><span class="p">)</span> <span class="o">/</span> <span class="mf">2.0</span>
+
+        <span class="k">if</span> <span class="n">base_x</span> <span class="o">==</span> <span class="s1">&#39;chebychev&#39;</span><span class="p">:</span>
+            <span class="n">y</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">Y</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="p">:]</span>
+            <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">base_y</span> <span class="o">==</span> <span class="s1">&#39;fft&#39;</span><span class="p">:</span>
+                <span class="bp">self</span><span class="o">.</span><span class="n">add_BC</span><span class="p">(</span><span class="n">component</span><span class="o">=</span><span class="s1">&#39;u&#39;</span><span class="p">,</span> <span class="n">equation</span><span class="o">=</span><span class="s1">&#39;u&#39;</span><span class="p">,</span> <span class="n">axis</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span> <span class="n">x</span><span class="o">=-</span><span class="mi">1</span><span class="p">,</span> <span class="n">v</span><span class="o">=</span><span class="n">beta</span> <span class="o">*</span> <span class="n">y</span> <span class="o">-</span> <span class="n">alpha</span> <span class="o">+</span> <span class="n">gamma</span><span class="p">,</span> <span class="n">kind</span><span class="o">=</span><span class="s1">&#39;Dirichlet&#39;</span><span class="p">)</span>
+            <span class="bp">self</span><span class="o">.</span><span class="n">add_BC</span><span class="p">(</span><span class="n">component</span><span class="o">=</span><span class="s1">&#39;ux&#39;</span><span class="p">,</span> <span class="n">equation</span><span class="o">=</span><span class="s1">&#39;ux&#39;</span><span class="p">,</span> <span class="n">axis</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span> <span class="n">v</span><span class="o">=</span><span class="mi">2</span> <span class="o">*</span> <span class="n">alpha</span><span class="p">,</span> <span class="n">kind</span><span class="o">=</span><span class="s1">&#39;integral&#39;</span><span class="p">)</span>
+        <span class="k">else</span><span class="p">:</span>
+            <span class="k">assert</span> <span class="n">a</span> <span class="o">==</span> <span class="n">b</span><span class="p">,</span> <span class="sa">f</span><span class="s1">&#39;Need periodic boundary conditions in x for </span><span class="si">{</span><span class="n">base_x</span><span class="si">}</span><span class="s1"> method!&#39;</span>
+        <span class="k">if</span> <span class="n">base_y</span> <span class="o">==</span> <span class="s1">&#39;chebychev&#39;</span><span class="p">:</span>
+            <span class="n">x</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">X</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">]</span>
+            <span class="bp">self</span><span class="o">.</span><span class="n">add_BC</span><span class="p">(</span><span class="n">component</span><span class="o">=</span><span class="s1">&#39;u&#39;</span><span class="p">,</span> <span class="n">equation</span><span class="o">=</span><span class="s1">&#39;u&#39;</span><span class="p">,</span> <span class="n">axis</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="o">=-</span><span class="mi">1</span><span class="p">,</span> <span class="n">v</span><span class="o">=</span><span class="n">alpha</span> <span class="o">*</span> <span class="n">x</span> <span class="o">-</span> <span class="n">beta</span> <span class="o">+</span> <span class="n">gamma</span><span class="p">,</span> <span class="n">kind</span><span class="o">=</span><span class="s1">&#39;Dirichlet&#39;</span><span class="p">)</span>
+            <span class="bp">self</span><span class="o">.</span><span class="n">add_BC</span><span class="p">(</span><span class="n">component</span><span class="o">=</span><span class="s1">&#39;uy&#39;</span><span class="p">,</span> <span class="n">equation</span><span class="o">=</span><span class="s1">&#39;uy&#39;</span><span class="p">,</span> <span class="n">axis</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">v</span><span class="o">=</span><span class="mi">2</span> <span class="o">*</span> <span class="n">beta</span><span class="p">,</span> <span class="n">kind</span><span class="o">=</span><span class="s1">&#39;integral&#39;</span><span class="p">)</span>
+        <span class="k">else</span><span class="p">:</span>
+            <span class="k">assert</span> <span class="n">c</span> <span class="o">==</span> <span class="n">b</span><span class="p">,</span> <span class="sa">f</span><span class="s1">&#39;Need periodic boundary conditions in y for </span><span class="si">{</span><span class="n">base_y</span><span class="si">}</span><span class="s1"> method!&#39;</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">setup_BCs</span><span class="p">()</span>
+
+<div class="viewcode-block" id="Heat2DChebychev.eval_f">
+<a class="viewcode-back" href="../../../pySDC/implementations.problem_classes.HeatEquation_Chebychev.html#implementations.problem_classes.HeatEquation_Chebychev.Heat2DChebychev.eval_f">[docs]</a>
+    <span class="k">def</span> <span class="nf">eval_f</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
+        <span class="n">f</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">f_init</span>
+        <span class="n">iu</span><span class="p">,</span> <span class="n">iux</span><span class="p">,</span> <span class="n">iuy</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">index</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">components</span><span class="p">)</span>
+
+        <span class="n">me_hat</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">u_init_forward</span>
+        <span class="n">me_hat</span><span class="p">[:]</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">transform</span><span class="p">(</span><span class="n">u</span><span class="p">)</span>
+        <span class="n">me_hat</span><span class="p">[</span><span class="n">iu</span><span class="p">]</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">nu</span> <span class="o">*</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">Dx</span> <span class="o">@</span> <span class="n">me_hat</span><span class="p">[</span><span class="n">iux</span><span class="p">]</span><span class="o">.</span><span class="n">flatten</span><span class="p">()</span> <span class="o">+</span> <span class="bp">self</span><span class="o">.</span><span class="n">Dy</span> <span class="o">@</span> <span class="n">me_hat</span><span class="p">[</span><span class="n">iuy</span><span class="p">]</span><span class="o">.</span><span class="n">flatten</span><span class="p">())</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span>
+            <span class="n">me_hat</span><span class="p">[</span><span class="n">iu</span><span class="p">]</span><span class="o">.</span><span class="n">shape</span>
+        <span class="p">)</span>
+        <span class="n">me</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">itransform</span><span class="p">(</span><span class="n">me_hat</span><span class="p">)</span>
+
+        <span class="n">f</span><span class="p">[</span><span class="bp">self</span><span class="o">.</span><span class="n">index</span><span class="p">(</span><span class="s2">&quot;u&quot;</span><span class="p">)]</span> <span class="o">=</span> <span class="n">me</span><span class="p">[</span><span class="n">iu</span><span class="p">]</span><span class="o">.</span><span class="n">real</span>
+        <span class="k">return</span> <span class="n">f</span></div>
+
+
+<div class="viewcode-block" id="Heat2DChebychev.u_exact">
+<a class="viewcode-back" href="../../../pySDC/implementations.problem_classes.HeatEquation_Chebychev.html#implementations.problem_classes.HeatEquation_Chebychev.Heat2DChebychev.u_exact">[docs]</a>
+    <span class="k">def</span> <span class="nf">u_exact</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">t</span><span class="p">):</span>
+        <span class="n">xp</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">xp</span>
+        <span class="n">iu</span><span class="p">,</span> <span class="n">iux</span><span class="p">,</span> <span class="n">iuy</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">index</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">components</span><span class="p">)</span>
+        <span class="n">u</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">u_init</span>
+
+        <span class="n">fx</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">fx</span> <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">base_x</span> <span class="o">==</span> <span class="s1">&#39;fft&#39;</span> <span class="k">else</span> <span class="n">np</span><span class="o">.</span><span class="n">pi</span> <span class="o">*</span> <span class="bp">self</span><span class="o">.</span><span class="n">fx</span>
+        <span class="n">fy</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">fy</span> <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">base_y</span> <span class="o">==</span> <span class="s1">&#39;fft&#39;</span> <span class="k">else</span> <span class="n">np</span><span class="o">.</span><span class="n">pi</span> <span class="o">*</span> <span class="bp">self</span><span class="o">.</span><span class="n">fy</span>
+
+        <span class="n">time_dep</span> <span class="o">=</span> <span class="n">xp</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="bp">self</span><span class="o">.</span><span class="n">nu</span> <span class="o">*</span> <span class="p">(</span><span class="n">fx</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">fy</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span> <span class="o">*</span> <span class="n">t</span><span class="p">)</span>
+
+        <span class="n">alpha</span> <span class="o">=</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">b</span> <span class="o">-</span> <span class="bp">self</span><span class="o">.</span><span class="n">a</span><span class="p">)</span> <span class="o">/</span> <span class="mf">2.0</span>
+        <span class="n">beta</span> <span class="o">=</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">c</span> <span class="o">-</span> <span class="bp">self</span><span class="o">.</span><span class="n">b</span><span class="p">)</span> <span class="o">/</span> <span class="mf">2.0</span>
+        <span class="n">gamma</span> <span class="o">=</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">c</span> <span class="o">+</span> <span class="bp">self</span><span class="o">.</span><span class="n">a</span><span class="p">)</span> <span class="o">/</span> <span class="mf">2.0</span>
+
+        <span class="n">u</span><span class="p">[</span><span class="n">iu</span><span class="p">]</span> <span class="o">=</span> <span class="n">xp</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="n">fx</span> <span class="o">*</span> <span class="bp">self</span><span class="o">.</span><span class="n">X</span><span class="p">)</span> <span class="o">*</span> <span class="n">xp</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="n">fy</span> <span class="o">*</span> <span class="bp">self</span><span class="o">.</span><span class="n">Y</span><span class="p">)</span> <span class="o">*</span> <span class="n">time_dep</span> <span class="o">+</span> <span class="n">alpha</span> <span class="o">*</span> <span class="bp">self</span><span class="o">.</span><span class="n">X</span> <span class="o">+</span> <span class="n">beta</span> <span class="o">*</span> <span class="bp">self</span><span class="o">.</span><span class="n">Y</span> <span class="o">+</span> <span class="n">gamma</span>
+        <span class="n">u</span><span class="p">[</span><span class="n">iux</span><span class="p">]</span> <span class="o">=</span> <span class="n">fx</span> <span class="o">*</span> <span class="n">xp</span><span class="o">.</span><span class="n">cos</span><span class="p">(</span><span class="n">fx</span> <span class="o">*</span> <span class="bp">self</span><span class="o">.</span><span class="n">X</span><span class="p">)</span> <span class="o">*</span> <span class="n">xp</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="n">fy</span> <span class="o">*</span> <span class="bp">self</span><span class="o">.</span><span class="n">Y</span><span class="p">)</span> <span class="o">*</span> <span class="n">time_dep</span> <span class="o">+</span> <span class="n">alpha</span>
+        <span class="n">u</span><span class="p">[</span><span class="n">iuy</span><span class="p">]</span> <span class="o">=</span> <span class="n">fy</span> <span class="o">*</span> <span class="n">xp</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="n">fx</span> <span class="o">*</span> <span class="bp">self</span><span class="o">.</span><span class="n">X</span><span class="p">)</span> <span class="o">*</span> <span class="n">xp</span><span class="o">.</span><span class="n">cos</span><span class="p">(</span><span class="n">fy</span> <span class="o">*</span> <span class="bp">self</span><span class="o">.</span><span class="n">Y</span><span class="p">)</span> <span class="o">*</span> <span class="n">time_dep</span> <span class="o">+</span> <span class="n">beta</span>
+
+        <span class="k">return</span> <span class="n">u</span></div>
+</div>
+
+
+
+<div class="viewcode-block" id="Heat2DUltraspherical">
+<a class="viewcode-back" href="../../../pySDC/implementations.problem_classes.HeatEquation_Chebychev.html#implementations.problem_classes.HeatEquation_Chebychev.Heat2DUltraspherical">[docs]</a>
+<span class="k">class</span> <span class="nc">Heat2DUltraspherical</span><span class="p">(</span><span class="n">GenericSpectralLinear</span><span class="p">):</span>
+<span class="w">    </span><span class="sd">&quot;&quot;&quot;</span>
+<span class="sd">    2D Heat equation with Dirichlet Boundary conditions discretized on (-1, 1)x(-1,1) using spectral methods based on FFT and Gegenbauer.</span>
+<span class="sd">    &quot;&quot;&quot;</span>
+
+    <span class="n">dtype_u</span> <span class="o">=</span> <span class="n">mesh</span>
+    <span class="n">dtype_f</span> <span class="o">=</span> <span class="n">mesh</span>
+
+    <span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span>
+        <span class="bp">self</span><span class="p">,</span> <span class="n">nx</span><span class="o">=</span><span class="mi">128</span><span class="p">,</span> <span class="n">ny</span><span class="o">=</span><span class="mi">128</span><span class="p">,</span> <span class="n">base_x</span><span class="o">=</span><span class="s1">&#39;fft&#39;</span><span class="p">,</span> <span class="n">base_y</span><span class="o">=</span><span class="s1">&#39;ultraspherical&#39;</span><span class="p">,</span> <span class="n">a</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span> <span class="n">b</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span> <span class="n">c</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span> <span class="n">fx</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">fy</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">nu</span><span class="o">=</span><span class="mf">1.0</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span>
+    <span class="p">):</span>
+<span class="w">        </span><span class="sd">&quot;&quot;&quot;</span>
+<span class="sd">        Constructor. `kwargs` are forwarded to parent class constructor.</span>
+
+<span class="sd">        Args:</span>
+<span class="sd">            nx (int): Resolution in x-direction</span>
+<span class="sd">            ny (int): Resolution in y-direction</span>
+<span class="sd">            base_x (str): Spectral base in x-direction</span>
+<span class="sd">            base_y (str): Spectral base in y-direction</span>
+<span class="sd">            a (float): BC at y=0 and x=-1</span>
+<span class="sd">            b (float): BC at y=0 and x=+1</span>
+<span class="sd">            c (float): BC at y=1 and x = -1</span>
+<span class="sd">            fx (int): Horizontal frequency of initial conditions</span>
+<span class="sd">            fy (int): Vertical frequency of initial conditions</span>
+<span class="sd">            nu (float): Diffusion parameter</span>
+<span class="sd">        &quot;&quot;&quot;</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">_makeAttributeAndRegister</span><span class="p">(</span>
+            <span class="s1">&#39;nx&#39;</span><span class="p">,</span> <span class="s1">&#39;ny&#39;</span><span class="p">,</span> <span class="s1">&#39;base_x&#39;</span><span class="p">,</span> <span class="s1">&#39;base_y&#39;</span><span class="p">,</span> <span class="s1">&#39;a&#39;</span><span class="p">,</span> <span class="s1">&#39;b&#39;</span><span class="p">,</span> <span class="s1">&#39;c&#39;</span><span class="p">,</span> <span class="s1">&#39;fx&#39;</span><span class="p">,</span> <span class="s1">&#39;fy&#39;</span><span class="p">,</span> <span class="s1">&#39;nu&#39;</span><span class="p">,</span> <span class="n">localVars</span><span class="o">=</span><span class="nb">locals</span><span class="p">(),</span> <span class="n">readOnly</span><span class="o">=</span><span class="kc">True</span>
+        <span class="p">)</span>
+
+        <span class="n">bases</span> <span class="o">=</span> <span class="p">[{</span><span class="s1">&#39;base&#39;</span><span class="p">:</span> <span class="n">base_x</span><span class="p">,</span> <span class="s1">&#39;N&#39;</span><span class="p">:</span> <span class="n">nx</span><span class="p">},</span> <span class="p">{</span><span class="s1">&#39;base&#39;</span><span class="p">:</span> <span class="n">base_y</span><span class="p">,</span> <span class="s1">&#39;N&#39;</span><span class="p">:</span> <span class="n">ny</span><span class="p">}]</span>
+        <span class="n">components</span> <span class="o">=</span> <span class="p">[</span><span class="s1">&#39;u&#39;</span><span class="p">]</span>
+
+        <span class="nb">super</span><span class="p">()</span><span class="o">.</span><span class="fm">__init__</span><span class="p">(</span><span class="n">bases</span><span class="p">,</span> <span class="n">components</span><span class="p">,</span> <span class="n">Dirichlet_recombination</span><span class="o">=</span><span class="kc">False</span><span class="p">,</span> <span class="n">spectral_space</span><span class="o">=</span><span class="kc">False</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
+
+        <span class="bp">self</span><span class="o">.</span><span class="n">Y</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">X</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">get_grid</span><span class="p">()</span>
+
+        <span class="bp">self</span><span class="o">.</span><span class="n">Dxx</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">get_differentiation_matrix</span><span class="p">(</span><span class="n">axes</span><span class="o">=</span><span class="p">(</span><span class="mi">0</span><span class="p">,),</span> <span class="n">p</span><span class="o">=</span><span class="mi">2</span><span class="p">)</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">Dyy</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">get_differentiation_matrix</span><span class="p">(</span><span class="n">axes</span><span class="o">=</span><span class="p">(</span><span class="mi">1</span><span class="p">,),</span> <span class="n">p</span><span class="o">=</span><span class="mi">2</span><span class="p">)</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">S2</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">get_basis_change_matrix</span><span class="p">(</span><span class="n">p</span><span class="o">=</span><span class="mi">2</span><span class="p">)</span>
+        <span class="n">I</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">get_Id</span><span class="p">()</span>
+
+        <span class="n">L_lhs</span> <span class="o">=</span> <span class="p">{</span>
+            <span class="s1">&#39;u&#39;</span><span class="p">:</span> <span class="p">{</span><span class="s1">&#39;u&#39;</span><span class="p">:</span> <span class="o">-</span><span class="n">nu</span> <span class="o">*</span> <span class="bp">self</span><span class="o">.</span><span class="n">Dxx</span> <span class="o">-</span> <span class="n">nu</span> <span class="o">*</span> <span class="bp">self</span><span class="o">.</span><span class="n">Dyy</span><span class="p">},</span>
+        <span class="p">}</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">setup_L</span><span class="p">(</span><span class="n">L_lhs</span><span class="p">)</span>
+
+        <span class="n">M_lhs</span> <span class="o">=</span> <span class="p">{</span><span class="s1">&#39;u&#39;</span><span class="p">:</span> <span class="p">{</span><span class="s1">&#39;u&#39;</span><span class="p">:</span> <span class="n">I</span><span class="p">}}</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">setup_M</span><span class="p">(</span><span class="n">M_lhs</span><span class="p">)</span>
+
+        <span class="k">for</span> <span class="n">base</span> <span class="ow">in</span> <span class="p">[</span><span class="n">base_x</span><span class="p">,</span> <span class="n">base_y</span><span class="p">]:</span>
+            <span class="k">assert</span> <span class="n">base</span> <span class="ow">in</span> <span class="p">[</span><span class="s1">&#39;ultraspherical&#39;</span><span class="p">,</span> <span class="s1">&#39;fft&#39;</span><span class="p">]</span>
+
+        <span class="n">alpha</span> <span class="o">=</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">b</span> <span class="o">-</span> <span class="bp">self</span><span class="o">.</span><span class="n">a</span><span class="p">)</span> <span class="o">/</span> <span class="mf">2.0</span>
+        <span class="n">beta</span> <span class="o">=</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">c</span> <span class="o">-</span> <span class="bp">self</span><span class="o">.</span><span class="n">b</span><span class="p">)</span> <span class="o">/</span> <span class="mf">2.0</span>
+        <span class="n">gamma</span> <span class="o">=</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">c</span> <span class="o">+</span> <span class="bp">self</span><span class="o">.</span><span class="n">a</span><span class="p">)</span> <span class="o">/</span> <span class="mf">2.0</span>
+
+        <span class="k">if</span> <span class="n">base_x</span> <span class="o">==</span> <span class="s1">&#39;ultraspherical&#39;</span><span class="p">:</span>
+            <span class="n">y</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">Y</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="p">:]</span>
+            <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">base_y</span> <span class="o">==</span> <span class="s1">&#39;fft&#39;</span><span class="p">:</span>
+                <span class="bp">self</span><span class="o">.</span><span class="n">add_BC</span><span class="p">(</span><span class="n">component</span><span class="o">=</span><span class="s1">&#39;u&#39;</span><span class="p">,</span> <span class="n">equation</span><span class="o">=</span><span class="s1">&#39;u&#39;</span><span class="p">,</span> <span class="n">axis</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span> <span class="n">x</span><span class="o">=-</span><span class="mi">1</span><span class="p">,</span> <span class="n">v</span><span class="o">=</span><span class="n">beta</span> <span class="o">*</span> <span class="n">y</span> <span class="o">-</span> <span class="n">alpha</span> <span class="o">+</span> <span class="n">gamma</span><span class="p">,</span> <span class="n">kind</span><span class="o">=</span><span class="s1">&#39;Dirichlet&#39;</span><span class="p">)</span>
+            <span class="bp">self</span><span class="o">.</span><span class="n">add_BC</span><span class="p">(</span><span class="n">component</span><span class="o">=</span><span class="s1">&#39;u&#39;</span><span class="p">,</span> <span class="n">equation</span><span class="o">=</span><span class="s1">&#39;u&#39;</span><span class="p">,</span> <span class="n">axis</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span> <span class="n">v</span><span class="o">=</span><span class="n">beta</span> <span class="o">*</span> <span class="n">y</span> <span class="o">+</span> <span class="n">alpha</span> <span class="o">+</span> <span class="n">gamma</span><span class="p">,</span> <span class="n">x</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">line</span><span class="o">=-</span><span class="mi">2</span><span class="p">,</span> <span class="n">kind</span><span class="o">=</span><span class="s1">&#39;Dirichlet&#39;</span><span class="p">)</span>
+        <span class="k">else</span><span class="p">:</span>
+            <span class="k">assert</span> <span class="n">a</span> <span class="o">==</span> <span class="n">b</span><span class="p">,</span> <span class="sa">f</span><span class="s1">&#39;Need periodic boundary conditions in x for </span><span class="si">{</span><span class="n">base_x</span><span class="si">}</span><span class="s1"> method!&#39;</span>
+        <span class="k">if</span> <span class="n">base_y</span> <span class="o">==</span> <span class="s1">&#39;ultraspherical&#39;</span><span class="p">:</span>
+            <span class="n">x</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">X</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">]</span>
+            <span class="bp">self</span><span class="o">.</span><span class="n">add_BC</span><span class="p">(</span>
+                <span class="n">component</span><span class="o">=</span><span class="s1">&#39;u&#39;</span><span class="p">,</span> <span class="n">equation</span><span class="o">=</span><span class="s1">&#39;u&#39;</span><span class="p">,</span> <span class="n">axis</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="o">=-</span><span class="mi">1</span><span class="p">,</span> <span class="n">v</span><span class="o">=</span><span class="n">alpha</span> <span class="o">*</span> <span class="n">x</span> <span class="o">-</span> <span class="n">beta</span> <span class="o">+</span> <span class="n">gamma</span><span class="p">,</span> <span class="n">kind</span><span class="o">=</span><span class="s1">&#39;Dirichlet&#39;</span><span class="p">,</span> <span class="n">line</span><span class="o">=-</span><span class="mi">1</span>
+            <span class="p">)</span>
+            <span class="bp">self</span><span class="o">.</span><span class="n">add_BC</span><span class="p">(</span>
+                <span class="n">component</span><span class="o">=</span><span class="s1">&#39;u&#39;</span><span class="p">,</span> <span class="n">equation</span><span class="o">=</span><span class="s1">&#39;u&#39;</span><span class="p">,</span> <span class="n">axis</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="o">=+</span><span class="mi">1</span><span class="p">,</span> <span class="n">v</span><span class="o">=</span><span class="n">alpha</span> <span class="o">*</span> <span class="n">x</span> <span class="o">+</span> <span class="n">beta</span> <span class="o">+</span> <span class="n">gamma</span><span class="p">,</span> <span class="n">kind</span><span class="o">=</span><span class="s1">&#39;Dirichlet&#39;</span><span class="p">,</span> <span class="n">line</span><span class="o">=-</span><span class="mi">2</span>
+            <span class="p">)</span>
+        <span class="k">else</span><span class="p">:</span>
+            <span class="k">assert</span> <span class="n">c</span> <span class="o">==</span> <span class="n">b</span><span class="p">,</span> <span class="sa">f</span><span class="s1">&#39;Need periodic boundary conditions in y for </span><span class="si">{</span><span class="n">base_y</span><span class="si">}</span><span class="s1"> method!&#39;</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">setup_BCs</span><span class="p">()</span>
+
+<div class="viewcode-block" id="Heat2DUltraspherical.eval_f">
+<a class="viewcode-back" href="../../../pySDC/implementations.problem_classes.HeatEquation_Chebychev.html#implementations.problem_classes.HeatEquation_Chebychev.Heat2DUltraspherical.eval_f">[docs]</a>
+    <span class="k">def</span> <span class="nf">eval_f</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
+        <span class="n">f</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">f_init</span>
+        <span class="n">iu</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">index</span><span class="p">(</span><span class="s1">&#39;u&#39;</span><span class="p">)</span>
+
+        <span class="n">me_hat</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">u_init_forward</span>
+        <span class="n">me_hat</span><span class="p">[:]</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">transform</span><span class="p">(</span><span class="n">u</span><span class="p">)</span>
+        <span class="n">me_hat</span><span class="p">[</span><span class="n">iu</span><span class="p">]</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">nu</span> <span class="o">*</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">S2</span> <span class="o">@</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">Dxx</span> <span class="o">+</span> <span class="bp">self</span><span class="o">.</span><span class="n">Dyy</span><span class="p">)</span> <span class="o">@</span> <span class="n">me_hat</span><span class="p">[</span><span class="n">iu</span><span class="p">]</span><span class="o">.</span><span class="n">flatten</span><span class="p">())</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="n">me_hat</span><span class="p">[</span><span class="n">iu</span><span class="p">]</span><span class="o">.</span><span class="n">shape</span><span class="p">)</span>
+        <span class="n">me</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">itransform</span><span class="p">(</span><span class="n">me_hat</span><span class="p">)</span>
+
+        <span class="n">f</span><span class="p">[</span><span class="n">iu</span><span class="p">]</span> <span class="o">=</span> <span class="n">me</span><span class="p">[</span><span class="n">iu</span><span class="p">]</span><span class="o">.</span><span class="n">real</span>
+        <span class="k">return</span> <span class="n">f</span></div>
+
+
+<div class="viewcode-block" id="Heat2DUltraspherical.u_exact">
+<a class="viewcode-back" href="../../../pySDC/implementations.problem_classes.HeatEquation_Chebychev.html#implementations.problem_classes.HeatEquation_Chebychev.Heat2DUltraspherical.u_exact">[docs]</a>
+    <span class="k">def</span> <span class="nf">u_exact</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">t</span><span class="p">):</span>
+        <span class="n">xp</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">xp</span>
+        <span class="n">iu</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">index</span><span class="p">(</span><span class="s1">&#39;u&#39;</span><span class="p">)</span>
+        <span class="n">u</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">u_init</span>
+
+        <span class="n">fx</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">fx</span> <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">base_x</span> <span class="o">==</span> <span class="s1">&#39;fft&#39;</span> <span class="k">else</span> <span class="n">np</span><span class="o">.</span><span class="n">pi</span> <span class="o">*</span> <span class="bp">self</span><span class="o">.</span><span class="n">fx</span>
+        <span class="n">fy</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">fy</span> <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">base_y</span> <span class="o">==</span> <span class="s1">&#39;fft&#39;</span> <span class="k">else</span> <span class="n">np</span><span class="o">.</span><span class="n">pi</span> <span class="o">*</span> <span class="bp">self</span><span class="o">.</span><span class="n">fy</span>
+
+        <span class="n">time_dep</span> <span class="o">=</span> <span class="n">xp</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="bp">self</span><span class="o">.</span><span class="n">nu</span> <span class="o">*</span> <span class="p">(</span><span class="n">fx</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">fy</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span> <span class="o">*</span> <span class="n">t</span><span class="p">)</span>
+
+        <span class="n">alpha</span> <span class="o">=</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">b</span> <span class="o">-</span> <span class="bp">self</span><span class="o">.</span><span class="n">a</span><span class="p">)</span> <span class="o">/</span> <span class="mf">2.0</span>
+        <span class="n">beta</span> <span class="o">=</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">c</span> <span class="o">-</span> <span class="bp">self</span><span class="o">.</span><span class="n">b</span><span class="p">)</span> <span class="o">/</span> <span class="mf">2.0</span>
+        <span class="n">gamma</span> <span class="o">=</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">c</span> <span class="o">+</span> <span class="bp">self</span><span class="o">.</span><span class="n">a</span><span class="p">)</span> <span class="o">/</span> <span class="mf">2.0</span>
+
+        <span class="n">u</span><span class="p">[</span><span class="n">iu</span><span class="p">]</span> <span class="o">=</span> <span class="n">xp</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="n">fx</span> <span class="o">*</span> <span class="bp">self</span><span class="o">.</span><span class="n">X</span><span class="p">)</span> <span class="o">*</span> <span class="n">xp</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="n">fy</span> <span class="o">*</span> <span class="bp">self</span><span class="o">.</span><span class="n">Y</span><span class="p">)</span> <span class="o">*</span> <span class="n">time_dep</span> <span class="o">+</span> <span class="n">alpha</span> <span class="o">*</span> <span class="bp">self</span><span class="o">.</span><span class="n">X</span> <span class="o">+</span> <span class="n">beta</span> <span class="o">*</span> <span class="bp">self</span><span class="o">.</span><span class="n">Y</span> <span class="o">+</span> <span class="n">gamma</span>
+
+        <span class="k">return</span> <span class="n">u</span></div>
+</div>
+
+</pre></div>
+
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diff --git a/_modules/implementations/problem_classes/RayleighBenard.html b/_modules/implementations/problem_classes/RayleighBenard.html
new file mode 100644
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+        <li class="nav-item nav-item-this"><a href="">implementations.problem_classes.RayleighBenard</a></li> 
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+            
+  <h1>Source code for implementations.problem_classes.RayleighBenard</h1><div class="highlight"><pre>
+<span></span><span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
+<span class="kn">from</span> <span class="nn">mpi4py</span> <span class="kn">import</span> <span class="n">MPI</span>
+
+<span class="kn">from</span> <span class="nn">pySDC.implementations.problem_classes.generic_spectral</span> <span class="kn">import</span> <span class="n">GenericSpectralLinear</span>
+<span class="kn">from</span> <span class="nn">pySDC.implementations.datatype_classes.mesh</span> <span class="kn">import</span> <span class="n">mesh</span><span class="p">,</span> <span class="n">imex_mesh</span>
+<span class="kn">from</span> <span class="nn">pySDC.core.convergence_controller</span> <span class="kn">import</span> <span class="n">ConvergenceController</span>
+<span class="kn">from</span> <span class="nn">pySDC.core.hooks</span> <span class="kn">import</span> <span class="n">Hooks</span>
+<span class="kn">from</span> <span class="nn">pySDC.implementations.convergence_controller_classes.check_convergence</span> <span class="kn">import</span> <span class="n">CheckConvergence</span>
+
+
+<div class="viewcode-block" id="RayleighBenard">
+<a class="viewcode-back" href="../../../pySDC/implementations.problem_classes.RayleighBenard.html#implementations.problem_classes.RayleighBenard.RayleighBenard">[docs]</a>
+<span class="k">class</span> <span class="nc">RayleighBenard</span><span class="p">(</span><span class="n">GenericSpectralLinear</span><span class="p">):</span>
+<span class="w">    </span><span class="sd">&quot;&quot;&quot;</span>
+<span class="sd">    Rayleigh-Benard Convection is a variation of incompressible Navier-Stokes.</span>
+
+<span class="sd">    The equations we solve are</span>
+
+<span class="sd">        u_x + v_z = 0</span>
+<span class="sd">        T_t - kappa (T_xx + T_zz) = -uT_x - vT_z</span>
+<span class="sd">        u_t - nu (u_xx + u_zz) + p_x = -uu_x - vu_z</span>
+<span class="sd">        v_t - nu (v_xx + v_zz) + p_z - T = -uv_x - vv_z</span>
+
+<span class="sd">    with u the horizontal velocity, v the vertical velocity (in z-direction), T the temperature, p the pressure, indices</span>
+<span class="sd">    denoting derivatives, kappa=(Rayleigh * Prandl)**(-1/2) and nu = (Rayleigh / Prandl)**(-1/2). Everything on the left</span>
+<span class="sd">    hand side, that is the viscous part, the pressure gradient and the buoyancy due to temperature are treated</span>
+<span class="sd">    implicitly, while the non-linear convection part on the right hand side is integrated explicitly.</span>
+
+<span class="sd">    The domain, vertical boundary conditions and pressure gauge are</span>
+
+<span class="sd">        Omega = [0, 8) x (-1, 1)</span>
+<span class="sd">        T(z=+1) = 0</span>
+<span class="sd">        T(z=-1) = 2</span>
+<span class="sd">        u(z=+-1) = v(z=+-1) = 0</span>
+<span class="sd">        integral over p = 0</span>
+
+<span class="sd">    The spectral discretization uses FFT horizontally, implying periodic BCs, and an ultraspherical method vertically to</span>
+<span class="sd">    facilitate the Dirichlet BCs.</span>
+
+<span class="sd">    Parameters:</span>
+<span class="sd">        Prandl (float): Prandl number</span>
+<span class="sd">        Rayleigh (float): Rayleigh number</span>
+<span class="sd">        nx (int): Horizontal resolution</span>
+<span class="sd">        nz (int): Vertical resolution</span>
+<span class="sd">        BCs (dict): Can specify boundary conditions here</span>
+<span class="sd">        dealiasing (float): Dealiasing factor for evaluating the non-linear part</span>
+<span class="sd">        comm (mpi4py.Intracomm): Space communicator</span>
+<span class="sd">    &quot;&quot;&quot;</span>
+
+    <span class="n">dtype_u</span> <span class="o">=</span> <span class="n">mesh</span>
+    <span class="n">dtype_f</span> <span class="o">=</span> <span class="n">imex_mesh</span>
+
+    <span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span>
+        <span class="bp">self</span><span class="p">,</span>
+        <span class="n">Prandl</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span>
+        <span class="n">Rayleigh</span><span class="o">=</span><span class="mf">2e6</span><span class="p">,</span>
+        <span class="n">nx</span><span class="o">=</span><span class="mi">256</span><span class="p">,</span>
+        <span class="n">nz</span><span class="o">=</span><span class="mi">64</span><span class="p">,</span>
+        <span class="n">BCs</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span>
+        <span class="n">dealiasing</span><span class="o">=</span><span class="mi">3</span> <span class="o">/</span> <span class="mi">2</span><span class="p">,</span>
+        <span class="n">comm</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span>
+        <span class="o">**</span><span class="n">kwargs</span><span class="p">,</span>
+    <span class="p">):</span>
+<span class="w">        </span><span class="sd">&quot;&quot;&quot;</span>
+<span class="sd">        Constructor. `kwargs` are forwarded to parent class constructor.</span>
+
+<span class="sd">        Args:</span>
+<span class="sd">            Prandl (float): Prandtl number</span>
+<span class="sd">            Rayleigh (float): Rayleigh number</span>
+<span class="sd">            nx (int): Resolution in x-direction</span>
+<span class="sd">            nz (int): Resolution in z direction</span>
+<span class="sd">            BCs (dict): Vertical boundary conditions</span>
+<span class="sd">            dealiasing (float): Dealiasing for evaluating the non-linear part in real space</span>
+<span class="sd">            comm (mpi4py.Intracomm): Space communicator</span>
+<span class="sd">        &quot;&quot;&quot;</span>
+        <span class="n">BCs</span> <span class="o">=</span> <span class="p">{}</span> <span class="k">if</span> <span class="n">BCs</span> <span class="ow">is</span> <span class="kc">None</span> <span class="k">else</span> <span class="n">BCs</span>
+        <span class="n">BCs</span> <span class="o">=</span> <span class="p">{</span>
+            <span class="s1">&#39;T_top&#39;</span><span class="p">:</span> <span class="mi">0</span><span class="p">,</span>
+            <span class="s1">&#39;T_bottom&#39;</span><span class="p">:</span> <span class="mi">2</span><span class="p">,</span>
+            <span class="s1">&#39;v_top&#39;</span><span class="p">:</span> <span class="mi">0</span><span class="p">,</span>
+            <span class="s1">&#39;v_bottom&#39;</span><span class="p">:</span> <span class="mi">0</span><span class="p">,</span>
+            <span class="s1">&#39;u_top&#39;</span><span class="p">:</span> <span class="mi">0</span><span class="p">,</span>
+            <span class="s1">&#39;u_bottom&#39;</span><span class="p">:</span> <span class="mi">0</span><span class="p">,</span>
+            <span class="s1">&#39;p_integral&#39;</span><span class="p">:</span> <span class="mi">0</span><span class="p">,</span>
+            <span class="o">**</span><span class="n">BCs</span><span class="p">,</span>
+        <span class="p">}</span>
+        <span class="k">if</span> <span class="n">comm</span> <span class="ow">is</span> <span class="kc">None</span><span class="p">:</span>
+            <span class="k">try</span><span class="p">:</span>
+                <span class="kn">from</span> <span class="nn">mpi4py</span> <span class="kn">import</span> <span class="n">MPI</span>
+
+                <span class="n">comm</span> <span class="o">=</span> <span class="n">MPI</span><span class="o">.</span><span class="n">COMM_WORLD</span>
+            <span class="k">except</span> <span class="ne">ModuleNotFoundError</span><span class="p">:</span>
+                <span class="k">pass</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">_makeAttributeAndRegister</span><span class="p">(</span>
+            <span class="s1">&#39;Prandl&#39;</span><span class="p">,</span>
+            <span class="s1">&#39;Rayleigh&#39;</span><span class="p">,</span>
+            <span class="s1">&#39;nx&#39;</span><span class="p">,</span>
+            <span class="s1">&#39;nz&#39;</span><span class="p">,</span>
+            <span class="s1">&#39;BCs&#39;</span><span class="p">,</span>
+            <span class="s1">&#39;dealiasing&#39;</span><span class="p">,</span>
+            <span class="s1">&#39;comm&#39;</span><span class="p">,</span>
+            <span class="n">localVars</span><span class="o">=</span><span class="nb">locals</span><span class="p">(),</span>
+            <span class="n">readOnly</span><span class="o">=</span><span class="kc">True</span><span class="p">,</span>
+        <span class="p">)</span>
+
+        <span class="n">bases</span> <span class="o">=</span> <span class="p">[{</span><span class="s1">&#39;base&#39;</span><span class="p">:</span> <span class="s1">&#39;fft&#39;</span><span class="p">,</span> <span class="s1">&#39;N&#39;</span><span class="p">:</span> <span class="n">nx</span><span class="p">,</span> <span class="s1">&#39;x0&#39;</span><span class="p">:</span> <span class="mi">0</span><span class="p">,</span> <span class="s1">&#39;x1&#39;</span><span class="p">:</span> <span class="mi">8</span><span class="p">},</span> <span class="p">{</span><span class="s1">&#39;base&#39;</span><span class="p">:</span> <span class="s1">&#39;ultraspherical&#39;</span><span class="p">,</span> <span class="s1">&#39;N&#39;</span><span class="p">:</span> <span class="n">nz</span><span class="p">}]</span>
+        <span class="n">components</span> <span class="o">=</span> <span class="p">[</span><span class="s1">&#39;u&#39;</span><span class="p">,</span> <span class="s1">&#39;v&#39;</span><span class="p">,</span> <span class="s1">&#39;T&#39;</span><span class="p">,</span> <span class="s1">&#39;p&#39;</span><span class="p">]</span>
+        <span class="nb">super</span><span class="p">()</span><span class="o">.</span><span class="fm">__init__</span><span class="p">(</span><span class="n">bases</span><span class="p">,</span> <span class="n">components</span><span class="p">,</span> <span class="n">comm</span><span class="o">=</span><span class="n">comm</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
+
+        <span class="bp">self</span><span class="o">.</span><span class="n">Z</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">X</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">get_grid</span><span class="p">()</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">Kz</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">Kx</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">get_wavenumbers</span><span class="p">()</span>
+
+        <span class="c1"># construct 2D matrices</span>
+        <span class="n">Dzz</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">get_differentiation_matrix</span><span class="p">(</span><span class="n">axes</span><span class="o">=</span><span class="p">(</span><span class="mi">1</span><span class="p">,),</span> <span class="n">p</span><span class="o">=</span><span class="mi">2</span><span class="p">)</span>
+        <span class="n">Dz</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">get_differentiation_matrix</span><span class="p">(</span><span class="n">axes</span><span class="o">=</span><span class="p">(</span><span class="mi">1</span><span class="p">,))</span>
+        <span class="n">Dx</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">get_differentiation_matrix</span><span class="p">(</span><span class="n">axes</span><span class="o">=</span><span class="p">(</span><span class="mi">0</span><span class="p">,))</span>
+        <span class="n">Dxx</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">get_differentiation_matrix</span><span class="p">(</span><span class="n">axes</span><span class="o">=</span><span class="p">(</span><span class="mi">0</span><span class="p">,),</span> <span class="n">p</span><span class="o">=</span><span class="mi">2</span><span class="p">)</span>
+        <span class="n">Id</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">get_Id</span><span class="p">()</span>
+
+        <span class="n">S1</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">get_basis_change_matrix</span><span class="p">(</span><span class="n">p_out</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span> <span class="n">p_in</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
+        <span class="n">S2</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">get_basis_change_matrix</span><span class="p">(</span><span class="n">p_out</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span> <span class="n">p_in</span><span class="o">=</span><span class="mi">2</span><span class="p">)</span>
+
+        <span class="n">U01</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">get_basis_change_matrix</span><span class="p">(</span><span class="n">p_in</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span> <span class="n">p_out</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
+        <span class="n">U12</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">get_basis_change_matrix</span><span class="p">(</span><span class="n">p_in</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">p_out</span><span class="o">=</span><span class="mi">2</span><span class="p">)</span>
+        <span class="n">U02</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">get_basis_change_matrix</span><span class="p">(</span><span class="n">p_in</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span> <span class="n">p_out</span><span class="o">=</span><span class="mi">2</span><span class="p">)</span>
+
+        <span class="bp">self</span><span class="o">.</span><span class="n">Dx</span> <span class="o">=</span> <span class="n">Dx</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">Dxx</span> <span class="o">=</span> <span class="n">Dxx</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">Dz</span> <span class="o">=</span> <span class="n">S1</span> <span class="o">@</span> <span class="n">Dz</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">Dzz</span> <span class="o">=</span> <span class="n">S2</span> <span class="o">@</span> <span class="n">Dzz</span>
+
+        <span class="n">kappa</span> <span class="o">=</span> <span class="p">(</span><span class="n">Rayleigh</span> <span class="o">*</span> <span class="n">Prandl</span><span class="p">)</span> <span class="o">**</span> <span class="p">(</span><span class="o">-</span><span class="mi">1</span> <span class="o">/</span> <span class="mf">2.0</span><span class="p">)</span>
+        <span class="n">nu</span> <span class="o">=</span> <span class="p">(</span><span class="n">Rayleigh</span> <span class="o">/</span> <span class="n">Prandl</span><span class="p">)</span> <span class="o">**</span> <span class="p">(</span><span class="o">-</span><span class="mi">1</span> <span class="o">/</span> <span class="mf">2.0</span><span class="p">)</span>
+
+        <span class="c1"># construct operators</span>
+        <span class="n">L_lhs</span> <span class="o">=</span> <span class="p">{</span>
+            <span class="s1">&#39;p&#39;</span><span class="p">:</span> <span class="p">{</span><span class="s1">&#39;u&#39;</span><span class="p">:</span> <span class="n">U01</span> <span class="o">@</span> <span class="n">Dx</span><span class="p">,</span> <span class="s1">&#39;v&#39;</span><span class="p">:</span> <span class="n">Dz</span><span class="p">},</span>  <span class="c1"># divergence free constraint</span>
+            <span class="s1">&#39;u&#39;</span><span class="p">:</span> <span class="p">{</span><span class="s1">&#39;p&#39;</span><span class="p">:</span> <span class="n">U02</span> <span class="o">@</span> <span class="n">Dx</span><span class="p">,</span> <span class="s1">&#39;u&#39;</span><span class="p">:</span> <span class="o">-</span><span class="n">nu</span> <span class="o">*</span> <span class="p">(</span><span class="n">U02</span> <span class="o">@</span> <span class="n">Dxx</span> <span class="o">+</span> <span class="n">Dzz</span><span class="p">)},</span>
+            <span class="s1">&#39;v&#39;</span><span class="p">:</span> <span class="p">{</span><span class="s1">&#39;p&#39;</span><span class="p">:</span> <span class="n">U12</span> <span class="o">@</span> <span class="n">Dz</span><span class="p">,</span> <span class="s1">&#39;v&#39;</span><span class="p">:</span> <span class="o">-</span><span class="n">nu</span> <span class="o">*</span> <span class="p">(</span><span class="n">U02</span> <span class="o">@</span> <span class="n">Dxx</span> <span class="o">+</span> <span class="n">Dzz</span><span class="p">),</span> <span class="s1">&#39;T&#39;</span><span class="p">:</span> <span class="o">-</span><span class="n">U02</span> <span class="o">@</span> <span class="n">Id</span><span class="p">},</span>
+            <span class="s1">&#39;T&#39;</span><span class="p">:</span> <span class="p">{</span><span class="s1">&#39;T&#39;</span><span class="p">:</span> <span class="o">-</span><span class="n">kappa</span> <span class="o">*</span> <span class="p">(</span><span class="n">U02</span> <span class="o">@</span> <span class="n">Dxx</span> <span class="o">+</span> <span class="n">Dzz</span><span class="p">)},</span>
+        <span class="p">}</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">setup_L</span><span class="p">(</span><span class="n">L_lhs</span><span class="p">)</span>
+
+        <span class="c1"># mass matrix</span>
+        <span class="n">M_lhs</span> <span class="o">=</span> <span class="p">{</span><span class="n">i</span><span class="p">:</span> <span class="p">{</span><span class="n">i</span><span class="p">:</span> <span class="n">U02</span> <span class="o">@</span> <span class="n">Id</span><span class="p">}</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="p">[</span><span class="s1">&#39;u&#39;</span><span class="p">,</span> <span class="s1">&#39;v&#39;</span><span class="p">,</span> <span class="s1">&#39;T&#39;</span><span class="p">]}</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">setup_M</span><span class="p">(</span><span class="n">M_lhs</span><span class="p">)</span>
+
+        <span class="c1"># Prepare going from second (first for divergence free equation) derivative basis back to Chebychov-T</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">base_change</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_setup_operator</span><span class="p">({</span><span class="o">**</span><span class="p">{</span><span class="n">comp</span><span class="p">:</span> <span class="p">{</span><span class="n">comp</span><span class="p">:</span> <span class="n">S2</span><span class="p">}</span> <span class="k">for</span> <span class="n">comp</span> <span class="ow">in</span> <span class="p">[</span><span class="s1">&#39;u&#39;</span><span class="p">,</span> <span class="s1">&#39;v&#39;</span><span class="p">,</span> <span class="s1">&#39;T&#39;</span><span class="p">]},</span> <span class="s1">&#39;p&#39;</span><span class="p">:</span> <span class="p">{</span><span class="s1">&#39;p&#39;</span><span class="p">:</span> <span class="n">S1</span><span class="p">}})</span>
+
+        <span class="c1"># BCs</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">add_BC</span><span class="p">(</span>
+            <span class="n">component</span><span class="o">=</span><span class="s1">&#39;p&#39;</span><span class="p">,</span> <span class="n">equation</span><span class="o">=</span><span class="s1">&#39;p&#39;</span><span class="p">,</span> <span class="n">axis</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">v</span><span class="o">=</span><span class="bp">self</span><span class="o">.</span><span class="n">BCs</span><span class="p">[</span><span class="s1">&#39;p_integral&#39;</span><span class="p">],</span> <span class="n">kind</span><span class="o">=</span><span class="s1">&#39;integral&#39;</span><span class="p">,</span> <span class="n">line</span><span class="o">=-</span><span class="mi">1</span><span class="p">,</span> <span class="n">scalar</span><span class="o">=</span><span class="kc">True</span>
+        <span class="p">)</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">add_BC</span><span class="p">(</span><span class="n">component</span><span class="o">=</span><span class="s1">&#39;T&#39;</span><span class="p">,</span> <span class="n">equation</span><span class="o">=</span><span class="s1">&#39;T&#39;</span><span class="p">,</span> <span class="n">axis</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="o">=-</span><span class="mi">1</span><span class="p">,</span> <span class="n">v</span><span class="o">=</span><span class="bp">self</span><span class="o">.</span><span class="n">BCs</span><span class="p">[</span><span class="s1">&#39;T_bottom&#39;</span><span class="p">],</span> <span class="n">kind</span><span class="o">=</span><span class="s1">&#39;Dirichlet&#39;</span><span class="p">,</span> <span class="n">line</span><span class="o">=-</span><span class="mi">1</span><span class="p">)</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">add_BC</span><span class="p">(</span><span class="n">component</span><span class="o">=</span><span class="s1">&#39;T&#39;</span><span class="p">,</span> <span class="n">equation</span><span class="o">=</span><span class="s1">&#39;T&#39;</span><span class="p">,</span> <span class="n">axis</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">v</span><span class="o">=</span><span class="bp">self</span><span class="o">.</span><span class="n">BCs</span><span class="p">[</span><span class="s1">&#39;T_top&#39;</span><span class="p">],</span> <span class="n">kind</span><span class="o">=</span><span class="s1">&#39;Dirichlet&#39;</span><span class="p">,</span> <span class="n">line</span><span class="o">=-</span><span class="mi">2</span><span class="p">)</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">add_BC</span><span class="p">(</span><span class="n">component</span><span class="o">=</span><span class="s1">&#39;v&#39;</span><span class="p">,</span> <span class="n">equation</span><span class="o">=</span><span class="s1">&#39;v&#39;</span><span class="p">,</span> <span class="n">axis</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">v</span><span class="o">=</span><span class="bp">self</span><span class="o">.</span><span class="n">BCs</span><span class="p">[</span><span class="s1">&#39;v_bottom&#39;</span><span class="p">],</span> <span class="n">kind</span><span class="o">=</span><span class="s1">&#39;Dirichlet&#39;</span><span class="p">,</span> <span class="n">line</span><span class="o">=-</span><span class="mi">1</span><span class="p">)</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">add_BC</span><span class="p">(</span><span class="n">component</span><span class="o">=</span><span class="s1">&#39;v&#39;</span><span class="p">,</span> <span class="n">equation</span><span class="o">=</span><span class="s1">&#39;v&#39;</span><span class="p">,</span> <span class="n">axis</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="o">=-</span><span class="mi">1</span><span class="p">,</span> <span class="n">v</span><span class="o">=</span><span class="bp">self</span><span class="o">.</span><span class="n">BCs</span><span class="p">[</span><span class="s1">&#39;v_bottom&#39;</span><span class="p">],</span> <span class="n">kind</span><span class="o">=</span><span class="s1">&#39;Dirichlet&#39;</span><span class="p">,</span> <span class="n">line</span><span class="o">=-</span><span class="mi">2</span><span class="p">)</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">remove_BC</span><span class="p">(</span><span class="n">component</span><span class="o">=</span><span class="s1">&#39;v&#39;</span><span class="p">,</span> <span class="n">equation</span><span class="o">=</span><span class="s1">&#39;v&#39;</span><span class="p">,</span> <span class="n">axis</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="o">=-</span><span class="mi">1</span><span class="p">,</span> <span class="n">kind</span><span class="o">=</span><span class="s1">&#39;Dirichlet&#39;</span><span class="p">,</span> <span class="n">line</span><span class="o">=-</span><span class="mi">2</span><span class="p">,</span> <span class="n">scalar</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">add_BC</span><span class="p">(</span><span class="n">component</span><span class="o">=</span><span class="s1">&#39;u&#39;</span><span class="p">,</span> <span class="n">equation</span><span class="o">=</span><span class="s1">&#39;u&#39;</span><span class="p">,</span> <span class="n">axis</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">v</span><span class="o">=</span><span class="bp">self</span><span class="o">.</span><span class="n">BCs</span><span class="p">[</span><span class="s1">&#39;u_top&#39;</span><span class="p">],</span> <span class="n">x</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">kind</span><span class="o">=</span><span class="s1">&#39;Dirichlet&#39;</span><span class="p">,</span> <span class="n">line</span><span class="o">=-</span><span class="mi">2</span><span class="p">)</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">add_BC</span><span class="p">(</span>
+            <span class="n">component</span><span class="o">=</span><span class="s1">&#39;u&#39;</span><span class="p">,</span>
+            <span class="n">equation</span><span class="o">=</span><span class="s1">&#39;u&#39;</span><span class="p">,</span>
+            <span class="n">axis</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span>
+            <span class="n">v</span><span class="o">=</span><span class="bp">self</span><span class="o">.</span><span class="n">BCs</span><span class="p">[</span><span class="s1">&#39;u_bottom&#39;</span><span class="p">],</span>
+            <span class="n">x</span><span class="o">=-</span><span class="mi">1</span><span class="p">,</span>
+            <span class="n">kind</span><span class="o">=</span><span class="s1">&#39;Dirichlet&#39;</span><span class="p">,</span>
+            <span class="n">line</span><span class="o">=-</span><span class="mi">1</span><span class="p">,</span>
+        <span class="p">)</span>
+
+        <span class="c1"># eliminate Nyquist mode if needed</span>
+        <span class="k">if</span> <span class="n">nx</span> <span class="o">%</span> <span class="mi">2</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
+            <span class="n">Nyquist_mode_index</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">get_Nyquist_mode_index</span><span class="p">()</span>
+            <span class="k">for</span> <span class="n">component</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">components</span><span class="p">:</span>
+                <span class="bp">self</span><span class="o">.</span><span class="n">add_BC</span><span class="p">(</span>
+                    <span class="n">component</span><span class="o">=</span><span class="n">component</span><span class="p">,</span> <span class="n">equation</span><span class="o">=</span><span class="n">component</span><span class="p">,</span> <span class="n">axis</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span> <span class="n">kind</span><span class="o">=</span><span class="s1">&#39;Nyquist&#39;</span><span class="p">,</span> <span class="n">line</span><span class="o">=</span><span class="nb">int</span><span class="p">(</span><span class="n">Nyquist_mode_index</span><span class="p">),</span> <span class="n">v</span><span class="o">=</span><span class="mi">0</span>
+                <span class="p">)</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">setup_BCs</span><span class="p">()</span>
+
+<div class="viewcode-block" id="RayleighBenard.eval_f">
+<a class="viewcode-back" href="../../../pySDC/implementations.problem_classes.RayleighBenard.html#implementations.problem_classes.RayleighBenard.RayleighBenard.eval_f">[docs]</a>
+    <span class="k">def</span> <span class="nf">eval_f</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
+        <span class="n">f</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">f_init</span>
+
+        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">spectral_space</span><span class="p">:</span>
+            <span class="n">u_hat</span> <span class="o">=</span> <span class="n">u</span><span class="o">.</span><span class="n">copy</span><span class="p">()</span>
+        <span class="k">else</span><span class="p">:</span>
+            <span class="n">u_hat</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">transform</span><span class="p">(</span><span class="n">u</span><span class="p">)</span>
+
+        <span class="n">f_impl_hat</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">u_init_forward</span>
+
+        <span class="n">Dz</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">Dz</span>
+        <span class="n">Dx</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">Dx</span>
+
+        <span class="n">iu</span><span class="p">,</span> <span class="n">iv</span><span class="p">,</span> <span class="n">iT</span><span class="p">,</span> <span class="n">ip</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">index</span><span class="p">([</span><span class="s1">&#39;u&#39;</span><span class="p">,</span> <span class="s1">&#39;v&#39;</span><span class="p">,</span> <span class="s1">&#39;T&#39;</span><span class="p">,</span> <span class="s1">&#39;p&#39;</span><span class="p">])</span>
+
+        <span class="c1"># evaluate implicit terms</span>
+        <span class="n">f_impl_hat</span> <span class="o">=</span> <span class="o">-</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">base_change</span> <span class="o">@</span> <span class="bp">self</span><span class="o">.</span><span class="n">L</span> <span class="o">@</span> <span class="n">u_hat</span><span class="o">.</span><span class="n">flatten</span><span class="p">())</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="n">u_hat</span><span class="o">.</span><span class="n">shape</span><span class="p">)</span>
+
+        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">spectral_space</span><span class="p">:</span>
+            <span class="n">f</span><span class="o">.</span><span class="n">impl</span><span class="p">[:]</span> <span class="o">=</span> <span class="n">f_impl_hat</span>
+        <span class="k">else</span><span class="p">:</span>
+            <span class="n">f</span><span class="o">.</span><span class="n">impl</span><span class="p">[:]</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">itransform</span><span class="p">(</span><span class="n">f_impl_hat</span><span class="p">)</span><span class="o">.</span><span class="n">real</span>
+
+        <span class="c1"># treat convection explicitly with dealiasing</span>
+        <span class="n">Dx_u_hat</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">u_init_forward</span>
+        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="p">[</span><span class="n">iu</span><span class="p">,</span> <span class="n">iv</span><span class="p">,</span> <span class="n">iT</span><span class="p">]:</span>
+            <span class="n">Dx_u_hat</span><span class="p">[</span><span class="n">i</span><span class="p">][:]</span> <span class="o">=</span> <span class="p">(</span><span class="n">Dx</span> <span class="o">@</span> <span class="n">u_hat</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">.</span><span class="n">flatten</span><span class="p">())</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="n">Dx_u_hat</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">.</span><span class="n">shape</span><span class="p">)</span>
+        <span class="n">Dz_u_hat</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">u_init_forward</span>
+        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="p">[</span><span class="n">iu</span><span class="p">,</span> <span class="n">iv</span><span class="p">,</span> <span class="n">iT</span><span class="p">]:</span>
+            <span class="n">Dz_u_hat</span><span class="p">[</span><span class="n">i</span><span class="p">][:]</span> <span class="o">=</span> <span class="p">(</span><span class="n">Dz</span> <span class="o">@</span> <span class="n">u_hat</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">.</span><span class="n">flatten</span><span class="p">())</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="n">Dz_u_hat</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">.</span><span class="n">shape</span><span class="p">)</span>
+
+        <span class="n">padding</span> <span class="o">=</span> <span class="p">[</span><span class="bp">self</span><span class="o">.</span><span class="n">dealiasing</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">dealiasing</span><span class="p">]</span>
+        <span class="n">Dx_u_pad</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">itransform</span><span class="p">(</span><span class="n">Dx_u_hat</span><span class="p">,</span> <span class="n">padding</span><span class="o">=</span><span class="n">padding</span><span class="p">)</span><span class="o">.</span><span class="n">real</span>
+        <span class="n">Dz_u_pad</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">itransform</span><span class="p">(</span><span class="n">Dz_u_hat</span><span class="p">,</span> <span class="n">padding</span><span class="o">=</span><span class="n">padding</span><span class="p">)</span><span class="o">.</span><span class="n">real</span>
+        <span class="n">u_pad</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">itransform</span><span class="p">(</span><span class="n">u_hat</span><span class="p">,</span> <span class="n">padding</span><span class="o">=</span><span class="n">padding</span><span class="p">)</span><span class="o">.</span><span class="n">real</span>
+
+        <span class="n">fexpl_pad</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">xp</span><span class="o">.</span><span class="n">zeros_like</span><span class="p">(</span><span class="n">u_pad</span><span class="p">)</span>
+        <span class="n">fexpl_pad</span><span class="p">[</span><span class="n">iu</span><span class="p">][:]</span> <span class="o">=</span> <span class="o">-</span><span class="p">(</span><span class="n">u_pad</span><span class="p">[</span><span class="n">iu</span><span class="p">]</span> <span class="o">*</span> <span class="n">Dx_u_pad</span><span class="p">[</span><span class="n">iu</span><span class="p">]</span> <span class="o">+</span> <span class="n">u_pad</span><span class="p">[</span><span class="n">iv</span><span class="p">]</span> <span class="o">*</span> <span class="n">Dz_u_pad</span><span class="p">[</span><span class="n">iu</span><span class="p">])</span>
+        <span class="n">fexpl_pad</span><span class="p">[</span><span class="n">iv</span><span class="p">][:]</span> <span class="o">=</span> <span class="o">-</span><span class="p">(</span><span class="n">u_pad</span><span class="p">[</span><span class="n">iu</span><span class="p">]</span> <span class="o">*</span> <span class="n">Dx_u_pad</span><span class="p">[</span><span class="n">iv</span><span class="p">]</span> <span class="o">+</span> <span class="n">u_pad</span><span class="p">[</span><span class="n">iv</span><span class="p">]</span> <span class="o">*</span> <span class="n">Dz_u_pad</span><span class="p">[</span><span class="n">iv</span><span class="p">])</span>
+        <span class="n">fexpl_pad</span><span class="p">[</span><span class="n">iT</span><span class="p">][:]</span> <span class="o">=</span> <span class="o">-</span><span class="p">(</span><span class="n">u_pad</span><span class="p">[</span><span class="n">iu</span><span class="p">]</span> <span class="o">*</span> <span class="n">Dx_u_pad</span><span class="p">[</span><span class="n">iT</span><span class="p">]</span> <span class="o">+</span> <span class="n">u_pad</span><span class="p">[</span><span class="n">iv</span><span class="p">]</span> <span class="o">*</span> <span class="n">Dz_u_pad</span><span class="p">[</span><span class="n">iT</span><span class="p">])</span>
+
+        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">spectral_space</span><span class="p">:</span>
+            <span class="n">f</span><span class="o">.</span><span class="n">expl</span><span class="p">[:]</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">transform</span><span class="p">(</span><span class="n">fexpl_pad</span><span class="p">,</span> <span class="n">padding</span><span class="o">=</span><span class="n">padding</span><span class="p">)</span>
+        <span class="k">else</span><span class="p">:</span>
+            <span class="n">f</span><span class="o">.</span><span class="n">expl</span><span class="p">[:]</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">itransform</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">transform</span><span class="p">(</span><span class="n">fexpl_pad</span><span class="p">,</span> <span class="n">padding</span><span class="o">=</span><span class="n">padding</span><span class="p">))</span><span class="o">.</span><span class="n">real</span>
+
+        <span class="k">return</span> <span class="n">f</span></div>
+
+
+<div class="viewcode-block" id="RayleighBenard.u_exact">
+<a class="viewcode-back" href="../../../pySDC/implementations.problem_classes.RayleighBenard.html#implementations.problem_classes.RayleighBenard.RayleighBenard.u_exact">[docs]</a>
+    <span class="k">def</span> <span class="nf">u_exact</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">t</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span> <span class="n">noise_level</span><span class="o">=</span><span class="mf">1e-3</span><span class="p">,</span> <span class="n">seed</span><span class="o">=</span><span class="mi">99</span><span class="p">):</span>
+        <span class="k">assert</span> <span class="n">t</span> <span class="o">==</span> <span class="mi">0</span>
+        <span class="k">assert</span> <span class="p">(</span>
+            <span class="bp">self</span><span class="o">.</span><span class="n">BCs</span><span class="p">[</span><span class="s1">&#39;v_top&#39;</span><span class="p">]</span> <span class="o">==</span> <span class="bp">self</span><span class="o">.</span><span class="n">BCs</span><span class="p">[</span><span class="s1">&#39;v_bottom&#39;</span><span class="p">]</span>
+        <span class="p">),</span> <span class="s1">&#39;Initial conditions are only implemented for zero velocity gradient&#39;</span>
+
+        <span class="n">me</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">spectral</span><span class="o">.</span><span class="n">u_init</span>
+        <span class="n">iu</span><span class="p">,</span> <span class="n">iv</span><span class="p">,</span> <span class="n">iT</span><span class="p">,</span> <span class="n">ip</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">index</span><span class="p">([</span><span class="s1">&#39;u&#39;</span><span class="p">,</span> <span class="s1">&#39;v&#39;</span><span class="p">,</span> <span class="s1">&#39;T&#39;</span><span class="p">,</span> <span class="s1">&#39;p&#39;</span><span class="p">])</span>
+
+        <span class="c1"># linear temperature gradient</span>
+        <span class="k">for</span> <span class="n">comp</span> <span class="ow">in</span> <span class="p">[</span><span class="s1">&#39;T&#39;</span><span class="p">,</span> <span class="s1">&#39;v&#39;</span><span class="p">,</span> <span class="s1">&#39;u&#39;</span><span class="p">]:</span>
+            <span class="n">a</span> <span class="o">=</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">BCs</span><span class="p">[</span><span class="sa">f</span><span class="s1">&#39;</span><span class="si">{</span><span class="n">comp</span><span class="si">}</span><span class="s1">_top&#39;</span><span class="p">]</span> <span class="o">-</span> <span class="bp">self</span><span class="o">.</span><span class="n">BCs</span><span class="p">[</span><span class="sa">f</span><span class="s1">&#39;</span><span class="si">{</span><span class="n">comp</span><span class="si">}</span><span class="s1">_bottom&#39;</span><span class="p">])</span> <span class="o">/</span> <span class="mi">2</span>
+            <span class="n">b</span> <span class="o">=</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">BCs</span><span class="p">[</span><span class="sa">f</span><span class="s1">&#39;</span><span class="si">{</span><span class="n">comp</span><span class="si">}</span><span class="s1">_top&#39;</span><span class="p">]</span> <span class="o">+</span> <span class="bp">self</span><span class="o">.</span><span class="n">BCs</span><span class="p">[</span><span class="sa">f</span><span class="s1">&#39;</span><span class="si">{</span><span class="n">comp</span><span class="si">}</span><span class="s1">_bottom&#39;</span><span class="p">])</span> <span class="o">/</span> <span class="mi">2</span>
+            <span class="n">me</span><span class="p">[</span><span class="bp">self</span><span class="o">.</span><span class="n">index</span><span class="p">(</span><span class="n">comp</span><span class="p">)]</span> <span class="o">=</span> <span class="n">a</span> <span class="o">*</span> <span class="bp">self</span><span class="o">.</span><span class="n">Z</span> <span class="o">+</span> <span class="n">b</span>
+
+        <span class="c1"># perturb slightly</span>
+        <span class="n">rng</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">xp</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">default_rng</span><span class="p">(</span><span class="n">seed</span><span class="o">=</span><span class="n">seed</span><span class="p">)</span>
+
+        <span class="n">noise</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">spectral</span><span class="o">.</span><span class="n">u_init</span>
+        <span class="n">noise</span><span class="p">[</span><span class="n">iT</span><span class="p">]</span> <span class="o">=</span> <span class="n">rng</span><span class="o">.</span><span class="n">random</span><span class="p">(</span><span class="n">size</span><span class="o">=</span><span class="n">me</span><span class="p">[</span><span class="n">iT</span><span class="p">]</span><span class="o">.</span><span class="n">shape</span><span class="p">)</span>
+
+        <span class="n">me</span><span class="p">[</span><span class="n">iT</span><span class="p">]</span> <span class="o">+=</span> <span class="n">noise</span><span class="p">[</span><span class="n">iT</span><span class="p">]</span><span class="o">.</span><span class="n">real</span> <span class="o">*</span> <span class="n">noise_level</span> <span class="o">*</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">Z</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span> <span class="o">*</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">Z</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
+
+        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">spectral_space</span><span class="p">:</span>
+            <span class="n">me_hat</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">spectral</span><span class="o">.</span><span class="n">u_init_forward</span>
+            <span class="n">me_hat</span><span class="p">[:]</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">transform</span><span class="p">(</span><span class="n">me</span><span class="p">)</span>
+            <span class="k">return</span> <span class="n">me_hat</span>
+        <span class="k">else</span><span class="p">:</span>
+            <span class="k">return</span> <span class="n">me</span></div>
+
+
+<div class="viewcode-block" id="RayleighBenard.get_fig">
+<a class="viewcode-back" href="../../../pySDC/implementations.problem_classes.RayleighBenard.html#implementations.problem_classes.RayleighBenard.RayleighBenard.get_fig">[docs]</a>
+    <span class="k">def</span> <span class="nf">get_fig</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>  <span class="c1"># pragma: no cover</span>
+<span class="w">        </span><span class="sd">&quot;&quot;&quot;</span>
+<span class="sd">        Get a figure suitable to plot the solution of this problem</span>
+
+<span class="sd">        Returns</span>
+<span class="sd">        -------</span>
+<span class="sd">        self.fig : matplotlib.pyplot.figure.Figure</span>
+<span class="sd">        &quot;&quot;&quot;</span>
+        <span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
+        <span class="kn">from</span> <span class="nn">mpl_toolkits.axes_grid1</span> <span class="kn">import</span> <span class="n">make_axes_locatable</span>
+
+        <span class="n">plt</span><span class="o">.</span><span class="n">rcParams</span><span class="p">[</span><span class="s1">&#39;figure.constrained_layout.use&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="kc">True</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">fig</span><span class="p">,</span> <span class="n">axs</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">sharex</span><span class="o">=</span><span class="kc">True</span><span class="p">,</span> <span class="n">sharey</span><span class="o">=</span><span class="kc">True</span><span class="p">,</span> <span class="n">figsize</span><span class="o">=</span><span class="p">((</span><span class="mi">10</span><span class="p">,</span> <span class="mi">5</span><span class="p">)))</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">cax</span> <span class="o">=</span> <span class="p">[]</span>
+        <span class="n">divider</span> <span class="o">=</span> <span class="n">make_axes_locatable</span><span class="p">(</span><span class="n">axs</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">cax</span> <span class="o">+=</span> <span class="p">[</span><span class="n">divider</span><span class="o">.</span><span class="n">append_axes</span><span class="p">(</span><span class="s1">&#39;right&#39;</span><span class="p">,</span> <span class="n">size</span><span class="o">=</span><span class="s1">&#39;3%&#39;</span><span class="p">,</span> <span class="n">pad</span><span class="o">=</span><span class="mf">0.03</span><span class="p">)]</span>
+        <span class="n">divider2</span> <span class="o">=</span> <span class="n">make_axes_locatable</span><span class="p">(</span><span class="n">axs</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">cax</span> <span class="o">+=</span> <span class="p">[</span><span class="n">divider2</span><span class="o">.</span><span class="n">append_axes</span><span class="p">(</span><span class="s1">&#39;right&#39;</span><span class="p">,</span> <span class="n">size</span><span class="o">=</span><span class="s1">&#39;3%&#39;</span><span class="p">,</span> <span class="n">pad</span><span class="o">=</span><span class="mf">0.03</span><span class="p">)]</span>
+        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">fig</span></div>
+
+
+<div class="viewcode-block" id="RayleighBenard.plot">
+<a class="viewcode-back" href="../../../pySDC/implementations.problem_classes.RayleighBenard.html#implementations.problem_classes.RayleighBenard.RayleighBenard.plot">[docs]</a>
+    <span class="k">def</span> <span class="nf">plot</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">t</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">fig</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">quantity</span><span class="o">=</span><span class="s1">&#39;T&#39;</span><span class="p">):</span>  <span class="c1"># pragma: no cover</span>
+<span class="w">        </span><span class="sa">r</span><span class="sd">&quot;&quot;&quot;</span>
+<span class="sd">        Plot the solution.</span>
+
+<span class="sd">        Parameters</span>
+<span class="sd">        ----------</span>
+<span class="sd">        u : dtype_u</span>
+<span class="sd">            Solution to be plotted</span>
+<span class="sd">        t : float</span>
+<span class="sd">            Time to display at the top of the figure</span>
+<span class="sd">        fig : matplotlib.pyplot.figure.Figure</span>
+<span class="sd">            Figure with the same structure as a figure generated by `self.get_fig`. If none is supplied, a new figure will be generated.</span>
+<span class="sd">        quantity : (str)</span>
+<span class="sd">            quantity you want to plot</span>
+
+<span class="sd">        Returns</span>
+<span class="sd">        -------</span>
+<span class="sd">        None</span>
+<span class="sd">        &quot;&quot;&quot;</span>
+        <span class="n">fig</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">get_fig</span><span class="p">()</span> <span class="k">if</span> <span class="n">fig</span> <span class="ow">is</span> <span class="kc">None</span> <span class="k">else</span> <span class="n">fig</span>
+        <span class="n">axs</span> <span class="o">=</span> <span class="n">fig</span><span class="o">.</span><span class="n">axes</span>
+
+        <span class="n">imV</span> <span class="o">=</span> <span class="n">axs</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">pcolormesh</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">X</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">Z</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">compute_vorticity</span><span class="p">(</span><span class="n">u</span><span class="p">)</span><span class="o">.</span><span class="n">real</span><span class="p">)</span>
+
+        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">spectral_space</span><span class="p">:</span>
+            <span class="n">u</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">itransform</span><span class="p">(</span><span class="n">u</span><span class="p">)</span>
+
+        <span class="n">imT</span> <span class="o">=</span> <span class="n">axs</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">pcolormesh</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">X</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">Z</span><span class="p">,</span> <span class="n">u</span><span class="p">[</span><span class="bp">self</span><span class="o">.</span><span class="n">index</span><span class="p">(</span><span class="n">quantity</span><span class="p">)]</span><span class="o">.</span><span class="n">real</span><span class="p">)</span>
+
+        <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">label</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="sa">rf</span><span class="s1">&#39;$</span><span class="si">{</span><span class="n">quantity</span><span class="si">}</span><span class="s1">$&#39;</span><span class="p">,</span> <span class="s1">&#39;vorticity&#39;</span><span class="p">]):</span>
+            <span class="n">axs</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">.</span><span class="n">set_aspect</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
+            <span class="n">axs</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="n">label</span><span class="p">)</span>
+
+        <span class="k">if</span> <span class="n">t</span> <span class="ow">is</span> <span class="ow">not</span> <span class="kc">None</span><span class="p">:</span>
+            <span class="n">fig</span><span class="o">.</span><span class="n">suptitle</span><span class="p">(</span><span class="sa">f</span><span class="s1">&#39;t = </span><span class="si">{</span><span class="n">t</span><span class="si">:</span><span class="s1">.2f</span><span class="si">}</span><span class="s1">&#39;</span><span class="p">)</span>
+        <span class="n">axs</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="sa">r</span><span class="s1">&#39;$x$&#39;</span><span class="p">)</span>
+        <span class="n">axs</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="sa">r</span><span class="s1">&#39;$z$&#39;</span><span class="p">)</span>
+        <span class="n">fig</span><span class="o">.</span><span class="n">colorbar</span><span class="p">(</span><span class="n">imT</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">cax</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
+        <span class="n">fig</span><span class="o">.</span><span class="n">colorbar</span><span class="p">(</span><span class="n">imV</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">cax</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span></div>
+
+
+<div class="viewcode-block" id="RayleighBenard.compute_vorticity">
+<a class="viewcode-back" href="../../../pySDC/implementations.problem_classes.RayleighBenard.html#implementations.problem_classes.RayleighBenard.RayleighBenard.compute_vorticity">[docs]</a>
+    <span class="k">def</span> <span class="nf">compute_vorticity</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">u</span><span class="p">):</span>
+        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">spectral_space</span><span class="p">:</span>
+            <span class="n">u_hat</span> <span class="o">=</span> <span class="n">u</span><span class="o">.</span><span class="n">copy</span><span class="p">()</span>
+        <span class="k">else</span><span class="p">:</span>
+            <span class="n">u_hat</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">transform</span><span class="p">(</span><span class="n">u</span><span class="p">)</span>
+        <span class="n">Dz</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">Dz</span>
+        <span class="n">Dx</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">Dx</span>
+        <span class="n">iu</span><span class="p">,</span> <span class="n">iv</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">index</span><span class="p">([</span><span class="s1">&#39;u&#39;</span><span class="p">,</span> <span class="s1">&#39;v&#39;</span><span class="p">])</span>
+
+        <span class="n">vorticity_hat</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">spectral</span><span class="o">.</span><span class="n">u_init_forward</span>
+        <span class="n">vorticity_hat</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="p">(</span><span class="n">Dx</span> <span class="o">*</span> <span class="n">u_hat</span><span class="p">[</span><span class="n">iv</span><span class="p">]</span><span class="o">.</span><span class="n">flatten</span><span class="p">()</span> <span class="o">+</span> <span class="n">Dz</span> <span class="o">@</span> <span class="n">u_hat</span><span class="p">[</span><span class="n">iu</span><span class="p">]</span><span class="o">.</span><span class="n">flatten</span><span class="p">())</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="n">u</span><span class="p">[</span><span class="n">iu</span><span class="p">]</span><span class="o">.</span><span class="n">shape</span><span class="p">)</span>
+        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">itransform</span><span class="p">(</span><span class="n">vorticity_hat</span><span class="p">)[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">real</span></div>
+
+
+<div class="viewcode-block" id="RayleighBenard.compute_Nusselt_numbers">
+<a class="viewcode-back" href="../../../pySDC/implementations.problem_classes.RayleighBenard.html#implementations.problem_classes.RayleighBenard.RayleighBenard.compute_Nusselt_numbers">[docs]</a>
+    <span class="k">def</span> <span class="nf">compute_Nusselt_numbers</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">u</span><span class="p">):</span>
+<span class="w">        </span><span class="sd">&quot;&quot;&quot;</span>
+<span class="sd">        Compute the various versions of the Nusselt number. This reflects the type of heat transport.</span>
+<span class="sd">        If the Nusselt number is equal to one, it indicates heat transport due to conduction. If it is larger,</span>
+<span class="sd">        advection is present.</span>
+<span class="sd">        Computing the Nusselt number at various places can be used to check the code.</span>
+
+<span class="sd">        Args:</span>
+<span class="sd">            u: The solution you want to compute the Nusselt numbers of</span>
+
+<span class="sd">        Returns:</span>
+<span class="sd">            dict: Nusselt number averaged over the entire volume and horizontally averaged at the top and bottom.</span>
+<span class="sd">        &quot;&quot;&quot;</span>
+        <span class="n">iv</span><span class="p">,</span> <span class="n">iT</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">index</span><span class="p">([</span><span class="s1">&#39;v&#39;</span><span class="p">,</span> <span class="s1">&#39;T&#39;</span><span class="p">])</span>
+
+        <span class="n">DzT_hat</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">spectral</span><span class="o">.</span><span class="n">u_init_forward</span>
+
+        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">spectral_space</span><span class="p">:</span>
+            <span class="n">u_hat</span> <span class="o">=</span> <span class="n">u</span><span class="o">.</span><span class="n">copy</span><span class="p">()</span>
+        <span class="k">else</span><span class="p">:</span>
+            <span class="n">u_hat</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">transform</span><span class="p">(</span><span class="n">u</span><span class="p">)</span>
+
+        <span class="n">DzT_hat</span><span class="p">[</span><span class="n">iT</span><span class="p">]</span> <span class="o">=</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">Dz</span> <span class="o">@</span> <span class="n">u_hat</span><span class="p">[</span><span class="n">iT</span><span class="p">]</span><span class="o">.</span><span class="n">flatten</span><span class="p">())</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="n">DzT_hat</span><span class="p">[</span><span class="n">iT</span><span class="p">]</span><span class="o">.</span><span class="n">shape</span><span class="p">)</span>
+
+        <span class="c1"># compute vT with dealiasing</span>
+        <span class="n">padding</span> <span class="o">=</span> <span class="p">[</span><span class="bp">self</span><span class="o">.</span><span class="n">dealiasing</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">dealiasing</span><span class="p">]</span>
+        <span class="n">u_pad</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">itransform</span><span class="p">(</span><span class="n">u_hat</span><span class="p">,</span> <span class="n">padding</span><span class="o">=</span><span class="n">padding</span><span class="p">)</span><span class="o">.</span><span class="n">real</span>
+        <span class="n">_me</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">xp</span><span class="o">.</span><span class="n">zeros_like</span><span class="p">(</span><span class="n">u_pad</span><span class="p">)</span>
+        <span class="n">_me</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="n">u_pad</span><span class="p">[</span><span class="n">iv</span><span class="p">]</span> <span class="o">*</span> <span class="n">u_pad</span><span class="p">[</span><span class="n">iT</span><span class="p">]</span>
+        <span class="n">vT_hat</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">transform</span><span class="p">(</span><span class="n">_me</span><span class="p">,</span> <span class="n">padding</span><span class="o">=</span><span class="n">padding</span><span class="p">)</span>
+
+        <span class="n">nusselt_hat</span> <span class="o">=</span> <span class="p">(</span><span class="n">vT_hat</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">-</span> <span class="n">DzT_hat</span><span class="p">[</span><span class="n">iT</span><span class="p">])</span> <span class="o">/</span> <span class="bp">self</span><span class="o">.</span><span class="n">nx</span>
+        <span class="n">nusselt_no_v_hat</span> <span class="o">=</span> <span class="p">(</span><span class="o">-</span><span class="n">DzT_hat</span><span class="p">[</span><span class="n">iT</span><span class="p">])</span> <span class="o">/</span> <span class="bp">self</span><span class="o">.</span><span class="n">nx</span>
+
+        <span class="n">integral_z</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">xp</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">nusselt_hat</span> <span class="o">*</span> <span class="bp">self</span><span class="o">.</span><span class="n">spectral</span><span class="o">.</span><span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">get_BC</span><span class="p">(</span><span class="n">kind</span><span class="o">=</span><span class="s1">&#39;integral&#39;</span><span class="p">),</span> <span class="n">axis</span><span class="o">=-</span><span class="mi">1</span><span class="p">)</span><span class="o">.</span><span class="n">real</span>
+        <span class="n">integral_V</span> <span class="o">=</span> <span class="p">(</span>
+            <span class="n">integral_z</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">*</span> <span class="bp">self</span><span class="o">.</span><span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">L</span>
+        <span class="p">)</span>  <span class="c1"># only the first Fourier mode has non-zero integral with periodic BCs</span>
+        <span class="n">Nusselt_V</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">comm</span><span class="o">.</span><span class="n">bcast</span><span class="p">(</span><span class="n">integral_V</span> <span class="o">/</span> <span class="bp">self</span><span class="o">.</span><span class="n">spectral</span><span class="o">.</span><span class="n">V</span><span class="p">,</span> <span class="n">root</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>
+
+        <span class="n">Nusselt_t</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">comm</span><span class="o">.</span><span class="n">bcast</span><span class="p">(</span>
+            <span class="bp">self</span><span class="o">.</span><span class="n">xp</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">nusselt_hat</span> <span class="o">*</span> <span class="bp">self</span><span class="o">.</span><span class="n">spectral</span><span class="o">.</span><span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">get_BC</span><span class="p">(</span><span class="n">kind</span><span class="o">=</span><span class="s1">&#39;Dirichlet&#39;</span><span class="p">,</span> <span class="n">x</span><span class="o">=</span><span class="mi">1</span><span class="p">),</span> <span class="n">axis</span><span class="o">=-</span><span class="mi">1</span><span class="p">)</span><span class="o">.</span><span class="n">real</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">root</span><span class="o">=</span><span class="mi">0</span>
+        <span class="p">)</span>
+        <span class="n">Nusselt_b</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">comm</span><span class="o">.</span><span class="n">bcast</span><span class="p">(</span>
+            <span class="bp">self</span><span class="o">.</span><span class="n">xp</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">nusselt_hat</span> <span class="o">*</span> <span class="bp">self</span><span class="o">.</span><span class="n">spectral</span><span class="o">.</span><span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">get_BC</span><span class="p">(</span><span class="n">kind</span><span class="o">=</span><span class="s1">&#39;Dirichlet&#39;</span><span class="p">,</span> <span class="n">x</span><span class="o">=-</span><span class="mi">1</span><span class="p">),</span> <span class="n">axis</span><span class="o">=-</span><span class="mi">1</span><span class="p">)</span><span class="o">.</span><span class="n">real</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">root</span><span class="o">=</span><span class="mi">0</span>
+        <span class="p">)</span>
+        <span class="n">Nusselt_no_v_t</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">comm</span><span class="o">.</span><span class="n">bcast</span><span class="p">(</span>
+            <span class="bp">self</span><span class="o">.</span><span class="n">xp</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">nusselt_no_v_hat</span> <span class="o">*</span> <span class="bp">self</span><span class="o">.</span><span class="n">spectral</span><span class="o">.</span><span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">get_BC</span><span class="p">(</span><span class="n">kind</span><span class="o">=</span><span class="s1">&#39;Dirichlet&#39;</span><span class="p">,</span> <span class="n">x</span><span class="o">=</span><span class="mi">1</span><span class="p">),</span> <span class="n">axis</span><span class="o">=-</span><span class="mi">1</span><span class="p">)</span><span class="o">.</span><span class="n">real</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">root</span><span class="o">=</span><span class="mi">0</span>
+        <span class="p">)</span>
+        <span class="n">Nusselt_no_v_b</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">comm</span><span class="o">.</span><span class="n">bcast</span><span class="p">(</span>
+            <span class="bp">self</span><span class="o">.</span><span class="n">xp</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">nusselt_no_v_hat</span> <span class="o">*</span> <span class="bp">self</span><span class="o">.</span><span class="n">spectral</span><span class="o">.</span><span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">get_BC</span><span class="p">(</span><span class="n">kind</span><span class="o">=</span><span class="s1">&#39;Dirichlet&#39;</span><span class="p">,</span> <span class="n">x</span><span class="o">=-</span><span class="mi">1</span><span class="p">),</span> <span class="n">axis</span><span class="o">=-</span><span class="mi">1</span><span class="p">)</span><span class="o">.</span><span class="n">real</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span>
+            <span class="n">root</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span>
+        <span class="p">)</span>
+
+        <span class="k">return</span> <span class="p">{</span>
+            <span class="s1">&#39;V&#39;</span><span class="p">:</span> <span class="n">Nusselt_V</span><span class="p">,</span>
+            <span class="s1">&#39;t&#39;</span><span class="p">:</span> <span class="n">Nusselt_t</span><span class="p">,</span>
+            <span class="s1">&#39;b&#39;</span><span class="p">:</span> <span class="n">Nusselt_b</span><span class="p">,</span>
+            <span class="s1">&#39;t_no_v&#39;</span><span class="p">:</span> <span class="n">Nusselt_no_v_t</span><span class="p">,</span>
+            <span class="s1">&#39;b_no_v&#39;</span><span class="p">:</span> <span class="n">Nusselt_no_v_b</span><span class="p">,</span>
+        <span class="p">}</span></div>
+
+
+<div class="viewcode-block" id="RayleighBenard.compute_viscous_dissipation">
+<a class="viewcode-back" href="../../../pySDC/implementations.problem_classes.RayleighBenard.html#implementations.problem_classes.RayleighBenard.RayleighBenard.compute_viscous_dissipation">[docs]</a>
+    <span class="k">def</span> <span class="nf">compute_viscous_dissipation</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">u</span><span class="p">):</span>
+        <span class="n">iu</span><span class="p">,</span> <span class="n">iv</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">index</span><span class="p">([</span><span class="s1">&#39;u&#39;</span><span class="p">,</span> <span class="s1">&#39;v&#39;</span><span class="p">])</span>
+
+        <span class="n">Lap_u_hat</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">spectral</span><span class="o">.</span><span class="n">u_init_forward</span>
+
+        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">spectral_space</span><span class="p">:</span>
+            <span class="n">u_hat</span> <span class="o">=</span> <span class="n">u</span><span class="o">.</span><span class="n">copy</span><span class="p">()</span>
+        <span class="k">else</span><span class="p">:</span>
+            <span class="n">u_hat</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">transform</span><span class="p">(</span><span class="n">u</span><span class="p">)</span>
+        <span class="n">Lap_u_hat</span><span class="p">[</span><span class="n">iu</span><span class="p">]</span> <span class="o">=</span> <span class="p">((</span><span class="bp">self</span><span class="o">.</span><span class="n">Dzz</span> <span class="o">+</span> <span class="bp">self</span><span class="o">.</span><span class="n">Dxx</span><span class="p">)</span> <span class="o">@</span> <span class="n">u_hat</span><span class="p">[</span><span class="n">iu</span><span class="p">]</span><span class="o">.</span><span class="n">flatten</span><span class="p">())</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="n">u_hat</span><span class="p">[</span><span class="n">iu</span><span class="p">]</span><span class="o">.</span><span class="n">shape</span><span class="p">)</span>
+        <span class="n">Lap_u_hat</span><span class="p">[</span><span class="n">iv</span><span class="p">]</span> <span class="o">=</span> <span class="p">((</span><span class="bp">self</span><span class="o">.</span><span class="n">Dzz</span> <span class="o">+</span> <span class="bp">self</span><span class="o">.</span><span class="n">Dxx</span><span class="p">)</span> <span class="o">@</span> <span class="n">u_hat</span><span class="p">[</span><span class="n">iv</span><span class="p">]</span><span class="o">.</span><span class="n">flatten</span><span class="p">())</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="n">u_hat</span><span class="p">[</span><span class="n">iu</span><span class="p">]</span><span class="o">.</span><span class="n">shape</span><span class="p">)</span>
+        <span class="n">Lap_u</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">itransform</span><span class="p">(</span><span class="n">Lap_u_hat</span><span class="p">)</span>
+
+        <span class="k">return</span> <span class="nb">abs</span><span class="p">(</span><span class="n">u</span><span class="p">[</span><span class="n">iu</span><span class="p">]</span> <span class="o">*</span> <span class="n">Lap_u</span><span class="p">[</span><span class="n">iu</span><span class="p">]</span> <span class="o">+</span> <span class="n">u</span><span class="p">[</span><span class="n">iv</span><span class="p">]</span> <span class="o">*</span> <span class="n">Lap_u</span><span class="p">[</span><span class="n">iv</span><span class="p">])</span></div>
+
+
+<div class="viewcode-block" id="RayleighBenard.compute_buoyancy_generation">
+<a class="viewcode-back" href="../../../pySDC/implementations.problem_classes.RayleighBenard.html#implementations.problem_classes.RayleighBenard.RayleighBenard.compute_buoyancy_generation">[docs]</a>
+    <span class="k">def</span> <span class="nf">compute_buoyancy_generation</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">u</span><span class="p">):</span>
+        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">spectral_space</span><span class="p">:</span>
+            <span class="n">u</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">itransform</span><span class="p">(</span><span class="n">u</span><span class="p">)</span>
+        <span class="n">iv</span><span class="p">,</span> <span class="n">iT</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">index</span><span class="p">([</span><span class="s1">&#39;v&#39;</span><span class="p">,</span> <span class="s1">&#39;T&#39;</span><span class="p">])</span>
+        <span class="k">return</span> <span class="nb">abs</span><span class="p">(</span><span class="n">u</span><span class="p">[</span><span class="n">iv</span><span class="p">]</span> <span class="o">*</span> <span class="bp">self</span><span class="o">.</span><span class="n">Rayleigh</span> <span class="o">*</span> <span class="n">u</span><span class="p">[</span><span class="n">iT</span><span class="p">])</span></div>
+</div>
+
+
+
+<div class="viewcode-block" id="CFLLimit">
+<a class="viewcode-back" href="../../../pySDC/implementations.problem_classes.RayleighBenard.html#implementations.problem_classes.RayleighBenard.CFLLimit">[docs]</a>
+<span class="k">class</span> <span class="nc">CFLLimit</span><span class="p">(</span><span class="n">ConvergenceController</span><span class="p">):</span>
+
+<div class="viewcode-block" id="CFLLimit.dependencies">
+<a class="viewcode-back" href="../../../pySDC/implementations.problem_classes.RayleighBenard.html#implementations.problem_classes.RayleighBenard.CFLLimit.dependencies">[docs]</a>
+    <span class="k">def</span> <span class="nf">dependencies</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">controller</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
+        <span class="kn">from</span> <span class="nn">pySDC.implementations.hooks.log_step_size</span> <span class="kn">import</span> <span class="n">LogStepSize</span>
+
+        <span class="n">controller</span><span class="o">.</span><span class="n">add_hook</span><span class="p">(</span><span class="n">LogCFL</span><span class="p">)</span>
+        <span class="n">controller</span><span class="o">.</span><span class="n">add_hook</span><span class="p">(</span><span class="n">LogStepSize</span><span class="p">)</span></div>
+
+
+<div class="viewcode-block" id="CFLLimit.setup_status_variables">
+<a class="viewcode-back" href="../../../pySDC/implementations.problem_classes.RayleighBenard.html#implementations.problem_classes.RayleighBenard.CFLLimit.setup_status_variables">[docs]</a>
+    <span class="k">def</span> <span class="nf">setup_status_variables</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">controller</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
+<span class="w">        </span><span class="sd">&quot;&quot;&quot;</span>
+<span class="sd">        Add the embedded error variable to the error function.</span>
+
+<span class="sd">        Args:</span>
+<span class="sd">            controller (pySDC.Controller): The controller</span>
+<span class="sd">        &quot;&quot;&quot;</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">add_status_variable_to_level</span><span class="p">(</span><span class="s1">&#39;CFL_limit&#39;</span><span class="p">)</span></div>
+
+
+<div class="viewcode-block" id="CFLLimit.setup">
+<a class="viewcode-back" href="../../../pySDC/implementations.problem_classes.RayleighBenard.html#implementations.problem_classes.RayleighBenard.CFLLimit.setup">[docs]</a>
+    <span class="k">def</span> <span class="nf">setup</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">controller</span><span class="p">,</span> <span class="n">params</span><span class="p">,</span> <span class="n">description</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
+<span class="w">        </span><span class="sd">&quot;&quot;&quot;</span>
+<span class="sd">        Define default parameters here.</span>
+
+<span class="sd">        Default parameters are:</span>
+<span class="sd">         - control_order (int): The order relative to other convergence controllers</span>
+<span class="sd">         - dt_max (float): maximal step size</span>
+<span class="sd">         - dt_min (float): minimal step size</span>
+
+<span class="sd">        Args:</span>
+<span class="sd">            controller (pySDC.Controller): The controller</span>
+<span class="sd">            params (dict): The params passed for this specific convergence controller</span>
+<span class="sd">            description (dict): The description object used to instantiate the controller</span>
+
+<span class="sd">        Returns:</span>
+<span class="sd">            (dict): The updated params dictionary</span>
+<span class="sd">        &quot;&quot;&quot;</span>
+        <span class="n">defaults</span> <span class="o">=</span> <span class="p">{</span>
+            <span class="s2">&quot;control_order&quot;</span><span class="p">:</span> <span class="o">-</span><span class="mi">50</span><span class="p">,</span>
+            <span class="s2">&quot;dt_max&quot;</span><span class="p">:</span> <span class="n">np</span><span class="o">.</span><span class="n">inf</span><span class="p">,</span>
+            <span class="s2">&quot;dt_min&quot;</span><span class="p">:</span> <span class="mi">0</span><span class="p">,</span>
+            <span class="s2">&quot;cfl&quot;</span><span class="p">:</span> <span class="mf">0.4</span><span class="p">,</span>
+        <span class="p">}</span>
+        <span class="k">return</span> <span class="p">{</span><span class="o">**</span><span class="n">defaults</span><span class="p">,</span> <span class="o">**</span><span class="nb">super</span><span class="p">()</span><span class="o">.</span><span class="n">setup</span><span class="p">(</span><span class="n">controller</span><span class="p">,</span> <span class="n">params</span><span class="p">,</span> <span class="n">description</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)}</span></div>
+
+
+<div class="viewcode-block" id="CFLLimit.compute_max_step_size">
+<a class="viewcode-back" href="../../../pySDC/implementations.problem_classes.RayleighBenard.html#implementations.problem_classes.RayleighBenard.CFLLimit.compute_max_step_size">[docs]</a>
+    <span class="nd">@staticmethod</span>
+    <span class="k">def</span> <span class="nf">compute_max_step_size</span><span class="p">(</span><span class="n">P</span><span class="p">,</span> <span class="n">u</span><span class="p">):</span>
+        <span class="n">grid_spacing_x</span> <span class="o">=</span> <span class="n">P</span><span class="o">.</span><span class="n">X</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span> <span class="o">-</span> <span class="n">P</span><span class="o">.</span><span class="n">X</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span>
+
+        <span class="n">cell_wallz</span> <span class="o">=</span> <span class="n">P</span><span class="o">.</span><span class="n">xp</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="n">P</span><span class="o">.</span><span class="n">nz</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
+        <span class="n">cell_wallz</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="mi">1</span>
+        <span class="n">cell_wallz</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="o">-</span><span class="mi">1</span>
+        <span class="n">cell_wallz</span><span class="p">[</span><span class="mi">1</span><span class="p">:</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="p">(</span><span class="n">P</span><span class="o">.</span><span class="n">Z</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="p">:</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="o">+</span> <span class="n">P</span><span class="o">.</span><span class="n">Z</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">:])</span> <span class="o">/</span> <span class="mi">2</span>
+        <span class="n">grid_spacing_z</span> <span class="o">=</span> <span class="n">cell_wallz</span><span class="p">[:</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="o">-</span> <span class="n">cell_wallz</span><span class="p">[</span><span class="mi">1</span><span class="p">:]</span>
+
+        <span class="n">iu</span><span class="p">,</span> <span class="n">iv</span> <span class="o">=</span> <span class="n">P</span><span class="o">.</span><span class="n">index</span><span class="p">([</span><span class="s1">&#39;u&#39;</span><span class="p">,</span> <span class="s1">&#39;v&#39;</span><span class="p">])</span>
+
+        <span class="k">if</span> <span class="n">P</span><span class="o">.</span><span class="n">spectral_space</span><span class="p">:</span>
+            <span class="n">u</span> <span class="o">=</span> <span class="n">P</span><span class="o">.</span><span class="n">itransform</span><span class="p">(</span><span class="n">u</span><span class="p">)</span>
+
+        <span class="n">max_step_size_x</span> <span class="o">=</span> <span class="n">P</span><span class="o">.</span><span class="n">xp</span><span class="o">.</span><span class="n">min</span><span class="p">(</span><span class="n">grid_spacing_x</span> <span class="o">/</span> <span class="n">P</span><span class="o">.</span><span class="n">xp</span><span class="o">.</span><span class="n">abs</span><span class="p">(</span><span class="n">u</span><span class="p">[</span><span class="n">iu</span><span class="p">]))</span>
+        <span class="n">max_step_size_z</span> <span class="o">=</span> <span class="n">P</span><span class="o">.</span><span class="n">xp</span><span class="o">.</span><span class="n">min</span><span class="p">(</span><span class="n">grid_spacing_z</span> <span class="o">/</span> <span class="n">P</span><span class="o">.</span><span class="n">xp</span><span class="o">.</span><span class="n">abs</span><span class="p">(</span><span class="n">u</span><span class="p">[</span><span class="n">iv</span><span class="p">]))</span>
+        <span class="n">max_step_size</span> <span class="o">=</span> <span class="nb">min</span><span class="p">([</span><span class="n">max_step_size_x</span><span class="p">,</span> <span class="n">max_step_size_z</span><span class="p">])</span>
+
+        <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">P</span><span class="p">,</span> <span class="s1">&#39;comm&#39;</span><span class="p">):</span>
+            <span class="n">max_step_size</span> <span class="o">=</span> <span class="n">P</span><span class="o">.</span><span class="n">comm</span><span class="o">.</span><span class="n">allreduce</span><span class="p">(</span><span class="n">max_step_size</span><span class="p">,</span> <span class="n">op</span><span class="o">=</span><span class="n">MPI</span><span class="o">.</span><span class="n">MIN</span><span class="p">)</span>
+        <span class="k">return</span> <span class="nb">float</span><span class="p">(</span><span class="n">max_step_size</span><span class="p">)</span></div>
+
+
+<div class="viewcode-block" id="CFLLimit.get_new_step_size">
+<a class="viewcode-back" href="../../../pySDC/implementations.problem_classes.RayleighBenard.html#implementations.problem_classes.RayleighBenard.CFLLimit.get_new_step_size">[docs]</a>
+    <span class="k">def</span> <span class="nf">get_new_step_size</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">controller</span><span class="p">,</span> <span class="n">step</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
+        <span class="k">if</span> <span class="ow">not</span> <span class="n">CheckConvergence</span><span class="o">.</span><span class="n">check_convergence</span><span class="p">(</span><span class="n">step</span><span class="p">):</span>
+            <span class="k">return</span> <span class="kc">None</span>
+
+        <span class="n">L</span> <span class="o">=</span> <span class="n">step</span><span class="o">.</span><span class="n">levels</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
+        <span class="n">P</span> <span class="o">=</span> <span class="n">step</span><span class="o">.</span><span class="n">levels</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">prob</span>
+
+        <span class="n">L</span><span class="o">.</span><span class="n">sweep</span><span class="o">.</span><span class="n">compute_end_point</span><span class="p">()</span>
+        <span class="n">max_step_size</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">compute_max_step_size</span><span class="p">(</span><span class="n">P</span><span class="p">,</span> <span class="n">L</span><span class="o">.</span><span class="n">uend</span><span class="p">)</span>
+
+        <span class="n">L</span><span class="o">.</span><span class="n">status</span><span class="o">.</span><span class="n">CFL_limit</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">params</span><span class="o">.</span><span class="n">cfl</span> <span class="o">*</span> <span class="n">max_step_size</span>
+
+        <span class="n">dt_new</span> <span class="o">=</span> <span class="n">L</span><span class="o">.</span><span class="n">status</span><span class="o">.</span><span class="n">dt_new</span> <span class="k">if</span> <span class="n">L</span><span class="o">.</span><span class="n">status</span><span class="o">.</span><span class="n">dt_new</span> <span class="k">else</span> <span class="nb">max</span><span class="p">([</span><span class="bp">self</span><span class="o">.</span><span class="n">params</span><span class="o">.</span><span class="n">dt_max</span><span class="p">,</span> <span class="n">L</span><span class="o">.</span><span class="n">params</span><span class="o">.</span><span class="n">dt</span><span class="p">])</span>
+        <span class="n">L</span><span class="o">.</span><span class="n">status</span><span class="o">.</span><span class="n">dt_new</span> <span class="o">=</span> <span class="nb">min</span><span class="p">([</span><span class="n">dt_new</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">params</span><span class="o">.</span><span class="n">cfl</span> <span class="o">*</span> <span class="n">max_step_size</span><span class="p">])</span>
+        <span class="n">L</span><span class="o">.</span><span class="n">status</span><span class="o">.</span><span class="n">dt_new</span> <span class="o">=</span> <span class="nb">max</span><span class="p">([</span><span class="bp">self</span><span class="o">.</span><span class="n">params</span><span class="o">.</span><span class="n">dt_min</span><span class="p">,</span> <span class="n">L</span><span class="o">.</span><span class="n">status</span><span class="o">.</span><span class="n">dt_new</span><span class="p">])</span>
+
+        <span class="bp">self</span><span class="o">.</span><span class="n">log</span><span class="p">(</span><span class="sa">f</span><span class="s1">&#39;dt max: </span><span class="si">{</span><span class="n">max_step_size</span><span class="si">:</span><span class="s1">.2e</span><span class="si">}</span><span class="s1"> -&gt; New step size: </span><span class="si">{</span><span class="n">L</span><span class="o">.</span><span class="n">status</span><span class="o">.</span><span class="n">dt_new</span><span class="si">:</span><span class="s1">.2e</span><span class="si">}</span><span class="s1">&#39;</span><span class="p">,</span> <span class="n">step</span><span class="p">)</span></div>
+</div>
+
+
+
+<div class="viewcode-block" id="LogCFL">
+<a class="viewcode-back" href="../../../pySDC/implementations.problem_classes.RayleighBenard.html#implementations.problem_classes.RayleighBenard.LogCFL">[docs]</a>
+<span class="k">class</span> <span class="nc">LogCFL</span><span class="p">(</span><span class="n">Hooks</span><span class="p">):</span>
+
+<div class="viewcode-block" id="LogCFL.post_step">
+<a class="viewcode-back" href="../../../pySDC/implementations.problem_classes.RayleighBenard.html#implementations.problem_classes.RayleighBenard.LogCFL.post_step">[docs]</a>
+    <span class="k">def</span> <span class="nf">post_step</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">step</span><span class="p">,</span> <span class="n">level_number</span><span class="p">):</span>
+<span class="w">        </span><span class="sd">&quot;&quot;&quot;</span>
+<span class="sd">        Record CFL limit.</span>
+
+<span class="sd">        Args:</span>
+<span class="sd">            step (pySDC.Step.step): the current step</span>
+<span class="sd">            level_number (int): the current level number</span>
+
+<span class="sd">        Returns:</span>
+<span class="sd">            None</span>
+<span class="sd">        &quot;&quot;&quot;</span>
+        <span class="nb">super</span><span class="p">()</span><span class="o">.</span><span class="n">post_step</span><span class="p">(</span><span class="n">step</span><span class="p">,</span> <span class="n">level_number</span><span class="p">)</span>
+
+        <span class="n">L</span> <span class="o">=</span> <span class="n">step</span><span class="o">.</span><span class="n">levels</span><span class="p">[</span><span class="n">level_number</span><span class="p">]</span>
+
+        <span class="bp">self</span><span class="o">.</span><span class="n">add_to_stats</span><span class="p">(</span>
+            <span class="n">process</span><span class="o">=</span><span class="n">step</span><span class="o">.</span><span class="n">status</span><span class="o">.</span><span class="n">slot</span><span class="p">,</span>
+            <span class="n">time</span><span class="o">=</span><span class="n">L</span><span class="o">.</span><span class="n">time</span> <span class="o">+</span> <span class="n">L</span><span class="o">.</span><span class="n">dt</span><span class="p">,</span>
+            <span class="n">level</span><span class="o">=</span><span class="n">L</span><span class="o">.</span><span class="n">level_index</span><span class="p">,</span>
+            <span class="nb">iter</span><span class="o">=</span><span class="n">step</span><span class="o">.</span><span class="n">status</span><span class="o">.</span><span class="n">iter</span><span class="p">,</span>
+            <span class="n">sweep</span><span class="o">=</span><span class="n">L</span><span class="o">.</span><span class="n">status</span><span class="o">.</span><span class="n">sweep</span><span class="p">,</span>
+            <span class="nb">type</span><span class="o">=</span><span class="s1">&#39;CFL_limit&#39;</span><span class="p">,</span>
+            <span class="n">value</span><span class="o">=</span><span class="n">L</span><span class="o">.</span><span class="n">status</span><span class="o">.</span><span class="n">CFL_limit</span><span class="p">,</span>
+        <span class="p">)</span></div>
+</div>
+
+
+
+<div class="viewcode-block" id="LogAnalysisVariables">
+<a class="viewcode-back" href="../../../pySDC/implementations.problem_classes.RayleighBenard.html#implementations.problem_classes.RayleighBenard.LogAnalysisVariables">[docs]</a>
+<span class="k">class</span> <span class="nc">LogAnalysisVariables</span><span class="p">(</span><span class="n">Hooks</span><span class="p">):</span>
+
+<div class="viewcode-block" id="LogAnalysisVariables.post_step">
+<a class="viewcode-back" href="../../../pySDC/implementations.problem_classes.RayleighBenard.html#implementations.problem_classes.RayleighBenard.LogAnalysisVariables.post_step">[docs]</a>
+    <span class="k">def</span> <span class="nf">post_step</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">step</span><span class="p">,</span> <span class="n">level_number</span><span class="p">):</span>
+<span class="w">        </span><span class="sd">&quot;&quot;&quot;</span>
+<span class="sd">        Record Nusselt numbers.</span>
+
+<span class="sd">        Args:</span>
+<span class="sd">            step (pySDC.Step.step): the current step</span>
+<span class="sd">            level_number (int): the current level number</span>
+
+<span class="sd">        Returns:</span>
+<span class="sd">            None</span>
+<span class="sd">        &quot;&quot;&quot;</span>
+        <span class="nb">super</span><span class="p">()</span><span class="o">.</span><span class="n">post_step</span><span class="p">(</span><span class="n">step</span><span class="p">,</span> <span class="n">level_number</span><span class="p">)</span>
+
+        <span class="n">L</span> <span class="o">=</span> <span class="n">step</span><span class="o">.</span><span class="n">levels</span><span class="p">[</span><span class="n">level_number</span><span class="p">]</span>
+        <span class="n">P</span> <span class="o">=</span> <span class="n">L</span><span class="o">.</span><span class="n">prob</span>
+
+        <span class="n">L</span><span class="o">.</span><span class="n">sweep</span><span class="o">.</span><span class="n">compute_end_point</span><span class="p">()</span>
+        <span class="n">Nusselt</span> <span class="o">=</span> <span class="n">P</span><span class="o">.</span><span class="n">compute_Nusselt_numbers</span><span class="p">(</span><span class="n">L</span><span class="o">.</span><span class="n">uend</span><span class="p">)</span>
+        <span class="n">buoyancy_production</span> <span class="o">=</span> <span class="n">P</span><span class="o">.</span><span class="n">compute_buoyancy_generation</span><span class="p">(</span><span class="n">L</span><span class="o">.</span><span class="n">uend</span><span class="p">)</span>
+        <span class="n">viscous_dissipation</span> <span class="o">=</span> <span class="n">P</span><span class="o">.</span><span class="n">compute_viscous_dissipation</span><span class="p">(</span><span class="n">L</span><span class="o">.</span><span class="n">uend</span><span class="p">)</span>
+
+        <span class="k">for</span> <span class="n">key</span><span class="p">,</span> <span class="n">value</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span>
+            <span class="p">[</span><span class="s1">&#39;Nusselt&#39;</span><span class="p">,</span> <span class="s1">&#39;buoyancy_production&#39;</span><span class="p">,</span> <span class="s1">&#39;viscous_dissipation&#39;</span><span class="p">],</span>
+            <span class="p">[</span><span class="n">Nusselt</span><span class="p">,</span> <span class="n">buoyancy_production</span><span class="p">,</span> <span class="n">viscous_dissipation</span><span class="p">],</span>
+        <span class="p">):</span>
+            <span class="bp">self</span><span class="o">.</span><span class="n">add_to_stats</span><span class="p">(</span>
+                <span class="n">process</span><span class="o">=</span><span class="n">step</span><span class="o">.</span><span class="n">status</span><span class="o">.</span><span class="n">slot</span><span class="p">,</span>
+                <span class="n">time</span><span class="o">=</span><span class="n">L</span><span class="o">.</span><span class="n">time</span> <span class="o">+</span> <span class="n">L</span><span class="o">.</span><span class="n">dt</span><span class="p">,</span>
+                <span class="n">level</span><span class="o">=</span><span class="n">L</span><span class="o">.</span><span class="n">level_index</span><span class="p">,</span>
+                <span class="nb">iter</span><span class="o">=</span><span class="n">step</span><span class="o">.</span><span class="n">status</span><span class="o">.</span><span class="n">iter</span><span class="p">,</span>
+                <span class="n">sweep</span><span class="o">=</span><span class="n">L</span><span class="o">.</span><span class="n">status</span><span class="o">.</span><span class="n">sweep</span><span class="p">,</span>
+                <span class="nb">type</span><span class="o">=</span><span class="n">key</span><span class="p">,</span>
+                <span class="n">value</span><span class="o">=</span><span class="n">value</span><span class="p">,</span>
+            <span class="p">)</span></div>
+</div>
+
+</pre></div>
+
+            <div class="clearer"></div>
+          </div>
+        </div>
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diff --git a/_modules/implementations/problem_classes/generic_spectral.html b/_modules/implementations/problem_classes/generic_spectral.html
new file mode 100644
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+          <div class="body" role="main">
+            
+  <h1>Source code for implementations.problem_classes.generic_spectral</h1><div class="highlight"><pre>
+<span></span><span class="kn">from</span> <span class="nn">pySDC.core.problem</span> <span class="kn">import</span> <span class="n">Problem</span><span class="p">,</span> <span class="n">WorkCounter</span>
+<span class="kn">from</span> <span class="nn">pySDC.helpers.spectral_helper</span> <span class="kn">import</span> <span class="n">SpectralHelper</span>
+<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
+<span class="kn">from</span> <span class="nn">pySDC.core.errors</span> <span class="kn">import</span> <span class="n">ParameterError</span>
+
+
+<div class="viewcode-block" id="GenericSpectralLinear">
+<a class="viewcode-back" href="../../../pySDC/implementations.problem_classes.generic_spectral.html#implementations.problem_classes.generic_spectral.GenericSpectralLinear">[docs]</a>
+<span class="k">class</span> <span class="nc">GenericSpectralLinear</span><span class="p">(</span><span class="n">Problem</span><span class="p">):</span>
+<span class="w">    </span><span class="sd">&quot;&quot;&quot;</span>
+<span class="sd">    Generic class to solve problems of the form M u_t + L u = y, with mass matrix M, linear operator L and some right</span>
+<span class="sd">    hand side y using spectral methods.</span>
+<span class="sd">    L may contain algebraic conditions, as long as (M + dt L) is invertible.</span>
+
+<span class="sd">    Note that the `__getattr__` method is overloaded to pass requests on to the spectral helper if they are not</span>
+<span class="sd">    attributes of this class itself. For instance, you can add a BC by calling `self.spectral.add_BC` or equivalently</span>
+<span class="sd">    `self.add_BC`.</span>
+
+<span class="sd">    You can port problems derived from this more or less seamlessly to GPU by using the numerical libraries that are</span>
+<span class="sd">    class attributes of the spectral helper. This class will automatically switch the datatype using the `setup_GPU` class method.</span>
+
+<span class="sd">    Attributes:</span>
+<span class="sd">        spectral (pySDC.helpers.spectral_helper.SpectralHelper): Spectral helper</span>
+<span class="sd">        work_counters (dict): Dictionary for counting work</span>
+<span class="sd">        cached_factorizations (dict): Dictionary of cached matrix factorizations for solving</span>
+<span class="sd">        L (sparse matrix): Linear operator</span>
+<span class="sd">        M (sparse matrix): Mass matrix</span>
+<span class="sd">        diff_mask (list): Mask for separating differential and algebraic terms</span>
+<span class="sd">        Pl (sparse matrix): Left preconditioner</span>
+<span class="sd">        Pr (sparse matrix): Right preconditioner</span>
+<span class="sd">    &quot;&quot;&quot;</span>
+
+<div class="viewcode-block" id="GenericSpectralLinear.setup_GPU">
+<a class="viewcode-back" href="../../../pySDC/implementations.problem_classes.generic_spectral.html#implementations.problem_classes.generic_spectral.GenericSpectralLinear.setup_GPU">[docs]</a>
+    <span class="nd">@classmethod</span>
+    <span class="k">def</span> <span class="nf">setup_GPU</span><span class="p">(</span><span class="bp">cls</span><span class="p">):</span>
+<span class="w">        </span><span class="sd">&quot;&quot;&quot;switch to GPU modules&quot;&quot;&quot;</span>
+        <span class="kn">import</span> <span class="nn">cupy</span> <span class="k">as</span> <span class="nn">cp</span>
+        <span class="kn">from</span> <span class="nn">pySDC.implementations.datatype_classes.cupy_mesh</span> <span class="kn">import</span> <span class="n">cupy_mesh</span><span class="p">,</span> <span class="n">imex_cupy_mesh</span>
+        <span class="kn">from</span> <span class="nn">pySDC.implementations.datatype_classes.mesh</span> <span class="kn">import</span> <span class="n">mesh</span><span class="p">,</span> <span class="n">imex_mesh</span>
+
+        <span class="bp">cls</span><span class="o">.</span><span class="n">dtype_u</span> <span class="o">=</span> <span class="n">cupy_mesh</span>
+
+        <span class="n">GPU_versions</span> <span class="o">=</span> <span class="p">{</span>
+            <span class="n">mesh</span><span class="p">:</span> <span class="n">cupy_mesh</span><span class="p">,</span>
+            <span class="n">imex_mesh</span><span class="p">:</span> <span class="n">imex_cupy_mesh</span><span class="p">,</span>
+        <span class="p">}</span>
+
+        <span class="bp">cls</span><span class="o">.</span><span class="n">dtype_f</span> <span class="o">=</span> <span class="n">GPU_versions</span><span class="p">[</span><span class="bp">cls</span><span class="o">.</span><span class="n">dtype_f</span><span class="p">]</span></div>
+
+
+    <span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span>
+        <span class="bp">self</span><span class="p">,</span>
+        <span class="n">bases</span><span class="p">,</span>
+        <span class="n">components</span><span class="p">,</span>
+        <span class="n">comm</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span>
+        <span class="n">Dirichlet_recombination</span><span class="o">=</span><span class="kc">True</span><span class="p">,</span>
+        <span class="n">left_preconditioner</span><span class="o">=</span><span class="kc">True</span><span class="p">,</span>
+        <span class="n">solver_type</span><span class="o">=</span><span class="s1">&#39;cached_direct&#39;</span><span class="p">,</span>
+        <span class="n">solver_args</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span>
+        <span class="n">useGPU</span><span class="o">=</span><span class="kc">False</span><span class="p">,</span>
+        <span class="n">max_cached_factorizations</span><span class="o">=</span><span class="mi">12</span><span class="p">,</span>
+        <span class="n">spectral_space</span><span class="o">=</span><span class="kc">True</span><span class="p">,</span>
+        <span class="n">real_spectral_coefficients</span><span class="o">=</span><span class="kc">False</span><span class="p">,</span>
+        <span class="n">debug</span><span class="o">=</span><span class="kc">False</span><span class="p">,</span>
+    <span class="p">):</span>
+<span class="w">        </span><span class="sd">&quot;&quot;&quot;</span>
+<span class="sd">        Base class for problems discretized with spectral methods.</span>
+
+<span class="sd">        Args:</span>
+<span class="sd">            bases (list of dictionaries): 1D Bases</span>
+<span class="sd">            components (list of strings): Components of the equations</span>
+<span class="sd">            comm (mpi4py.Intracomm or None): MPI communicator</span>
+<span class="sd">            Dirichlet_recombination (bool): Use Dirichlet recombination in the last axis as right preconditioner</span>
+<span class="sd">            left_preconditioner (bool): Reverse the Kronecker product if yes</span>
+<span class="sd">            solver_type (str): Solver for linear systems</span>
+<span class="sd">            solver_args (dict): Arguments for linear solver</span>
+<span class="sd">            useGPU (bool): Run on GPU or CPU</span>
+<span class="sd">            max_cached_factorizations (int): Number of matrix decompositions to cache before starting eviction</span>
+<span class="sd">            spectral_space (bool): If yes, the solution will not be transformed back after solving and evaluating the RHS, and is expected as input in spectral space to these functions</span>
+<span class="sd">            real_spectral_coefficients (bool): If yes, allow only real values in spectral space, otherwise, allow complex.</span>
+<span class="sd">            debug (bool): Make additional tests at extra computational cost</span>
+<span class="sd">        &quot;&quot;&quot;</span>
+        <span class="n">solver_args</span> <span class="o">=</span> <span class="p">{}</span> <span class="k">if</span> <span class="n">solver_args</span> <span class="ow">is</span> <span class="kc">None</span> <span class="k">else</span> <span class="n">solver_args</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">_makeAttributeAndRegister</span><span class="p">(</span>
+            <span class="s1">&#39;max_cached_factorizations&#39;</span><span class="p">,</span>
+            <span class="s1">&#39;useGPU&#39;</span><span class="p">,</span>
+            <span class="s1">&#39;solver_type&#39;</span><span class="p">,</span>
+            <span class="s1">&#39;solver_args&#39;</span><span class="p">,</span>
+            <span class="s1">&#39;left_preconditioner&#39;</span><span class="p">,</span>
+            <span class="s1">&#39;Dirichlet_recombination&#39;</span><span class="p">,</span>
+            <span class="s1">&#39;comm&#39;</span><span class="p">,</span>
+            <span class="s1">&#39;spectral_space&#39;</span><span class="p">,</span>
+            <span class="s1">&#39;real_spectral_coefficients&#39;</span><span class="p">,</span>
+            <span class="s1">&#39;debug&#39;</span><span class="p">,</span>
+            <span class="n">localVars</span><span class="o">=</span><span class="nb">locals</span><span class="p">(),</span>
+        <span class="p">)</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">spectral</span> <span class="o">=</span> <span class="n">SpectralHelper</span><span class="p">(</span><span class="n">comm</span><span class="o">=</span><span class="n">comm</span><span class="p">,</span> <span class="n">useGPU</span><span class="o">=</span><span class="n">useGPU</span><span class="p">,</span> <span class="n">debug</span><span class="o">=</span><span class="n">debug</span><span class="p">)</span>
+
+        <span class="k">if</span> <span class="n">useGPU</span><span class="p">:</span>
+            <span class="bp">self</span><span class="o">.</span><span class="n">setup_GPU</span><span class="p">()</span>
+
+        <span class="k">for</span> <span class="n">base</span> <span class="ow">in</span> <span class="n">bases</span><span class="p">:</span>
+            <span class="bp">self</span><span class="o">.</span><span class="n">spectral</span><span class="o">.</span><span class="n">add_axis</span><span class="p">(</span><span class="o">**</span><span class="n">base</span><span class="p">)</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">spectral</span><span class="o">.</span><span class="n">add_component</span><span class="p">(</span><span class="n">components</span><span class="p">)</span>
+
+        <span class="bp">self</span><span class="o">.</span><span class="n">spectral</span><span class="o">.</span><span class="n">setup_fft</span><span class="p">(</span><span class="n">real_spectral_coefficients</span><span class="p">)</span>
+
+        <span class="nb">super</span><span class="p">()</span><span class="o">.</span><span class="fm">__init__</span><span class="p">(</span><span class="n">init</span><span class="o">=</span><span class="bp">self</span><span class="o">.</span><span class="n">spectral</span><span class="o">.</span><span class="n">init_forward</span> <span class="k">if</span> <span class="n">spectral_space</span> <span class="k">else</span> <span class="bp">self</span><span class="o">.</span><span class="n">spectral</span><span class="o">.</span><span class="n">init</span><span class="p">)</span>
+
+        <span class="bp">self</span><span class="o">.</span><span class="n">work_counters</span><span class="p">[</span><span class="n">solver_type</span><span class="p">]</span> <span class="o">=</span> <span class="n">WorkCounter</span><span class="p">()</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">work_counters</span><span class="p">[</span><span class="s1">&#39;factorizations&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="n">WorkCounter</span><span class="p">()</span>
+
+        <span class="bp">self</span><span class="o">.</span><span class="n">setup_preconditioner</span><span class="p">(</span><span class="n">Dirichlet_recombination</span><span class="p">,</span> <span class="n">left_preconditioner</span><span class="p">)</span>
+
+        <span class="bp">self</span><span class="o">.</span><span class="n">cached_factorizations</span> <span class="o">=</span> <span class="p">{}</span>
+
+    <span class="k">def</span> <span class="fm">__getattr__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">name</span><span class="p">):</span>
+<span class="w">        </span><span class="sd">&quot;&quot;&quot;</span>
+<span class="sd">        Pass requests on to the helper if they are not directly attributes of this class for convenience.</span>
+
+<span class="sd">        Args:</span>
+<span class="sd">            name (str): Name of the attribute you want</span>
+
+<span class="sd">        Returns:</span>
+<span class="sd">            request</span>
+<span class="sd">        &quot;&quot;&quot;</span>
+        <span class="k">return</span> <span class="nb">getattr</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">spectral</span><span class="p">,</span> <span class="n">name</span><span class="p">)</span>
+
+    <span class="k">def</span> <span class="nf">_setup_operator</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">LHS</span><span class="p">):</span>
+<span class="w">        </span><span class="sd">&quot;&quot;&quot;</span>
+<span class="sd">        Setup a sparse linear operator by adding relationships. See documentation for ``GenericSpectralLinear.setup_L`` to learn more.</span>
+
+<span class="sd">        Args:</span>
+<span class="sd">            LHS (dict): Equations to be added to the operator</span>
+
+<span class="sd">        Returns:</span>
+<span class="sd">            sparse linear operator</span>
+<span class="sd">        &quot;&quot;&quot;</span>
+        <span class="n">operator</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">spectral</span><span class="o">.</span><span class="n">get_empty_operator_matrix</span><span class="p">()</span>
+        <span class="k">for</span> <span class="n">line</span><span class="p">,</span> <span class="n">equation</span> <span class="ow">in</span> <span class="n">LHS</span><span class="o">.</span><span class="n">items</span><span class="p">():</span>
+            <span class="bp">self</span><span class="o">.</span><span class="n">spectral</span><span class="o">.</span><span class="n">add_equation_lhs</span><span class="p">(</span><span class="n">operator</span><span class="p">,</span> <span class="n">line</span><span class="p">,</span> <span class="n">equation</span><span class="p">)</span>
+        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">spectral</span><span class="o">.</span><span class="n">convert_operator_matrix_to_operator</span><span class="p">(</span><span class="n">operator</span><span class="p">)</span>
+
+<div class="viewcode-block" id="GenericSpectralLinear.setup_L">
+<a class="viewcode-back" href="../../../pySDC/implementations.problem_classes.generic_spectral.html#implementations.problem_classes.generic_spectral.GenericSpectralLinear.setup_L">[docs]</a>
+    <span class="k">def</span> <span class="nf">setup_L</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">LHS</span><span class="p">):</span>
+<span class="w">        </span><span class="sd">&quot;&quot;&quot;</span>
+<span class="sd">        Setup the left hand side of the linear operator L and store it in ``self.L``.</span>
+
+<span class="sd">        The argument is meant to be a dictionary with the line you want to write the equation in as the key and the relationship between components as another dictionary. For instance, you can add an algebraic condition capturing a first derivative relationship between u and ux as follows:</span>
+
+<span class="sd">        ```</span>
+<span class="sd">        Dx = self.get_differentiation_matrix(axes=(0,))</span>
+<span class="sd">        I = self.get_Id()</span>
+<span class="sd">        LHS = {&#39;ux&#39;: {&#39;u&#39;: Dx, &#39;ux&#39;: -I}}</span>
+<span class="sd">        self.setup_L(LHS)</span>
+<span class="sd">        ```</span>
+
+<span class="sd">        If you put zero as right hand side for the solver in the line for ux, ux will contain the x-derivative of u afterwards.</span>
+
+<span class="sd">        Args:</span>
+<span class="sd">            LHS (dict): Dictionary containing the equations.</span>
+<span class="sd">        &quot;&quot;&quot;</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">L</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_setup_operator</span><span class="p">(</span><span class="n">LHS</span><span class="p">)</span></div>
+
+
+<div class="viewcode-block" id="GenericSpectralLinear.setup_M">
+<a class="viewcode-back" href="../../../pySDC/implementations.problem_classes.generic_spectral.html#implementations.problem_classes.generic_spectral.GenericSpectralLinear.setup_M">[docs]</a>
+    <span class="k">def</span> <span class="nf">setup_M</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">LHS</span><span class="p">):</span>
+<span class="w">        </span><span class="sd">&#39;&#39;&#39;</span>
+<span class="sd">        Setup mass matrix, see documentation of ``GenericSpectralLinear.setup_L``.</span>
+<span class="sd">        &#39;&#39;&#39;</span>
+        <span class="n">diff_index</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">LHS</span><span class="o">.</span><span class="n">keys</span><span class="p">())</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">diff_mask</span> <span class="o">=</span> <span class="p">[</span><span class="n">me</span> <span class="ow">in</span> <span class="n">diff_index</span> <span class="k">for</span> <span class="n">me</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">components</span><span class="p">]</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">M</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_setup_operator</span><span class="p">(</span><span class="n">LHS</span><span class="p">)</span></div>
+
+
+<div class="viewcode-block" id="GenericSpectralLinear.setup_preconditioner">
+<a class="viewcode-back" href="../../../pySDC/implementations.problem_classes.generic_spectral.html#implementations.problem_classes.generic_spectral.GenericSpectralLinear.setup_preconditioner">[docs]</a>
+    <span class="k">def</span> <span class="nf">setup_preconditioner</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">Dirichlet_recombination</span><span class="o">=</span><span class="kc">True</span><span class="p">,</span> <span class="n">left_preconditioner</span><span class="o">=</span><span class="kc">True</span><span class="p">):</span>
+<span class="w">        </span><span class="sd">&quot;&quot;&quot;</span>
+<span class="sd">        Get left and right preconditioners.</span>
+
+<span class="sd">        Args:</span>
+<span class="sd">            Dirichlet_recombination (bool): Basis conversion for right preconditioner. Useful for Chebychev and Ultraspherical methods. 10/10 would recommend.</span>
+<span class="sd">            left_preconditioner (bool): If True, it will interleave the variables and reverse the Kronecker product</span>
+<span class="sd">        &quot;&quot;&quot;</span>
+        <span class="n">sp</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">spectral</span><span class="o">.</span><span class="n">sparse_lib</span>
+        <span class="n">N</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">prod</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">init</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">1</span><span class="p">:])</span>
+
+        <span class="n">Id</span> <span class="o">=</span> <span class="n">sp</span><span class="o">.</span><span class="n">eye</span><span class="p">(</span><span class="n">N</span><span class="p">)</span>
+        <span class="n">Pl_lhs</span> <span class="o">=</span> <span class="p">{</span><span class="n">comp</span><span class="p">:</span> <span class="p">{</span><span class="n">comp</span><span class="p">:</span> <span class="n">Id</span><span class="p">}</span> <span class="k">for</span> <span class="n">comp</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">components</span><span class="p">}</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">Pl</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_setup_operator</span><span class="p">(</span><span class="n">Pl_lhs</span><span class="p">)</span>
+
+        <span class="k">if</span> <span class="n">left_preconditioner</span><span class="p">:</span>
+            <span class="c1"># reverse Kronecker product</span>
+
+            <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">spectral</span><span class="o">.</span><span class="n">useGPU</span><span class="p">:</span>
+                <span class="n">R</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">Pl</span><span class="o">.</span><span class="n">get</span><span class="p">()</span><span class="o">.</span><span class="n">tolil</span><span class="p">()</span> <span class="o">*</span> <span class="mi">0</span>
+            <span class="k">else</span><span class="p">:</span>
+                <span class="n">R</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">Pl</span><span class="o">.</span><span class="n">tolil</span><span class="p">()</span> <span class="o">*</span> <span class="mi">0</span>
+
+            <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">ncomponents</span><span class="p">):</span>
+                <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">N</span><span class="p">):</span>
+                    <span class="n">R</span><span class="p">[</span><span class="n">i</span> <span class="o">*</span> <span class="bp">self</span><span class="o">.</span><span class="n">ncomponents</span> <span class="o">+</span> <span class="n">j</span><span class="p">,</span> <span class="n">j</span> <span class="o">*</span> <span class="n">N</span> <span class="o">+</span> <span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="mf">1.0</span>
+
+            <span class="bp">self</span><span class="o">.</span><span class="n">Pl</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">spectral</span><span class="o">.</span><span class="n">sparse_lib</span><span class="o">.</span><span class="n">csc_matrix</span><span class="p">(</span><span class="n">R</span><span class="p">)</span>
+
+        <span class="k">if</span> <span class="n">Dirichlet_recombination</span> <span class="ow">and</span> <span class="nb">type</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">axes</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">])</span><span class="o">.</span><span class="vm">__name__</span> <span class="ow">in</span> <span class="p">[</span><span class="s1">&#39;ChebychevHelper, Ultraspherical&#39;</span><span class="p">]:</span>
+            <span class="n">_Pr</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">spectral</span><span class="o">.</span><span class="n">get_Dirichlet_recombination_matrix</span><span class="p">(</span><span class="n">axis</span><span class="o">=-</span><span class="mi">1</span><span class="p">)</span>
+        <span class="k">else</span><span class="p">:</span>
+            <span class="n">_Pr</span> <span class="o">=</span> <span class="n">Id</span>
+
+        <span class="n">Pr_lhs</span> <span class="o">=</span> <span class="p">{</span><span class="n">comp</span><span class="p">:</span> <span class="p">{</span><span class="n">comp</span><span class="p">:</span> <span class="n">_Pr</span><span class="p">}</span> <span class="k">for</span> <span class="n">comp</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">components</span><span class="p">}</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">Pr</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_setup_operator</span><span class="p">(</span><span class="n">Pr_lhs</span><span class="p">)</span> <span class="o">@</span> <span class="bp">self</span><span class="o">.</span><span class="n">Pl</span><span class="o">.</span><span class="n">T</span></div>
+
+
+<div class="viewcode-block" id="GenericSpectralLinear.solve_system">
+<a class="viewcode-back" href="../../../pySDC/implementations.problem_classes.generic_spectral.html#implementations.problem_classes.generic_spectral.GenericSpectralLinear.solve_system">[docs]</a>
+    <span class="k">def</span> <span class="nf">solve_system</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">rhs</span><span class="p">,</span> <span class="n">dt</span><span class="p">,</span> <span class="n">u0</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="n">skip_itransform</span><span class="o">=</span><span class="kc">False</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
+<span class="w">        </span><span class="sd">&quot;&quot;&quot;</span>
+<span class="sd">        Do an implicit Euler step to solve M u_t + Lu = rhs, with M the mass matrix and L the linear operator as setup by</span>
+<span class="sd">        ``GenericSpectralLinear.setup_L`` and ``GenericSpectralLinear.setup_M``.</span>
+
+<span class="sd">        The implicit Euler step is (M - dt L) u = M rhs. Note that M need not be invertible as long as (M + dt*L) is.</span>
+<span class="sd">        This means solving with dt=0 to mimic explicit methods does not work for all problems, in particular simple DAEs.</span>
+
+<span class="sd">        Note that by putting M rhs on the right hand side, this function can only solve algebraic conditions equal to</span>
+<span class="sd">        zero. If you want something else, it should be easy to overload this function.</span>
+<span class="sd">        &quot;&quot;&quot;</span>
+
+        <span class="n">sp</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">spectral</span><span class="o">.</span><span class="n">sparse_lib</span>
+
+        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">spectral_space</span><span class="p">:</span>
+            <span class="n">rhs_hat</span> <span class="o">=</span> <span class="n">rhs</span><span class="o">.</span><span class="n">copy</span><span class="p">()</span>
+        <span class="k">else</span><span class="p">:</span>
+            <span class="n">rhs_hat</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">spectral</span><span class="o">.</span><span class="n">transform</span><span class="p">(</span><span class="n">rhs</span><span class="p">)</span>
+
+        <span class="n">rhs_hat</span> <span class="o">=</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">M</span> <span class="o">@</span> <span class="n">rhs_hat</span><span class="o">.</span><span class="n">flatten</span><span class="p">())</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="n">rhs_hat</span><span class="o">.</span><span class="n">shape</span><span class="p">)</span>
+        <span class="n">rhs_hat</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">spectral</span><span class="o">.</span><span class="n">put_BCs_in_rhs_hat</span><span class="p">(</span><span class="n">rhs_hat</span><span class="p">)</span>
+        <span class="n">rhs_hat</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">Pl</span> <span class="o">@</span> <span class="n">rhs_hat</span><span class="o">.</span><span class="n">flatten</span><span class="p">()</span>
+
+        <span class="k">if</span> <span class="n">dt</span> <span class="ow">not</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">cached_factorizations</span><span class="o">.</span><span class="n">keys</span><span class="p">():</span>
+            <span class="n">A</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">M</span> <span class="o">+</span> <span class="n">dt</span> <span class="o">*</span> <span class="bp">self</span><span class="o">.</span><span class="n">L</span>
+            <span class="n">A</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">Pl</span> <span class="o">@</span> <span class="bp">self</span><span class="o">.</span><span class="n">spectral</span><span class="o">.</span><span class="n">put_BCs_in_matrix</span><span class="p">(</span><span class="n">A</span><span class="p">)</span> <span class="o">@</span> <span class="bp">self</span><span class="o">.</span><span class="n">Pr</span>
+
+        <span class="c1"># import numpy as np</span>
+        <span class="c1"># if A.shape[0] &lt; 200:</span>
+        <span class="c1">#     import matplotlib.pyplot as plt</span>
+
+        <span class="c1">#     # M = self.spectral.put_BCs_in_matrix(self.L.copy())</span>
+        <span class="c1">#     M = A  # self.L</span>
+        <span class="c1">#     im = plt.imshow((M / abs(M)).real)</span>
+        <span class="c1">#     # im = plt.imshow(np.log10(abs(A.toarray())).real)</span>
+        <span class="c1">#     # im = plt.imshow(((A.toarray())).real)</span>
+        <span class="c1">#     plt.colorbar(im)</span>
+        <span class="c1">#     plt.show()</span>
+
+        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">solver_type</span><span class="o">.</span><span class="n">lower</span><span class="p">()</span> <span class="o">==</span> <span class="s1">&#39;cached_direct&#39;</span><span class="p">:</span>
+            <span class="k">if</span> <span class="n">dt</span> <span class="ow">not</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">cached_factorizations</span><span class="o">.</span><span class="n">keys</span><span class="p">():</span>
+                <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">cached_factorizations</span><span class="p">)</span> <span class="o">&gt;=</span> <span class="bp">self</span><span class="o">.</span><span class="n">max_cached_factorizations</span><span class="p">:</span>
+                    <span class="bp">self</span><span class="o">.</span><span class="n">cached_factorizations</span><span class="o">.</span><span class="n">pop</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">cached_factorizations</span><span class="o">.</span><span class="n">keys</span><span class="p">())[</span><span class="mi">0</span><span class="p">])</span>
+                    <span class="bp">self</span><span class="o">.</span><span class="n">logger</span><span class="o">.</span><span class="n">debug</span><span class="p">(</span><span class="sa">f</span><span class="s1">&#39;Evicted matrix factorization for </span><span class="si">{</span><span class="n">dt</span><span class="si">=:</span><span class="s1">.6f</span><span class="si">}</span><span class="s1"> from cache&#39;</span><span class="p">)</span>
+                <span class="bp">self</span><span class="o">.</span><span class="n">cached_factorizations</span><span class="p">[</span><span class="n">dt</span><span class="p">]</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">spectral</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">factorized</span><span class="p">(</span><span class="n">A</span><span class="p">)</span>
+                <span class="bp">self</span><span class="o">.</span><span class="n">logger</span><span class="o">.</span><span class="n">debug</span><span class="p">(</span><span class="sa">f</span><span class="s1">&#39;Cached matrix factorization for </span><span class="si">{</span><span class="n">dt</span><span class="si">=:</span><span class="s1">.6f</span><span class="si">}</span><span class="s1">&#39;</span><span class="p">)</span>
+                <span class="bp">self</span><span class="o">.</span><span class="n">work_counters</span><span class="p">[</span><span class="s1">&#39;factorizations&#39;</span><span class="p">]()</span>
+
+            <span class="n">_sol_hat</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">cached_factorizations</span><span class="p">[</span><span class="n">dt</span><span class="p">](</span><span class="n">rhs_hat</span><span class="p">)</span>
+            <span class="bp">self</span><span class="o">.</span><span class="n">logger</span><span class="o">.</span><span class="n">debug</span><span class="p">(</span><span class="sa">f</span><span class="s1">&#39;Used cached matrix factorization for </span><span class="si">{</span><span class="n">dt</span><span class="si">=:</span><span class="s1">.6f</span><span class="si">}</span><span class="s1">&#39;</span><span class="p">)</span>
+
+        <span class="k">elif</span> <span class="bp">self</span><span class="o">.</span><span class="n">solver_type</span><span class="o">.</span><span class="n">lower</span><span class="p">()</span> <span class="o">==</span> <span class="s1">&#39;direct&#39;</span><span class="p">:</span>
+            <span class="n">_sol_hat</span> <span class="o">=</span> <span class="n">sp</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">spsolve</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">rhs_hat</span><span class="p">)</span>
+        <span class="k">elif</span> <span class="bp">self</span><span class="o">.</span><span class="n">solver_type</span><span class="o">.</span><span class="n">lower</span><span class="p">()</span> <span class="o">==</span> <span class="s1">&#39;gmres&#39;</span><span class="p">:</span>
+            <span class="n">_sol_hat</span><span class="p">,</span> <span class="n">_</span> <span class="o">=</span> <span class="n">sp</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">gmres</span><span class="p">(</span>
+                <span class="n">A</span><span class="p">,</span>
+                <span class="n">rhs_hat</span><span class="p">,</span>
+                <span class="n">x0</span><span class="o">=</span><span class="n">u0</span><span class="o">.</span><span class="n">flatten</span><span class="p">(),</span>
+                <span class="o">**</span><span class="bp">self</span><span class="o">.</span><span class="n">solver_args</span><span class="p">,</span>
+                <span class="n">callback</span><span class="o">=</span><span class="bp">self</span><span class="o">.</span><span class="n">work_counters</span><span class="p">[</span><span class="bp">self</span><span class="o">.</span><span class="n">solver_type</span><span class="p">],</span>
+                <span class="n">callback_type</span><span class="o">=</span><span class="s1">&#39;legacy&#39;</span><span class="p">,</span>
+            <span class="p">)</span>
+        <span class="k">elif</span> <span class="bp">self</span><span class="o">.</span><span class="n">solver_type</span><span class="o">.</span><span class="n">lower</span><span class="p">()</span> <span class="o">==</span> <span class="s1">&#39;cg&#39;</span><span class="p">:</span>
+            <span class="n">_sol_hat</span><span class="p">,</span> <span class="n">_</span> <span class="o">=</span> <span class="n">sp</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">cg</span><span class="p">(</span>
+                <span class="n">A</span><span class="p">,</span> <span class="n">rhs_hat</span><span class="p">,</span> <span class="n">x0</span><span class="o">=</span><span class="n">u0</span><span class="o">.</span><span class="n">flatten</span><span class="p">(),</span> <span class="o">**</span><span class="bp">self</span><span class="o">.</span><span class="n">solver_args</span><span class="p">,</span> <span class="n">callback</span><span class="o">=</span><span class="bp">self</span><span class="o">.</span><span class="n">work_counters</span><span class="p">[</span><span class="bp">self</span><span class="o">.</span><span class="n">solver_type</span><span class="p">]</span>
+            <span class="p">)</span>
+        <span class="k">else</span><span class="p">:</span>
+            <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span><span class="sa">f</span><span class="s1">&#39;Solver </span><span class="si">{</span><span class="bp">self</span><span class="o">.</span><span class="n">solver_type</span><span class="si">:</span><span class="s1">!</span><span class="si">}</span><span class="s1"> not implemented in </span><span class="si">{</span><span class="nb">type</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span><span class="o">.</span><span class="vm">__name__</span><span class="si">}</span><span class="s1">!&#39;</span><span class="p">)</span>
+
+        <span class="n">sol_hat</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">spectral</span><span class="o">.</span><span class="n">u_init_forward</span>
+        <span class="n">sol_hat</span><span class="p">[</span><span class="o">...</span><span class="p">]</span> <span class="o">=</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">Pr</span> <span class="o">@</span> <span class="n">_sol_hat</span><span class="p">)</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="n">sol_hat</span><span class="o">.</span><span class="n">shape</span><span class="p">)</span>
+
+        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">spectral_space</span><span class="p">:</span>
+            <span class="k">return</span> <span class="n">sol_hat</span>
+        <span class="k">else</span><span class="p">:</span>
+            <span class="n">sol</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">spectral</span><span class="o">.</span><span class="n">u_init</span>
+            <span class="n">sol</span><span class="p">[:]</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">spectral</span><span class="o">.</span><span class="n">itransform</span><span class="p">(</span><span class="n">sol_hat</span><span class="p">)</span><span class="o">.</span><span class="n">real</span>
+
+            <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">spectral</span><span class="o">.</span><span class="n">debug</span><span class="p">:</span>
+                <span class="bp">self</span><span class="o">.</span><span class="n">spectral</span><span class="o">.</span><span class="n">check_BCs</span><span class="p">(</span><span class="n">sol</span><span class="p">)</span>
+
+            <span class="k">return</span> <span class="n">sol</span></div>
+</div>
+
+
+
+<div class="viewcode-block" id="compute_residual_DAE">
+<a class="viewcode-back" href="../../../pySDC/implementations.problem_classes.generic_spectral.html#implementations.problem_classes.generic_spectral.compute_residual_DAE">[docs]</a>
+<span class="k">def</span> <span class="nf">compute_residual_DAE</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">stage</span><span class="o">=</span><span class="s1">&#39;&#39;</span><span class="p">):</span>
+<span class="w">    </span><span class="sd">&quot;&quot;&quot;</span>
+<span class="sd">    Computation of the residual that does not add u_0 - u_m in algebraic equations.</span>
+
+<span class="sd">    Args:</span>
+<span class="sd">        stage (str): The current stage of the step the level belongs to</span>
+<span class="sd">    &quot;&quot;&quot;</span>
+
+    <span class="c1"># get current level and problem description</span>
+    <span class="n">L</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">level</span>
+
+    <span class="c1"># Check if we want to skip the residual computation to gain performance</span>
+    <span class="c1"># Keep in mind that skipping any residual computation is likely to give incorrect outputs of the residual!</span>
+    <span class="k">if</span> <span class="n">stage</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">params</span><span class="o">.</span><span class="n">skip_residual_computation</span><span class="p">:</span>
+        <span class="n">L</span><span class="o">.</span><span class="n">status</span><span class="o">.</span><span class="n">residual</span> <span class="o">=</span> <span class="mf">0.0</span> <span class="k">if</span> <span class="n">L</span><span class="o">.</span><span class="n">status</span><span class="o">.</span><span class="n">residual</span> <span class="ow">is</span> <span class="kc">None</span> <span class="k">else</span> <span class="n">L</span><span class="o">.</span><span class="n">status</span><span class="o">.</span><span class="n">residual</span>
+        <span class="k">return</span> <span class="kc">None</span>
+
+    <span class="c1"># check if there are new values (e.g. from a sweep)</span>
+    <span class="c1"># assert L.status.updated</span>
+
+    <span class="c1"># compute the residual for each node</span>
+
+    <span class="c1"># build QF(u)</span>
+    <span class="n">res_norm</span> <span class="o">=</span> <span class="p">[]</span>
+    <span class="n">res</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">integrate</span><span class="p">()</span>
+    <span class="n">mask</span> <span class="o">=</span> <span class="n">L</span><span class="o">.</span><span class="n">prob</span><span class="o">.</span><span class="n">diff_mask</span>
+    <span class="k">for</span> <span class="n">m</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">coll</span><span class="o">.</span><span class="n">num_nodes</span><span class="p">):</span>
+        <span class="n">res</span><span class="p">[</span><span class="n">m</span><span class="p">][</span><span class="n">mask</span><span class="p">]</span> <span class="o">+=</span> <span class="n">L</span><span class="o">.</span><span class="n">u</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="n">mask</span><span class="p">]</span> <span class="o">-</span> <span class="n">L</span><span class="o">.</span><span class="n">u</span><span class="p">[</span><span class="n">m</span> <span class="o">+</span> <span class="mi">1</span><span class="p">][</span><span class="n">mask</span><span class="p">]</span>
+        <span class="c1"># add tau if associated</span>
+        <span class="k">if</span> <span class="n">L</span><span class="o">.</span><span class="n">tau</span><span class="p">[</span><span class="n">m</span><span class="p">]</span> <span class="ow">is</span> <span class="ow">not</span> <span class="kc">None</span><span class="p">:</span>
+            <span class="n">res</span><span class="p">[</span><span class="n">m</span><span class="p">]</span> <span class="o">+=</span> <span class="n">L</span><span class="o">.</span><span class="n">tau</span><span class="p">[</span><span class="n">m</span><span class="p">]</span>
+        <span class="c1"># use abs function from data type here</span>
+        <span class="n">res_norm</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="nb">abs</span><span class="p">(</span><span class="n">res</span><span class="p">[</span><span class="n">m</span><span class="p">]))</span>
+        <span class="c1"># print(m, [abs(me) for me in res[m]], [abs(me) for me in L.u[0] - L.u[m + 1]])</span>
+
+    <span class="c1"># find maximal residual over the nodes</span>
+    <span class="k">if</span> <span class="n">L</span><span class="o">.</span><span class="n">params</span><span class="o">.</span><span class="n">residual_type</span> <span class="o">==</span> <span class="s1">&#39;full_abs&#39;</span><span class="p">:</span>
+        <span class="n">L</span><span class="o">.</span><span class="n">status</span><span class="o">.</span><span class="n">residual</span> <span class="o">=</span> <span class="nb">max</span><span class="p">(</span><span class="n">res_norm</span><span class="p">)</span>
+    <span class="k">elif</span> <span class="n">L</span><span class="o">.</span><span class="n">params</span><span class="o">.</span><span class="n">residual_type</span> <span class="o">==</span> <span class="s1">&#39;last_abs&#39;</span><span class="p">:</span>
+        <span class="n">L</span><span class="o">.</span><span class="n">status</span><span class="o">.</span><span class="n">residual</span> <span class="o">=</span> <span class="n">res_norm</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
+    <span class="k">elif</span> <span class="n">L</span><span class="o">.</span><span class="n">params</span><span class="o">.</span><span class="n">residual_type</span> <span class="o">==</span> <span class="s1">&#39;full_rel&#39;</span><span class="p">:</span>
+        <span class="n">L</span><span class="o">.</span><span class="n">status</span><span class="o">.</span><span class="n">residual</span> <span class="o">=</span> <span class="nb">max</span><span class="p">(</span><span class="n">res_norm</span><span class="p">)</span> <span class="o">/</span> <span class="nb">abs</span><span class="p">(</span><span class="n">L</span><span class="o">.</span><span class="n">u</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
+    <span class="k">elif</span> <span class="n">L</span><span class="o">.</span><span class="n">params</span><span class="o">.</span><span class="n">residual_type</span> <span class="o">==</span> <span class="s1">&#39;last_rel&#39;</span><span class="p">:</span>
+        <span class="n">L</span><span class="o">.</span><span class="n">status</span><span class="o">.</span><span class="n">residual</span> <span class="o">=</span> <span class="n">res_norm</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="o">/</span> <span class="nb">abs</span><span class="p">(</span><span class="n">L</span><span class="o">.</span><span class="n">u</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
+    <span class="k">else</span><span class="p">:</span>
+        <span class="k">raise</span> <span class="n">ParameterError</span><span class="p">(</span>
+            <span class="sa">f</span><span class="s1">&#39;residual_type = </span><span class="si">{</span><span class="n">L</span><span class="o">.</span><span class="n">params</span><span class="o">.</span><span class="n">residual_type</span><span class="si">}</span><span class="s1"> not implemented, choose &#39;</span>
+            <span class="sa">f</span><span class="s1">&#39;full_abs, last_abs, full_rel or last_rel instead&#39;</span>
+        <span class="p">)</span>
+
+    <span class="c1"># indicate that the residual has seen the new values</span>
+    <span class="n">L</span><span class="o">.</span><span class="n">status</span><span class="o">.</span><span class="n">updated</span> <span class="o">=</span> <span class="kc">False</span>
+
+    <span class="k">return</span> <span class="kc">None</span></div>
+
+
+
+<div class="viewcode-block" id="compute_residual_DAE_MPI">
+<a class="viewcode-back" href="../../../pySDC/implementations.problem_classes.generic_spectral.html#implementations.problem_classes.generic_spectral.compute_residual_DAE_MPI">[docs]</a>
+<span class="k">def</span> <span class="nf">compute_residual_DAE_MPI</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">stage</span><span class="o">=</span><span class="kc">None</span><span class="p">):</span>
+<span class="w">    </span><span class="sd">&quot;&quot;&quot;</span>
+<span class="sd">    Computation of the residual using the collocation matrix Q</span>
+
+<span class="sd">    Args:</span>
+<span class="sd">        stage (str): The current stage of the step the level belongs to</span>
+<span class="sd">    &quot;&quot;&quot;</span>
+    <span class="kn">from</span> <span class="nn">mpi4py</span> <span class="kn">import</span> <span class="n">MPI</span>
+
+    <span class="n">L</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">level</span>
+
+    <span class="c1"># Check if we want to skip the residual computation to gain performance</span>
+    <span class="c1"># Keep in mind that skipping any residual computation is likely to give incorrect outputs of the residual!</span>
+    <span class="k">if</span> <span class="n">stage</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">params</span><span class="o">.</span><span class="n">skip_residual_computation</span><span class="p">:</span>
+        <span class="n">L</span><span class="o">.</span><span class="n">status</span><span class="o">.</span><span class="n">residual</span> <span class="o">=</span> <span class="mf">0.0</span> <span class="k">if</span> <span class="n">L</span><span class="o">.</span><span class="n">status</span><span class="o">.</span><span class="n">residual</span> <span class="ow">is</span> <span class="kc">None</span> <span class="k">else</span> <span class="n">L</span><span class="o">.</span><span class="n">status</span><span class="o">.</span><span class="n">residual</span>
+        <span class="k">return</span> <span class="kc">None</span>
+
+    <span class="c1"># compute the residual for each node</span>
+
+    <span class="c1"># build QF(u)</span>
+    <span class="n">res</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">integrate</span><span class="p">(</span><span class="n">last_only</span><span class="o">=</span><span class="n">L</span><span class="o">.</span><span class="n">params</span><span class="o">.</span><span class="n">residual_type</span><span class="p">[:</span><span class="mi">4</span><span class="p">]</span> <span class="o">==</span> <span class="s1">&#39;last&#39;</span><span class="p">)</span>
+    <span class="n">mask</span> <span class="o">=</span> <span class="n">L</span><span class="o">.</span><span class="n">prob</span><span class="o">.</span><span class="n">diff_mask</span>
+    <span class="n">res</span><span class="p">[</span><span class="n">mask</span><span class="p">]</span> <span class="o">+=</span> <span class="n">L</span><span class="o">.</span><span class="n">u</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="n">mask</span><span class="p">]</span> <span class="o">-</span> <span class="n">L</span><span class="o">.</span><span class="n">u</span><span class="p">[</span><span class="bp">self</span><span class="o">.</span><span class="n">rank</span> <span class="o">+</span> <span class="mi">1</span><span class="p">][</span><span class="n">mask</span><span class="p">]</span>
+    <span class="c1"># add tau if associated</span>
+    <span class="k">if</span> <span class="n">L</span><span class="o">.</span><span class="n">tau</span><span class="p">[</span><span class="bp">self</span><span class="o">.</span><span class="n">rank</span><span class="p">]</span> <span class="ow">is</span> <span class="ow">not</span> <span class="kc">None</span><span class="p">:</span>
+        <span class="n">res</span> <span class="o">+=</span> <span class="n">L</span><span class="o">.</span><span class="n">tau</span><span class="p">[</span><span class="bp">self</span><span class="o">.</span><span class="n">rank</span><span class="p">]</span>
+    <span class="c1"># use abs function from data type here</span>
+    <span class="n">res_norm</span> <span class="o">=</span> <span class="nb">abs</span><span class="p">(</span><span class="n">res</span><span class="p">)</span>
+
+    <span class="c1"># find maximal residual over the nodes</span>
+    <span class="k">if</span> <span class="n">L</span><span class="o">.</span><span class="n">params</span><span class="o">.</span><span class="n">residual_type</span> <span class="o">==</span> <span class="s1">&#39;full_abs&#39;</span><span class="p">:</span>
+        <span class="n">L</span><span class="o">.</span><span class="n">status</span><span class="o">.</span><span class="n">residual</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">comm</span><span class="o">.</span><span class="n">allreduce</span><span class="p">(</span><span class="n">res_norm</span><span class="p">,</span> <span class="n">op</span><span class="o">=</span><span class="n">MPI</span><span class="o">.</span><span class="n">MAX</span><span class="p">)</span>
+    <span class="k">elif</span> <span class="n">L</span><span class="o">.</span><span class="n">params</span><span class="o">.</span><span class="n">residual_type</span> <span class="o">==</span> <span class="s1">&#39;last_abs&#39;</span><span class="p">:</span>
+        <span class="n">L</span><span class="o">.</span><span class="n">status</span><span class="o">.</span><span class="n">residual</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">comm</span><span class="o">.</span><span class="n">bcast</span><span class="p">(</span><span class="n">res_norm</span><span class="p">,</span> <span class="n">root</span><span class="o">=</span><span class="bp">self</span><span class="o">.</span><span class="n">comm</span><span class="o">.</span><span class="n">size</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span>
+    <span class="k">elif</span> <span class="n">L</span><span class="o">.</span><span class="n">params</span><span class="o">.</span><span class="n">residual_type</span> <span class="o">==</span> <span class="s1">&#39;full_rel&#39;</span><span class="p">:</span>
+        <span class="n">L</span><span class="o">.</span><span class="n">status</span><span class="o">.</span><span class="n">residual</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">comm</span><span class="o">.</span><span class="n">allreduce</span><span class="p">(</span><span class="n">res_norm</span> <span class="o">/</span> <span class="nb">abs</span><span class="p">(</span><span class="n">L</span><span class="o">.</span><span class="n">u</span><span class="p">[</span><span class="mi">0</span><span class="p">]),</span> <span class="n">op</span><span class="o">=</span><span class="n">MPI</span><span class="o">.</span><span class="n">MAX</span><span class="p">)</span>
+    <span class="k">elif</span> <span class="n">L</span><span class="o">.</span><span class="n">params</span><span class="o">.</span><span class="n">residual_type</span> <span class="o">==</span> <span class="s1">&#39;last_rel&#39;</span><span class="p">:</span>
+        <span class="n">L</span><span class="o">.</span><span class="n">status</span><span class="o">.</span><span class="n">residual</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">comm</span><span class="o">.</span><span class="n">bcast</span><span class="p">(</span><span class="n">res_norm</span> <span class="o">/</span> <span class="nb">abs</span><span class="p">(</span><span class="n">L</span><span class="o">.</span><span class="n">u</span><span class="p">[</span><span class="mi">0</span><span class="p">]),</span> <span class="n">root</span><span class="o">=</span><span class="bp">self</span><span class="o">.</span><span class="n">comm</span><span class="o">.</span><span class="n">size</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span>
+    <span class="k">else</span><span class="p">:</span>
+        <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span><span class="sa">f</span><span class="s1">&#39;residual type </span><span class="se">\&quot;</span><span class="si">{</span><span class="n">L</span><span class="o">.</span><span class="n">params</span><span class="o">.</span><span class="n">residual_type</span><span class="si">}</span><span class="se">\&quot;</span><span class="s1"> not implemented!&#39;</span><span class="p">)</span>
+
+    <span class="c1"># indicate that the residual has seen the new values</span>
+    <span class="n">L</span><span class="o">.</span><span class="n">status</span><span class="o">.</span><span class="n">updated</span> <span class="o">=</span> <span class="kc">False</span>
+
+    <span class="k">return</span> <span class="kc">None</span></div>
+
+</pre></div>
+
+            <div class="clearer"></div>
+          </div>
+        </div>
+      </div>
+      <div class="sphinxsidebar" role="navigation" aria-label="Main">
+        <div class="sphinxsidebarwrapper">
+<search id="searchbox" style="display: none" role="search">
+  <h3 id="searchlabel">Quick search</h3>
+    <div class="searchformwrapper">
+    <form class="search" action="../../../search.html" method="get">
+      <input type="text" name="q" aria-labelledby="searchlabel" autocomplete="off" autocorrect="off" autocapitalize="off" spellcheck="false"/>
+      <input type="submit" value="Go" />
+    </form>
+    </div>
+</search>
+<script>document.getElementById('searchbox').style.display = "block"</script>
+        </div>
+      </div>
+      <div class="clearer"></div>
+    </div>
+    <div class="related" role="navigation" aria-label="Related">
+      <h3>Navigation</h3>
+      <ul>
+        <li class="right" style="margin-right: 10px">
+          <a href="../../../genindex.html" title="General Index"
+             >index</a></li>
+        <li class="right" >
+          <a href="../../../py-modindex.html" title="Python Module Index"
+             >modules</a> |</li>
+        <li class="nav-item nav-item-0"><a href="../../../index.html">pySDC 5.5.1 documentation</a> &#187;</li>
+          <li class="nav-item nav-item-1"><a href="../../index.html" >Module code</a> &#187;</li>
+        <li class="nav-item nav-item-this"><a href="">implementations.problem_classes.generic_spectral</a></li> 
+      </ul>
+    </div>
+    <div class="footer" role="contentinfo">
+    &#169; Copyright 2024, Robert Speck.
+      Created using <a href="https://www.sphinx-doc.org/">Sphinx</a> 8.0.2.
+    </div>
+  </body>
+</html>
\ No newline at end of file
diff --git a/_modules/index.html b/_modules/index.html
index 71cea0afab..fcee97368a 100644
--- a/_modules/index.html
+++ b/_modules/index.html
@@ -53,6 +53,7 @@ <h1>All modules for which code is available</h1>
 <li><a href="helpers/problem_helper.html">helpers.problem_helper</a></li>
 <li><a href="helpers/pysdc_helper.html">helpers.pysdc_helper</a></li>
 <li><a href="helpers/setup_helper.html">helpers.setup_helper</a></li>
+<li><a href="helpers/spectral_helper.html">helpers.spectral_helper</a></li>
 <li><a href="helpers/stats_helper.html">helpers.stats_helper</a></li>
 <li><a href="helpers/testing.html">helpers.testing</a></li>
 <li><a href="helpers/transfer_helper.html">helpers.transfer_helper</a></li>
@@ -95,6 +96,7 @@ <h1>All modules for which code is available</h1>
 <li><a href="implementations/problem_classes/Boussinesq_2D_FD_imex.html">implementations.problem_classes.Boussinesq_2D_FD_imex</a></li>
 <li><a href="implementations/problem_classes/Brusselator.html">implementations.problem_classes.Brusselator</a></li>
 <li><a href="implementations/problem_classes/BuckConverter.html">implementations.problem_classes.BuckConverter</a></li>
+<li><a href="implementations/problem_classes/Burgers.html">implementations.problem_classes.Burgers</a></li>
 <li><a href="implementations/problem_classes/DiscontinuousTestODE.html">implementations.problem_classes.DiscontinuousTestODE</a></li>
 <li><a href="implementations/problem_classes/FastWaveSlowWave_0D.html">implementations.problem_classes.FastWaveSlowWave_0D</a></li>
 <li><a href="implementations/problem_classes/FermiPastaUlamTsingou.html">implementations.problem_classes.FermiPastaUlamTsingou</a></li>
@@ -107,6 +109,7 @@ <h1>All modules for which code is available</h1>
 <li><a href="implementations/problem_classes/HarmonicOscillator.html">implementations.problem_classes.HarmonicOscillator</a></li>
 <li><a href="implementations/problem_classes/HeatEquation_1D_FEniCS_matrix_forced.html">implementations.problem_classes.HeatEquation_1D_FEniCS_matrix_forced</a></li>
 <li><a href="implementations/problem_classes/HeatEquation_2D_PETSc_forced.html">implementations.problem_classes.HeatEquation_2D_PETSc_forced</a></li>
+<li><a href="implementations/problem_classes/HeatEquation_Chebychev.html">implementations.problem_classes.HeatEquation_Chebychev</a></li>
 <li><a href="implementations/problem_classes/HeatEquation_ND_FD.html">implementations.problem_classes.HeatEquation_ND_FD</a></li>
 <li><a href="implementations/problem_classes/HeatEquation_ND_FD_CuPy.html">implementations.problem_classes.HeatEquation_ND_FD_CuPy</a></li>
 <li><a href="implementations/problem_classes/HenonHeiles.html">implementations.problem_classes.HenonHeiles</a></li>
@@ -117,6 +120,7 @@ <h1>All modules for which code is available</h1>
 <li><a href="implementations/problem_classes/PenningTrap_3D.html">implementations.problem_classes.PenningTrap_3D</a></li>
 <li><a href="implementations/problem_classes/Piline.html">implementations.problem_classes.Piline</a></li>
 <li><a href="implementations/problem_classes/Quench.html">implementations.problem_classes.Quench</a></li>
+<li><a href="implementations/problem_classes/RayleighBenard.html">implementations.problem_classes.RayleighBenard</a></li>
 <li><a href="implementations/problem_classes/TestEquation_0D.html">implementations.problem_classes.TestEquation_0D</a></li>
 <li><a href="implementations/problem_classes/Van_der_Pol_implicit.html">implementations.problem_classes.Van_der_Pol_implicit</a></li>
 <li><a href="implementations/problem_classes/VorticityVelocity_2D_FEniCS_periodic.html">implementations.problem_classes.VorticityVelocity_2D_FEniCS_periodic</a></li>
@@ -131,6 +135,7 @@ <h1>All modules for which code is available</h1>
 <li><a href="implementations/problem_classes/boussinesq_helpers/unflatten.html">implementations.problem_classes.boussinesq_helpers.unflatten</a></li>
 <li><a href="implementations/problem_classes/generic_MPIFFT_Laplacian.html">implementations.problem_classes.generic_MPIFFT_Laplacian</a></li>
 <li><a href="implementations/problem_classes/generic_ND_FD.html">implementations.problem_classes.generic_ND_FD</a></li>
+<li><a href="implementations/problem_classes/generic_spectral.html">implementations.problem_classes.generic_spectral</a></li>
 <li><a href="implementations/problem_classes/nonlinear_ODE_1.html">implementations.problem_classes.nonlinear_ODE_1</a></li>
 <li><a href="implementations/problem_classes/odeScalar.html">implementations.problem_classes.odeScalar</a></li>
 <li><a href="implementations/problem_classes/odeSystem.html">implementations.problem_classes.odeSystem</a></li>
diff --git a/_sources/pySDC/helpers.rst.txt b/_sources/pySDC/helpers.rst.txt
index 6cdef6896f..0a0ddd32ac 100644
--- a/_sources/pySDC/helpers.rst.txt
+++ b/_sources/pySDC/helpers.rst.txt
@@ -12,6 +12,7 @@ Submodules
    helpers.problem_helper
    helpers.pysdc_helper
    helpers.setup_helper
+   helpers.spectral_helper
    helpers.stats_helper
    helpers.testing
    helpers.transfer_helper
diff --git a/_sources/pySDC/helpers.spectral_helper.rst.txt b/_sources/pySDC/helpers.spectral_helper.rst.txt
new file mode 100644
index 0000000000..a7976a3ba0
--- /dev/null
+++ b/_sources/pySDC/helpers.spectral_helper.rst.txt
@@ -0,0 +1,7 @@
+helpers.spectral\_helper module
+===============================
+
+.. automodule:: helpers.spectral_helper
+   :members:
+   :undoc-members:
+   :show-inheritance:
diff --git a/_sources/pySDC/implementations.problem_classes.Burgers.rst.txt b/_sources/pySDC/implementations.problem_classes.Burgers.rst.txt
new file mode 100644
index 0000000000..693e9c7190
--- /dev/null
+++ b/_sources/pySDC/implementations.problem_classes.Burgers.rst.txt
@@ -0,0 +1,7 @@
+implementations.problem\_classes.Burgers module
+===============================================
+
+.. automodule:: implementations.problem_classes.Burgers
+   :members:
+   :undoc-members:
+   :show-inheritance:
diff --git a/_sources/pySDC/implementations.problem_classes.HeatEquation_Chebychev.rst.txt b/_sources/pySDC/implementations.problem_classes.HeatEquation_Chebychev.rst.txt
new file mode 100644
index 0000000000..a061a0e2e6
--- /dev/null
+++ b/_sources/pySDC/implementations.problem_classes.HeatEquation_Chebychev.rst.txt
@@ -0,0 +1,7 @@
+implementations.problem\_classes.HeatEquation\_Chebychev module
+===============================================================
+
+.. automodule:: implementations.problem_classes.HeatEquation_Chebychev
+   :members:
+   :undoc-members:
+   :show-inheritance:
diff --git a/_sources/pySDC/implementations.problem_classes.RayleighBenard.rst.txt b/_sources/pySDC/implementations.problem_classes.RayleighBenard.rst.txt
new file mode 100644
index 0000000000..53cf916671
--- /dev/null
+++ b/_sources/pySDC/implementations.problem_classes.RayleighBenard.rst.txt
@@ -0,0 +1,7 @@
+implementations.problem\_classes.RayleighBenard module
+======================================================
+
+.. automodule:: implementations.problem_classes.RayleighBenard
+   :members:
+   :undoc-members:
+   :show-inheritance:
diff --git a/_sources/pySDC/implementations.problem_classes.generic_spectral.rst.txt b/_sources/pySDC/implementations.problem_classes.generic_spectral.rst.txt
new file mode 100644
index 0000000000..539d7101a8
--- /dev/null
+++ b/_sources/pySDC/implementations.problem_classes.generic_spectral.rst.txt
@@ -0,0 +1,7 @@
+implementations.problem\_classes.generic\_spectral module
+=========================================================
+
+.. automodule:: implementations.problem_classes.generic_spectral
+   :members:
+   :undoc-members:
+   :show-inheritance:
diff --git a/_sources/pySDC/implementations.problem_classes.rst.txt b/_sources/pySDC/implementations.problem_classes.rst.txt
index 1e6f102608..534ed27beb 100644
--- a/_sources/pySDC/implementations.problem_classes.rst.txt
+++ b/_sources/pySDC/implementations.problem_classes.rst.txt
@@ -31,6 +31,7 @@ Submodules
    implementations.problem_classes.Boussinesq_2D_FD_imex
    implementations.problem_classes.Brusselator
    implementations.problem_classes.BuckConverter
+   implementations.problem_classes.Burgers
    implementations.problem_classes.DiscontinuousTestODE
    implementations.problem_classes.FastWaveSlowWave_0D
    implementations.problem_classes.FermiPastaUlamTsingou
@@ -43,6 +44,7 @@ Submodules
    implementations.problem_classes.HarmonicOscillator
    implementations.problem_classes.HeatEquation_1D_FEniCS_matrix_forced
    implementations.problem_classes.HeatEquation_2D_PETSc_forced
+   implementations.problem_classes.HeatEquation_Chebychev
    implementations.problem_classes.HeatEquation_ND_FD
    implementations.problem_classes.HeatEquation_ND_FD_CuPy
    implementations.problem_classes.HenonHeiles
@@ -53,11 +55,13 @@ Submodules
    implementations.problem_classes.PenningTrap_3D
    implementations.problem_classes.Piline
    implementations.problem_classes.Quench
+   implementations.problem_classes.RayleighBenard
    implementations.problem_classes.TestEquation_0D
    implementations.problem_classes.Van_der_Pol_implicit
    implementations.problem_classes.VorticityVelocity_2D_FEniCS_periodic
    implementations.problem_classes.generic_MPIFFT_Laplacian
    implementations.problem_classes.generic_ND_FD
+   implementations.problem_classes.generic_spectral
    implementations.problem_classes.nonlinear_ODE_1
    implementations.problem_classes.odeScalar
    implementations.problem_classes.odeSystem
diff --git a/coverage/class_index.html b/coverage/class_index.html
index d1fb4c18c1..3f05e6480c 100644
--- a/coverage/class_index.html
+++ b/coverage/class_index.html
@@ -11,7 +11,7 @@
 <header>
     <div class="content">
         <h1>Coverage report:
-            <span class="pc_cov">78%</span>
+            <span class="pc_cov">79%</span>
         </h1>
         <aside id="help_panel_wrapper">
             <input id="help_panel_state" type="checkbox">
@@ -55,7 +55,7 @@ <h2>
         </h2>
         <p class="text">
             <a class="nav" href="https://coverage.readthedocs.io/en/7.6.1">coverage.py v7.6.1</a>,
-            created at 2024-09-19 09:13 +0000
+            created at 2024-09-20 16:55 +0000
         </p>
     </div>
 </header>
@@ -408,6 +408,54 @@ <h2>
                 <td>2</td>
                 <td class="right" data-ratio="19 19">100%</td>
             </tr>
+            <tr class="region">
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html#t7">pySDC/helpers/spectral_helper.py</a></td>
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html#t7"><data value='SpectralHelper1D'>SpectralHelper1D</data></a></td>
+                <td>29</td>
+                <td>13</td>
+                <td>6</td>
+                <td class="right" data-ratio="16 29">55%</td>
+            </tr>
+            <tr class="region">
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html#t174">pySDC/helpers/spectral_helper.py</a></td>
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html#t174"><data value='ChebychevHelper'>ChebychevHelper</data></a></td>
+                <td>123</td>
+                <td>3</td>
+                <td>8</td>
+                <td class="right" data-ratio="120 123">98%</td>
+            </tr>
+            <tr class="region">
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html#t543">pySDC/helpers/spectral_helper.py</a></td>
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html#t543"><data value='UltrasphericalHelper'>UltrasphericalHelper</data></a></td>
+                <td>28</td>
+                <td>1</td>
+                <td>0</td>
+                <td class="right" data-ratio="27 28">96%</td>
+            </tr>
+            <tr class="region">
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html#t672">pySDC/helpers/spectral_helper.py</a></td>
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html#t672"><data value='FFTHelper'>FFTHelper</data></a></td>
+                <td>29</td>
+                <td>5</td>
+                <td>2</td>
+                <td class="right" data-ratio="24 29">83%</td>
+            </tr>
+            <tr class="region">
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html#t808">pySDC/helpers/spectral_helper.py</a></td>
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html#t808"><data value='SpectralHelper'>SpectralHelper</data></a></td>
+                <td>441</td>
+                <td>40</td>
+                <td>15</td>
+                <td class="right" data-ratio="401 441">91%</td>
+            </tr>
+            <tr class="region">
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html">pySDC/helpers/spectral_helper.py</a></td>
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html"><data value=''><span class='no-noun'>(no class)</span></data></a></td>
+                <td>114</td>
+                <td>0</td>
+                <td>0</td>
+                <td class="right" data-ratio="114 114">100%</td>
+            </tr>
             <tr class="region">
                 <td class="name left"><a href="z_41564483689e14fd_stats_helper_py.html">pySDC/helpers/stats_helper.py</a></td>
                 <td class="name left"><a href="z_41564483689e14fd_stats_helper_py.html"><data value=''><span class='no-noun'>(no class)</span></data></a></td>
@@ -940,9 +988,9 @@ <h2>
                 <td class="name left"><a href="z_3017878c2b4bc866_mesh_py.html#t12">pySDC/implementations/datatype_classes/mesh.py</a></td>
                 <td class="name left"><a href="z_3017878c2b4bc866_mesh_py.html#t12"><data value='mesh'>mesh</data></a></td>
                 <td>26</td>
+                <td>0</td>
                 <td>1</td>
-                <td>1</td>
-                <td class="right" data-ratio="25 26">96%</td>
+                <td class="right" data-ratio="26 26">100%</td>
             </tr>
             <tr class="region">
                 <td class="name left"><a href="z_3017878c2b4bc866_mesh_py.html#t128">pySDC/implementations/datatype_classes/mesh.py</a></td>
@@ -1600,6 +1648,30 @@ <h2>
                 <td>0</td>
                 <td class="right" data-ratio="10 10">100%</td>
             </tr>
+            <tr class="region">
+                <td class="name left"><a href="z_91faa57f8583c837_Burgers_py.html#t7">pySDC/implementations/problem_classes/Burgers.py</a></td>
+                <td class="name left"><a href="z_91faa57f8583c837_Burgers_py.html#t7"><data value='Burgers1D'>Burgers1D</data></a></td>
+                <td>31</td>
+                <td>0</td>
+                <td>39</td>
+                <td class="right" data-ratio="31 31">100%</td>
+            </tr>
+            <tr class="region">
+                <td class="name left"><a href="z_91faa57f8583c837_Burgers_py.html#t151">pySDC/implementations/problem_classes/Burgers.py</a></td>
+                <td class="name left"><a href="z_91faa57f8583c837_Burgers_py.html#t151"><data value='Burgers2D'>Burgers2D</data></a></td>
+                <td>49</td>
+                <td>5</td>
+                <td>57</td>
+                <td class="right" data-ratio="44 49">90%</td>
+            </tr>
+            <tr class="region">
+                <td class="name left"><a href="z_91faa57f8583c837_Burgers_py.html">pySDC/implementations/problem_classes/Burgers.py</a></td>
+                <td class="name left"><a href="z_91faa57f8583c837_Burgers_py.html"><data value=''><span class='no-noun'>(no class)</span></data></a></td>
+                <td>16</td>
+                <td>0</td>
+                <td>4</td>
+                <td class="right" data-ratio="16 16">100%</td>
+            </tr>
             <tr class="region">
                 <td class="name left"><a href="z_91faa57f8583c837_DiscontinuousTestODE_py.html#t8">pySDC/implementations/problem_classes/DiscontinuousTestODE.py</a></td>
                 <td class="name left"><a href="z_91faa57f8583c837_DiscontinuousTestODE_py.html#t8"><data value='DiscontinuousTestODE'>DiscontinuousTestODE</data></a></td>
@@ -1912,6 +1984,46 @@ <h2>
                 <td>0</td>
                 <td class="right" data-ratio="14 14">100%</td>
             </tr>
+            <tr class="region">
+                <td class="name left"><a href="z_91faa57f8583c837_HeatEquation_Chebychev_py.html#t9">pySDC/implementations/problem_classes/HeatEquation_Chebychev.py</a></td>
+                <td class="name left"><a href="z_91faa57f8583c837_HeatEquation_Chebychev_py.html#t9"><data value='Heat1DChebychev'>Heat1DChebychev</data></a></td>
+                <td>46</td>
+                <td>0</td>
+                <td>2</td>
+                <td class="right" data-ratio="46 46">100%</td>
+            </tr>
+            <tr class="region">
+                <td class="name left"><a href="z_91faa57f8583c837_HeatEquation_Chebychev_py.html#t124">pySDC/implementations/problem_classes/HeatEquation_Chebychev.py</a></td>
+                <td class="name left"><a href="z_91faa57f8583c837_HeatEquation_Chebychev_py.html#t124"><data value='Heat1DUltraspherical'>Heat1DUltraspherical</data></a></td>
+                <td>44</td>
+                <td>0</td>
+                <td>2</td>
+                <td class="right" data-ratio="44 44">100%</td>
+            </tr>
+            <tr class="region">
+                <td class="name left"><a href="z_91faa57f8583c837_HeatEquation_Chebychev_py.html#t232">pySDC/implementations/problem_classes/HeatEquation_Chebychev.py</a></td>
+                <td class="name left"><a href="z_91faa57f8583c837_HeatEquation_Chebychev_py.html#t232"><data value='Heat2DChebychev'>Heat2DChebychev</data></a></td>
+                <td>52</td>
+                <td>0</td>
+                <td>0</td>
+                <td class="right" data-ratio="52 52">100%</td>
+            </tr>
+            <tr class="region">
+                <td class="name left"><a href="z_91faa57f8583c837_HeatEquation_Chebychev_py.html#t340">pySDC/implementations/problem_classes/HeatEquation_Chebychev.py</a></td>
+                <td class="name left"><a href="z_91faa57f8583c837_HeatEquation_Chebychev_py.html#t340"><data value='Heat2DUltraspherical'>Heat2DUltraspherical</data></a></td>
+                <td>49</td>
+                <td>0</td>
+                <td>0</td>
+                <td class="right" data-ratio="49 49">100%</td>
+            </tr>
+            <tr class="region">
+                <td class="name left"><a href="z_91faa57f8583c837_HeatEquation_Chebychev_py.html">pySDC/implementations/problem_classes/HeatEquation_Chebychev.py</a></td>
+                <td class="name left"><a href="z_91faa57f8583c837_HeatEquation_Chebychev_py.html"><data value=''><span class='no-noun'>(no class)</span></data></a></td>
+                <td>29</td>
+                <td>0</td>
+                <td>0</td>
+                <td class="right" data-ratio="29 29">100%</td>
+            </tr>
             <tr class="region">
                 <td class="name left"><a href="z_91faa57f8583c837_HeatEquation_ND_FD_py.html#t7">pySDC/implementations/problem_classes/HeatEquation_ND_FD.py</a></td>
                 <td class="name left"><a href="z_91faa57f8583c837_HeatEquation_ND_FD_py.html#t7"><data value='heatNd_unforced'>heatNd_unforced</data></a></td>
@@ -2096,6 +2208,46 @@ <h2>
                 <td>0</td>
                 <td class="right" data-ratio="22 22">100%</td>
             </tr>
+            <tr class="region">
+                <td class="name left"><a href="z_91faa57f8583c837_RayleighBenard_py.html#t11">pySDC/implementations/problem_classes/RayleighBenard.py</a></td>
+                <td class="name left"><a href="z_91faa57f8583c837_RayleighBenard_py.html#t11"><data value='RayleighBenard'>RayleighBenard</data></a></td>
+                <td>137</td>
+                <td>17</td>
+                <td>57</td>
+                <td class="right" data-ratio="120 137">88%</td>
+            </tr>
+            <tr class="region">
+                <td class="name left"><a href="z_91faa57f8583c837_RayleighBenard_py.html#t413">pySDC/implementations/problem_classes/RayleighBenard.py</a></td>
+                <td class="name left"><a href="z_91faa57f8583c837_RayleighBenard_py.html#t413"><data value='CFLLimit'>CFLLimit</data></a></td>
+                <td>32</td>
+                <td>18</td>
+                <td>1</td>
+                <td class="right" data-ratio="14 32">44%</td>
+            </tr>
+            <tr class="region">
+                <td class="name left"><a href="z_91faa57f8583c837_RayleighBenard_py.html#t497">pySDC/implementations/problem_classes/RayleighBenard.py</a></td>
+                <td class="name left"><a href="z_91faa57f8583c837_RayleighBenard_py.html#t497"><data value='LogCFL'>LogCFL</data></a></td>
+                <td>3</td>
+                <td>3</td>
+                <td>0</td>
+                <td class="right" data-ratio="0 3">0%</td>
+            </tr>
+            <tr class="region">
+                <td class="name left"><a href="z_91faa57f8583c837_RayleighBenard_py.html#t525">pySDC/implementations/problem_classes/RayleighBenard.py</a></td>
+                <td class="name left"><a href="z_91faa57f8583c837_RayleighBenard_py.html#t525"><data value='LogAnalysisVariables'>LogAnalysisVariables</data></a></td>
+                <td>9</td>
+                <td>9</td>
+                <td>0</td>
+                <td class="right" data-ratio="0 9">0%</td>
+            </tr>
+            <tr class="region">
+                <td class="name left"><a href="z_91faa57f8583c837_RayleighBenard_py.html">pySDC/implementations/problem_classes/RayleighBenard.py</a></td>
+                <td class="name left"><a href="z_91faa57f8583c837_RayleighBenard_py.html"><data value=''><span class='no-noun'>(no class)</span></data></a></td>
+                <td>28</td>
+                <td>0</td>
+                <td>2</td>
+                <td class="right" data-ratio="28 28">100%</td>
+            </tr>
             <tr class="region">
                 <td class="name left"><a href="z_91faa57f8583c837_TestEquation_0D_py.html#t8">pySDC/implementations/problem_classes/TestEquation_0D.py</a></td>
                 <td class="name left"><a href="z_91faa57f8583c837_TestEquation_0D_py.html#t8"><data value='testequation0d'>testequation0d</data></a></td>
@@ -2116,9 +2268,9 @@ <h2>
                 <td class="name left"><a href="z_91faa57f8583c837_Van_der_Pol_implicit_py.html#t9">pySDC/implementations/problem_classes/Van_der_Pol_implicit.py</a></td>
                 <td class="name left"><a href="z_91faa57f8583c837_Van_der_Pol_implicit_py.html#t9"><data value='vanderpol'>vanderpol</data></a></td>
                 <td>46</td>
+                <td>3</td>
                 <td>2</td>
-                <td>2</td>
-                <td class="right" data-ratio="44 46">96%</td>
+                <td class="right" data-ratio="43 46">93%</td>
             </tr>
             <tr class="region">
                 <td class="name left"><a href="z_91faa57f8583c837_Van_der_Pol_implicit_py.html">pySDC/implementations/problem_classes/Van_der_Pol_implicit.py</a></td>
@@ -2352,6 +2504,22 @@ <h2>
                 <td>0</td>
                 <td class="right" data-ratio="21 21">100%</td>
             </tr>
+            <tr class="region">
+                <td class="name left"><a href="z_91faa57f8583c837_generic_spectral_py.html#t7">pySDC/implementations/problem_classes/generic_spectral.py</a></td>
+                <td class="name left"><a href="z_91faa57f8583c837_generic_spectral_py.html#t7"><data value='GenericSpectralLinear'>GenericSpectralLinear</data></a></td>
+                <td>82</td>
+                <td>17</td>
+                <td>1</td>
+                <td class="right" data-ratio="65 82">79%</td>
+            </tr>
+            <tr class="region">
+                <td class="name left"><a href="z_91faa57f8583c837_generic_spectral_py.html">pySDC/implementations/problem_classes/generic_spectral.py</a></td>
+                <td class="name left"><a href="z_91faa57f8583c837_generic_spectral_py.html"><data value=''><span class='no-noun'>(no class)</span></data></a></td>
+                <td>59</td>
+                <td>43</td>
+                <td>3</td>
+                <td class="right" data-ratio="16 59">27%</td>
+            </tr>
             <tr class="region">
                 <td class="name left"><a href="z_91faa57f8583c837_nonlinear_ODE_1_py.html#t9">pySDC/implementations/problem_classes/nonlinear_ODE_1.py</a></td>
                 <td class="name left"><a href="z_91faa57f8583c837_nonlinear_ODE_1_py.html#t9"><data value='nonlinear_ODE_1'>nonlinear_ODE_1</data></a></td>
@@ -5076,9 +5244,9 @@ <h2>
                 <td class="name left"><a href="z_064a9f2a35945611_generate_statistics_py.html">pySDC/projects/soft_failure/generate_statistics.py</a></td>
                 <td class="name left"><a href="z_064a9f2a35945611_generate_statistics_py.html"><data value=''><span class='no-noun'>(no class)</span></data></a></td>
                 <td>199</td>
-                <td>49</td>
+                <td>50</td>
                 <td>16</td>
-                <td class="right" data-ratio="150 199">75%</td>
+                <td class="right" data-ratio="149 199">75%</td>
             </tr>
             <tr class="region">
                 <td class="name left"><a href="z_064a9f2a35945611_implicit_sweeper_faults_py.html#t10">pySDC/projects/soft_failure/implicit_sweeper_faults.py</a></td>
@@ -5092,9 +5260,9 @@ <h2>
                 <td class="name left"><a href="z_064a9f2a35945611_implicit_sweeper_faults_py.html#t25">pySDC/projects/soft_failure/implicit_sweeper_faults.py</a></td>
                 <td class="name left"><a href="z_064a9f2a35945611_implicit_sweeper_faults_py.html#t25"><data value='implicit_sweeper_faults'>implicit_sweeper_faults</data></a></td>
                 <td>115</td>
-                <td>5</td>
+                <td>8</td>
                 <td>0</td>
-                <td class="right" data-ratio="110 115">96%</td>
+                <td class="right" data-ratio="107 115">93%</td>
             </tr>
             <tr class="region">
                 <td class="name left"><a href="z_064a9f2a35945611_implicit_sweeper_faults_py.html">pySDC/projects/soft_failure/implicit_sweeper_faults.py</a></td>
@@ -5389,10 +5557,10 @@ <h2>
             <tr class="total">
                 <td class="name left">Total</td>
                 <td class="name left">&nbsp;</td>
-                <td>26137</td>
-                <td>5718</td>
-                <td>4998</td>
-                <td class="right" data-ratio="20419 26137">78%</td>
+                <td>27567</td>
+                <td>5896</td>
+                <td>5197</td>
+                <td class="right" data-ratio="21671 27567">79%</td>
             </tr>
         </tfoot>
     </table>
@@ -5405,7 +5573,7 @@ <h2>
     <div class="content">
         <p>
             <a class="nav" href="https://coverage.readthedocs.io/en/7.6.1">coverage.py v7.6.1</a>,
-            created at 2024-09-19 09:13 +0000
+            created at 2024-09-20 16:55 +0000
         </p>
     </div>
     <aside class="hidden">
diff --git a/coverage/coverage-badge.svg b/coverage/coverage-badge.svg
index 5cb6ec5c45..69273e9c6e 100644
--- a/coverage/coverage-badge.svg
+++ b/coverage/coverage-badge.svg
@@ -1 +1 @@
-<svg xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" width="114" height="20" role="img" aria-label="coverage: 78.12%"><title>coverage: 78.12%</title><linearGradient id="s" x2="0" y2="100%"><stop offset="0" stop-color="#bbb" stop-opacity=".1"/><stop offset="1" stop-opacity=".1"/></linearGradient><clipPath id="r"><rect width="114" height="20" rx="3" fill="#fff"/></clipPath><g clip-path="url(#r)"><rect width="61" height="20" fill="#555"/><rect x="61" width="53" height="20" fill="#97ca00"/><rect width="114" height="20" fill="url(#s)"/></g><g fill="#fff" text-anchor="middle" font-family="Verdana,Geneva,DejaVu Sans,sans-serif" text-rendering="geometricPrecision" font-size="110"><text aria-hidden="true" x="315" y="150" fill="#010101" fill-opacity=".3" transform="scale(.1)" textLength="510">coverage</text><text x="315" y="140" transform="scale(.1)" fill="#fff" textLength="510">coverage</text><text aria-hidden="true" x="865" y="150" fill="#010101" fill-opacity=".3" transform="scale(.1)" textLength="430">78.12%</text><text x="865" y="140" transform="scale(.1)" fill="#fff" textLength="430">78.12%</text></g></svg>
\ No newline at end of file
+<svg xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" width="114" height="20" role="img" aria-label="coverage: 78.61%"><title>coverage: 78.61%</title><linearGradient id="s" x2="0" y2="100%"><stop offset="0" stop-color="#bbb" stop-opacity=".1"/><stop offset="1" stop-opacity=".1"/></linearGradient><clipPath id="r"><rect width="114" height="20" rx="3" fill="#fff"/></clipPath><g clip-path="url(#r)"><rect width="61" height="20" fill="#555"/><rect x="61" width="53" height="20" fill="#97ca00"/><rect width="114" height="20" fill="url(#s)"/></g><g fill="#fff" text-anchor="middle" font-family="Verdana,Geneva,DejaVu Sans,sans-serif" text-rendering="geometricPrecision" font-size="110"><text aria-hidden="true" x="315" y="150" fill="#010101" fill-opacity=".3" transform="scale(.1)" textLength="510">coverage</text><text x="315" y="140" transform="scale(.1)" fill="#fff" textLength="510">coverage</text><text aria-hidden="true" x="865" y="150" fill="#010101" fill-opacity=".3" transform="scale(.1)" textLength="430">78.61%</text><text x="865" y="140" transform="scale(.1)" fill="#fff" textLength="430">78.61%</text></g></svg>
\ No newline at end of file
diff --git a/coverage/function_index.html b/coverage/function_index.html
index a91686d515..452c1ef1f3 100644
--- a/coverage/function_index.html
+++ b/coverage/function_index.html
@@ -11,7 +11,7 @@
 <header>
     <div class="content">
         <h1>Coverage report:
-            <span class="pc_cov">78%</span>
+            <span class="pc_cov">79%</span>
         </h1>
         <aside id="help_panel_wrapper">
             <input id="help_panel_state" type="checkbox">
@@ -55,7 +55,7 @@ <h2>
         </h2>
         <p class="text">
             <a class="nav" href="https://coverage.readthedocs.io/en/7.6.1">coverage.py v7.6.1</a>,
-            created at 2024-09-19 09:13 +0000
+            created at 2024-09-20 16:55 +0000
         </p>
     </div>
 </header>
@@ -1256,6 +1256,670 @@ <h2>
                 <td>0</td>
                 <td class="right" data-ratio="1 1">100%</td>
             </tr>
+            <tr class="region">
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html#t29">pySDC/helpers/spectral_helper.py</a></td>
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html#t29"><data value='init__'>SpectralHelper1D.__init__</data></a></td>
+                <td>7</td>
+                <td>1</td>
+                <td>0</td>
+                <td class="right" data-ratio="6 7">86%</td>
+            </tr>
+            <tr class="region">
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html#t49">pySDC/helpers/spectral_helper.py</a></td>
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html#t49"><data value='setup_GPU'>SpectralHelper1D.setup_GPU</data></a></td>
+                <td>9</td>
+                <td>9</td>
+                <td>0</td>
+                <td class="right" data-ratio="0 9">0%</td>
+            </tr>
+            <tr class="region">
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html#t62">pySDC/helpers/spectral_helper.py</a></td>
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html#t62"><data value='get_Id'>SpectralHelper1D.get_Id</data></a></td>
+                <td>1</td>
+                <td>0</td>
+                <td>0</td>
+                <td class="right" data-ratio="1 1">100%</td>
+            </tr>
+            <tr class="region">
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html#t71">pySDC/helpers/spectral_helper.py</a></td>
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html#t71"><data value='get_zero'>SpectralHelper1D.get_zero</data></a></td>
+                <td>1</td>
+                <td>1</td>
+                <td>0</td>
+                <td class="right" data-ratio="0 1">0%</td>
+            </tr>
+            <tr class="region">
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html#t92">pySDC/helpers/spectral_helper.py</a></td>
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html#t92"><data value='get_empty_operator_matrix'>SpectralHelper1D.get_empty_operator_matrix</data></a></td>
+                <td>1</td>
+                <td>1</td>
+                <td>0</td>
+                <td class="right" data-ratio="0 1">0%</td>
+            </tr>
+            <tr class="region">
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html#t105">pySDC/helpers/spectral_helper.py</a></td>
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html#t105"><data value='get_basis_change_matrix'>SpectralHelper1D.get_basis_change_matrix</data></a></td>
+                <td>1</td>
+                <td>0</td>
+                <td>0</td>
+                <td class="right" data-ratio="1 1">100%</td>
+            </tr>
+            <tr class="region">
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html#t138">pySDC/helpers/spectral_helper.py</a></td>
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html#t138"><data value='get_filter_matrix'>SpectralHelper1D.get_filter_matrix</data></a></td>
+                <td>9</td>
+                <td>1</td>
+                <td>1</td>
+                <td class="right" data-ratio="8 9">89%</td>
+            </tr>
+            <tr class="region">
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html#t185">pySDC/helpers/spectral_helper.py</a></td>
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html#t185"><data value='init__'>ChebychevHelper.__init__</data></a></td>
+                <td>8</td>
+                <td>0</td>
+                <td>0</td>
+                <td class="right" data-ratio="8 8">100%</td>
+            </tr>
+            <tr class="region">
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html#t208">pySDC/helpers/spectral_helper.py</a></td>
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html#t208"><data value='get_1dgrid'>ChebychevHelper.get_1dgrid</data></a></td>
+                <td>1</td>
+                <td>0</td>
+                <td>0</td>
+                <td class="right" data-ratio="1 1">100%</td>
+            </tr>
+            <tr class="region">
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html#t219">pySDC/helpers/spectral_helper.py</a></td>
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html#t219"><data value='get_wavenumbers'>ChebychevHelper.get_wavenumbers</data></a></td>
+                <td>1</td>
+                <td>0</td>
+                <td>0</td>
+                <td class="right" data-ratio="1 1">100%</td>
+            </tr>
+            <tr class="region">
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html#t223">pySDC/helpers/spectral_helper.py</a></td>
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html#t223"><data value='get_conv'>ChebychevHelper.get_conv</data></a></td>
+                <td>18</td>
+                <td>2</td>
+                <td>1</td>
+                <td class="right" data-ratio="16 18">89%</td>
+            </tr>
+            <tr class="region">
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html#t247">pySDC/helpers/spectral_helper.py</a></td>
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html#t247"><data value='get_forward_conv'>ChebychevHelper.get_conv.get_forward_conv</data></a></td>
+                <td>8</td>
+                <td>0</td>
+                <td>1</td>
+                <td class="right" data-ratio="8 8">100%</td>
+            </tr>
+            <tr class="region">
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html#t276">pySDC/helpers/spectral_helper.py</a></td>
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html#t276"><data value='get_basis_change_matrix'>ChebychevHelper.get_basis_change_matrix</data></a></td>
+                <td>1</td>
+                <td>0</td>
+                <td>0</td>
+                <td class="right" data-ratio="1 1">100%</td>
+            </tr>
+            <tr class="region">
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html#t291">pySDC/helpers/spectral_helper.py</a></td>
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html#t291"><data value='get_integration_matrix'>ChebychevHelper.get_integration_matrix</data></a></td>
+                <td>6</td>
+                <td>0</td>
+                <td>1</td>
+                <td class="right" data-ratio="6 6">100%</td>
+            </tr>
+            <tr class="region">
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html#t314">pySDC/helpers/spectral_helper.py</a></td>
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html#t314"><data value='get_differentiation_matrix'>ChebychevHelper.get_differentiation_matrix</data></a></td>
+                <td>6</td>
+                <td>0</td>
+                <td>0</td>
+                <td class="right" data-ratio="6 6">100%</td>
+            </tr>
+            <tr class="region">
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html#t332">pySDC/helpers/spectral_helper.py</a></td>
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html#t332"><data value='get_norm'>ChebychevHelper.get_norm</data></a></td>
+                <td>4</td>
+                <td>0</td>
+                <td>0</td>
+                <td class="right" data-ratio="4 4">100%</td>
+            </tr>
+            <tr class="region">
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html#t347">pySDC/helpers/spectral_helper.py</a></td>
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html#t347"><data value='get_fft_shuffle'>ChebychevHelper.get_fft_shuffle</data></a></td>
+                <td>8</td>
+                <td>0</td>
+                <td>0</td>
+                <td class="right" data-ratio="8 8">100%</td>
+            </tr>
+            <tr class="region">
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html#t370">pySDC/helpers/spectral_helper.py</a></td>
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html#t370"><data value='get_fft_shift'>ChebychevHelper.get_fft_shift</data></a></td>
+                <td>8</td>
+                <td>0</td>
+                <td>0</td>
+                <td class="right" data-ratio="8 8">100%</td>
+            </tr>
+            <tr class="region">
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html#t391">pySDC/helpers/spectral_helper.py</a></td>
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html#t391"><data value='get_fft_utils'>ChebychevHelper.get_fft_utils</data></a></td>
+                <td>6</td>
+                <td>0</td>
+                <td>0</td>
+                <td class="right" data-ratio="6 6">100%</td>
+            </tr>
+            <tr class="region">
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html#t411">pySDC/helpers/spectral_helper.py</a></td>
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html#t411"><data value='transform'>ChebychevHelper.transform</data></a></td>
+                <td>13</td>
+                <td>0</td>
+                <td>2</td>
+                <td class="right" data-ratio="13 13">100%</td>
+            </tr>
+            <tr class="region">
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html#t444">pySDC/helpers/spectral_helper.py</a></td>
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html#t444"><data value='itransform'>ChebychevHelper.itransform</data></a></td>
+                <td>13</td>
+                <td>0</td>
+                <td>1</td>
+                <td class="right" data-ratio="13 13">100%</td>
+            </tr>
+            <tr class="region">
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html#t476">pySDC/helpers/spectral_helper.py</a></td>
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html#t476"><data value='get_BC'>ChebychevHelper.get_BC</data></a></td>
+                <td>5</td>
+                <td>1</td>
+                <td>1</td>
+                <td class="right" data-ratio="4 5">80%</td>
+            </tr>
+            <tr class="region">
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html#t492">pySDC/helpers/spectral_helper.py</a></td>
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html#t492"><data value='get_integ_BC_row'>ChebychevHelper.get_integ_BC_row</data></a></td>
+                <td>5</td>
+                <td>0</td>
+                <td>0</td>
+                <td class="right" data-ratio="5 5">100%</td>
+            </tr>
+            <tr class="region">
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html#t506">pySDC/helpers/spectral_helper.py</a></td>
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html#t506"><data value='get_Dirichlet_BC_row'>ChebychevHelper.get_Dirichlet_BC_row</data></a></td>
+                <td>8</td>
+                <td>0</td>
+                <td>1</td>
+                <td class="right" data-ratio="8 8">100%</td>
+            </tr>
+            <tr class="region">
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html#t528">pySDC/helpers/spectral_helper.py</a></td>
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html#t528"><data value='get_Dirichlet_recombination_matrix'>ChebychevHelper.get_Dirichlet_recombination_matrix</data></a></td>
+                <td>4</td>
+                <td>0</td>
+                <td>0</td>
+                <td class="right" data-ratio="4 4">100%</td>
+            </tr>
+            <tr class="region">
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html#t554">pySDC/helpers/spectral_helper.py</a></td>
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html#t554"><data value='get_differentiation_matrix'>UltrasphericalHelper.get_differentiation_matrix</data></a></td>
+                <td>5</td>
+                <td>0</td>
+                <td>0</td>
+                <td class="right" data-ratio="5 5">100%</td>
+            </tr>
+            <tr class="region">
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html#t570">pySDC/helpers/spectral_helper.py</a></td>
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html#t570"><data value='get_S'>UltrasphericalHelper.get_S</data></a></td>
+                <td>9</td>
+                <td>0</td>
+                <td>0</td>
+                <td class="right" data-ratio="9 9">100%</td>
+            </tr>
+            <tr class="region">
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html#t596">pySDC/helpers/spectral_helper.py</a></td>
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html#t596"><data value='get_basis_change_matrix'>UltrasphericalHelper.get_basis_change_matrix</data></a></td>
+                <td>10</td>
+                <td>1</td>
+                <td>0</td>
+                <td class="right" data-ratio="9 10">90%</td>
+            </tr>
+            <tr class="region">
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html#t622">pySDC/helpers/spectral_helper.py</a></td>
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html#t622"><data value='get_integration_matrix'>UltrasphericalHelper.get_integration_matrix</data></a></td>
+                <td>1</td>
+                <td>0</td>
+                <td>0</td>
+                <td class="right" data-ratio="1 1">100%</td>
+            </tr>
+            <tr class="region">
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html#t654">pySDC/helpers/spectral_helper.py</a></td>
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html#t654"><data value='get_integration_constant'>UltrasphericalHelper.get_integration_constant</data></a></td>
+                <td>3</td>
+                <td>0</td>
+                <td>0</td>
+                <td class="right" data-ratio="3 3">100%</td>
+            </tr>
+            <tr class="region">
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html#t673">pySDC/helpers/spectral_helper.py</a></td>
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html#t673"><data value='init__'>FFTHelper.__init__</data></a></td>
+                <td>1</td>
+                <td>0</td>
+                <td>0</td>
+                <td class="right" data-ratio="1 1">100%</td>
+            </tr>
+            <tr class="region">
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html#t687">pySDC/helpers/spectral_helper.py</a></td>
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html#t687"><data value='get_1dgrid'>FFTHelper.get_1dgrid</data></a></td>
+                <td>2</td>
+                <td>0</td>
+                <td>0</td>
+                <td class="right" data-ratio="2 2">100%</td>
+            </tr>
+            <tr class="region">
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html#t694">pySDC/helpers/spectral_helper.py</a></td>
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html#t694"><data value='get_wavenumbers'>FFTHelper.get_wavenumbers</data></a></td>
+                <td>1</td>
+                <td>0</td>
+                <td>0</td>
+                <td class="right" data-ratio="1 1">100%</td>
+            </tr>
+            <tr class="region">
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html#t700">pySDC/helpers/spectral_helper.py</a></td>
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html#t700"><data value='get_differentiation_matrix'>FFTHelper.get_differentiation_matrix</data></a></td>
+                <td>6</td>
+                <td>3</td>
+                <td>1</td>
+                <td class="right" data-ratio="3 6">50%</td>
+            </tr>
+            <tr class="region">
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html#t721">pySDC/helpers/spectral_helper.py</a></td>
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html#t721"><data value='get_integration_matrix'>FFTHelper.get_integration_matrix</data></a></td>
+                <td>3</td>
+                <td>0</td>
+                <td>0</td>
+                <td class="right" data-ratio="3 3">100%</td>
+            </tr>
+            <tr class="region">
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html#t735">pySDC/helpers/spectral_helper.py</a></td>
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html#t735"><data value='transform'>FFTHelper.transform</data></a></td>
+                <td>1</td>
+                <td>0</td>
+                <td>1</td>
+                <td class="right" data-ratio="1 1">100%</td>
+            </tr>
+            <tr class="region">
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html#t748">pySDC/helpers/spectral_helper.py</a></td>
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html#t748"><data value='itransform'>FFTHelper.itransform</data></a></td>
+                <td>1</td>
+                <td>0</td>
+                <td>0</td>
+                <td class="right" data-ratio="1 1">100%</td>
+            </tr>
+            <tr class="region">
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html#t761">pySDC/helpers/spectral_helper.py</a></td>
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html#t761"><data value='get_BC'>FFTHelper.get_BC</data></a></td>
+                <td>8</td>
+                <td>2</td>
+                <td>0</td>
+                <td class="right" data-ratio="6 8">75%</td>
+            </tr>
+            <tr class="region">
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html#t785">pySDC/helpers/spectral_helper.py</a></td>
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html#t785"><data value='get_Nyquist_mode_index'>FFTHelper.get_Nyquist_mode_index</data></a></td>
+                <td>3</td>
+                <td>0</td>
+                <td>0</td>
+                <td class="right" data-ratio="3 3">100%</td>
+            </tr>
+            <tr class="region">
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html#t799">pySDC/helpers/spectral_helper.py</a></td>
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html#t799"><data value='get_integ_BC_row'>FFTHelper.get_integ_BC_row</data></a></td>
+                <td>3</td>
+                <td>0</td>
+                <td>0</td>
+                <td class="right" data-ratio="3 3">100%</td>
+            </tr>
+            <tr class="region">
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html#t845">pySDC/helpers/spectral_helper.py</a></td>
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html#t845"><data value='setup_GPU'>SpectralHelper.setup_GPU</data></a></td>
+                <td>10</td>
+                <td>10</td>
+                <td>0</td>
+                <td class="right" data-ratio="0 10">0%</td>
+            </tr>
+            <tr class="region">
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html#t861">pySDC/helpers/spectral_helper.py</a></td>
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html#t861"><data value='init__'>SpectralHelper.__init__</data></a></td>
+                <td>11</td>
+                <td>1</td>
+                <td>0</td>
+                <td class="right" data-ratio="10 11">91%</td>
+            </tr>
+            <tr class="region">
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html#t887">pySDC/helpers/spectral_helper.py</a></td>
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html#t887"><data value='u_init'>SpectralHelper.u_init</data></a></td>
+                <td>1</td>
+                <td>0</td>
+                <td>0</td>
+                <td class="right" data-ratio="1 1">100%</td>
+            </tr>
+            <tr class="region">
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html#t894">pySDC/helpers/spectral_helper.py</a></td>
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html#t894"><data value='u_init_forward'>SpectralHelper.u_init_forward</data></a></td>
+                <td>1</td>
+                <td>0</td>
+                <td>0</td>
+                <td class="right" data-ratio="1 1">100%</td>
+            </tr>
+            <tr class="region">
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html#t901">pySDC/helpers/spectral_helper.py</a></td>
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html#t901"><data value='shape'>SpectralHelper.shape</data></a></td>
+                <td>1</td>
+                <td>1</td>
+                <td>0</td>
+                <td class="right" data-ratio="0 1">0%</td>
+            </tr>
+            <tr class="region">
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html#t908">pySDC/helpers/spectral_helper.py</a></td>
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html#t908"><data value='ndim'>SpectralHelper.ndim</data></a></td>
+                <td>1</td>
+                <td>0</td>
+                <td>0</td>
+                <td class="right" data-ratio="1 1">100%</td>
+            </tr>
+            <tr class="region">
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html#t912">pySDC/helpers/spectral_helper.py</a></td>
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html#t912"><data value='ncomponents'>SpectralHelper.ncomponents</data></a></td>
+                <td>1</td>
+                <td>0</td>
+                <td>0</td>
+                <td class="right" data-ratio="1 1">100%</td>
+            </tr>
+            <tr class="region">
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html#t916">pySDC/helpers/spectral_helper.py</a></td>
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html#t916"><data value='V'>SpectralHelper.V</data></a></td>
+                <td>1</td>
+                <td>0</td>
+                <td>0</td>
+                <td class="right" data-ratio="1 1">100%</td>
+            </tr>
+            <tr class="region">
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html#t922">pySDC/helpers/spectral_helper.py</a></td>
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html#t922"><data value='add_axis'>SpectralHelper.add_axis</data></a></td>
+                <td>10</td>
+                <td>0</td>
+                <td>1</td>
+                <td class="right" data-ratio="10 10">100%</td>
+            </tr>
+            <tr class="region">
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html#t945">pySDC/helpers/spectral_helper.py</a></td>
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html#t945"><data value='add_component'>SpectralHelper.add_component</data></a></td>
+                <td>6</td>
+                <td>0</td>
+                <td>2</td>
+                <td class="right" data-ratio="6 6">100%</td>
+            </tr>
+            <tr class="region">
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html#t962">pySDC/helpers/spectral_helper.py</a></td>
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html#t962"><data value='index'>SpectralHelper.index</data></a></td>
+                <td>4</td>
+                <td>0</td>
+                <td>1</td>
+                <td class="right" data-ratio="4 4">100%</td>
+            </tr>
+            <tr class="region">
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html#t979">pySDC/helpers/spectral_helper.py</a></td>
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html#t979"><data value='get_empty_operator_matrix'>SpectralHelper.get_empty_operator_matrix</data></a></td>
+                <td>3</td>
+                <td>0</td>
+                <td>0</td>
+                <td class="right" data-ratio="3 3">100%</td>
+            </tr>
+            <tr class="region">
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html#t990">pySDC/helpers/spectral_helper.py</a></td>
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html#t990"><data value='get_BC'>SpectralHelper.get_BC</data></a></td>
+                <td>21</td>
+                <td>2</td>
+                <td>0</td>
+                <td class="right" data-ratio="19 21">90%</td>
+            </tr>
+            <tr class="region">
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html#t1039">pySDC/helpers/spectral_helper.py</a></td>
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html#t1039"><data value='remove_BC'>SpectralHelper.remove_BC</data></a></td>
+                <td>9</td>
+                <td>1</td>
+                <td>0</td>
+                <td class="right" data-ratio="8 9">89%</td>
+            </tr>
+            <tr class="region">
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html#t1073">pySDC/helpers/spectral_helper.py</a></td>
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html#t1073"><data value='add_BC'>SpectralHelper.add_BC</data></a></td>
+                <td>15</td>
+                <td>1</td>
+                <td>0</td>
+                <td class="right" data-ratio="14 15">93%</td>
+            </tr>
+            <tr class="region">
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html#t1125">pySDC/helpers/spectral_helper.py</a></td>
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html#t1125"><data value='setup_BCs'>SpectralHelper.setup_BCs</data></a></td>
+                <td>8</td>
+                <td>0</td>
+                <td>0</td>
+                <td class="right" data-ratio="8 8">100%</td>
+            </tr>
+            <tr class="region">
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html#t1143">pySDC/helpers/spectral_helper.py</a></td>
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html#t1143"><data value='check_BCs'>SpectralHelper.check_BCs</data></a></td>
+                <td>13</td>
+                <td>2</td>
+                <td>0</td>
+                <td class="right" data-ratio="11 13">85%</td>
+            </tr>
+            <tr class="region">
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html#t1172">pySDC/helpers/spectral_helper.py</a></td>
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html#t1172"><data value='put_BCs_in_matrix'>SpectralHelper.put_BCs_in_matrix</data></a></td>
+                <td>1</td>
+                <td>0</td>
+                <td>0</td>
+                <td class="right" data-ratio="1 1">100%</td>
+            </tr>
+            <tr class="region">
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html#t1178">pySDC/helpers/spectral_helper.py</a></td>
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html#t1178"><data value='put_BCs_in_rhs_hat'>SpectralHelper.put_BCs_in_rhs_hat</data></a></td>
+                <td>11</td>
+                <td>0</td>
+                <td>0</td>
+                <td class="right" data-ratio="11 11">100%</td>
+            </tr>
+            <tr class="region">
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html#t1207">pySDC/helpers/spectral_helper.py</a></td>
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html#t1207"><data value='put_BCs_in_rhs'>SpectralHelper.put_BCs_in_rhs</data></a></td>
+                <td>14</td>
+                <td>0</td>
+                <td>0</td>
+                <td class="right" data-ratio="14 14">100%</td>
+            </tr>
+            <tr class="region">
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html#t1245">pySDC/helpers/spectral_helper.py</a></td>
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html#t1245"><data value='add_equation_lhs'>SpectralHelper.add_equation_lhs</data></a></td>
+                <td>2</td>
+                <td>0</td>
+                <td>0</td>
+                <td class="right" data-ratio="2 2">100%</td>
+            </tr>
+            <tr class="region">
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html#t1284">pySDC/helpers/spectral_helper.py</a></td>
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html#t1284"><data value='convert_operator_matrix_to_operator'>SpectralHelper.convert_operator_matrix_to_operator</data></a></td>
+                <td>3</td>
+                <td>0</td>
+                <td>0</td>
+                <td class="right" data-ratio="3 3">100%</td>
+            </tr>
+            <tr class="region">
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html#t1300">pySDC/helpers/spectral_helper.py</a></td>
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html#t1300"><data value='get_wavenumbers'>SpectralHelper.get_wavenumbers</data></a></td>
+                <td>2</td>
+                <td>0</td>
+                <td>0</td>
+                <td class="right" data-ratio="2 2">100%</td>
+            </tr>
+            <tr class="region">
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html#t1307">pySDC/helpers/spectral_helper.py</a></td>
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html#t1307"><data value='get_grid'>SpectralHelper.get_grid</data></a></td>
+                <td>2</td>
+                <td>0</td>
+                <td>0</td>
+                <td class="right" data-ratio="2 2">100%</td>
+            </tr>
+            <tr class="region">
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html#t1314">pySDC/helpers/spectral_helper.py</a></td>
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html#t1314"><data value='get_fft'>SpectralHelper.get_fft</data></a></td>
+                <td>22</td>
+                <td>0</td>
+                <td>0</td>
+                <td class="right" data-ratio="22 22">100%</td>
+            </tr>
+            <tr class="region">
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html#t1371">pySDC/helpers/spectral_helper.py</a></td>
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html#t1371"><data value='setup_fft'>SpectralHelper.setup_fft</data></a></td>
+                <td>12</td>
+                <td>0</td>
+                <td>0</td>
+                <td class="right" data-ratio="12 12">100%</td>
+            </tr>
+            <tr class="region">
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html#t1417">pySDC/helpers/spectral_helper.py</a></td>
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html#t1417"><data value='transform_fft'>SpectralHelper._transform_fft</data></a></td>
+                <td>2</td>
+                <td>0</td>
+                <td>0</td>
+                <td class="right" data-ratio="2 2">100%</td>
+            </tr>
+            <tr class="region">
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html#t1432">pySDC/helpers/spectral_helper.py</a></td>
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html#t1432"><data value='transform_dct'>SpectralHelper._transform_dct</data></a></td>
+                <td>27</td>
+                <td>0</td>
+                <td>2</td>
+                <td class="right" data-ratio="27 27">100%</td>
+            </tr>
+            <tr class="region">
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html#t1489">pySDC/helpers/spectral_helper.py</a></td>
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html#t1489"><data value='transform_single_component'>SpectralHelper.transform_single_component</data></a></td>
+                <td>30</td>
+                <td>1</td>
+                <td>0</td>
+                <td class="right" data-ratio="29 30">97%</td>
+            </tr>
+            <tr class="region">
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html#t1562">pySDC/helpers/spectral_helper.py</a></td>
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html#t1562"><data value='transform'>SpectralHelper.transform</data></a></td>
+                <td>7</td>
+                <td>0</td>
+                <td>0</td>
+                <td class="right" data-ratio="7 7">100%</td>
+            </tr>
+            <tr class="region">
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html#t1594">pySDC/helpers/spectral_helper.py</a></td>
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html#t1594"><data value='transform_ifft'>SpectralHelper._transform_ifft</data></a></td>
+                <td>2</td>
+                <td>0</td>
+                <td>0</td>
+                <td class="right" data-ratio="2 2">100%</td>
+            </tr>
+            <tr class="region">
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html#t1599">pySDC/helpers/spectral_helper.py</a></td>
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html#t1599"><data value='transform_idct'>SpectralHelper._transform_idct</data></a></td>
+                <td>30</td>
+                <td>0</td>
+                <td>2</td>
+                <td class="right" data-ratio="30 30">100%</td>
+            </tr>
+            <tr class="region">
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html#t1651">pySDC/helpers/spectral_helper.py</a></td>
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html#t1651"><data value='itransform_single_component'>SpectralHelper.itransform_single_component</data></a></td>
+                <td>29</td>
+                <td>1</td>
+                <td>0</td>
+                <td class="right" data-ratio="28 29">97%</td>
+            </tr>
+            <tr class="region">
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html#t1722">pySDC/helpers/spectral_helper.py</a></td>
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html#t1722"><data value='get_aligned'>SpectralHelper.get_aligned</data></a></td>
+                <td>33</td>
+                <td>4</td>
+                <td>0</td>
+                <td class="right" data-ratio="29 33">88%</td>
+            </tr>
+            <tr class="region">
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html#t1787">pySDC/helpers/spectral_helper.py</a></td>
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html#t1787"><data value='itransform'>SpectralHelper.itransform</data></a></td>
+                <td>7</td>
+                <td>0</td>
+                <td>0</td>
+                <td class="right" data-ratio="7 7">100%</td>
+            </tr>
+            <tr class="region">
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html#t1819">pySDC/helpers/spectral_helper.py</a></td>
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html#t1819"><data value='get_local_slice_of_1D_matrix'>SpectralHelper.get_local_slice_of_1D_matrix</data></a></td>
+                <td>1</td>
+                <td>0</td>
+                <td>0</td>
+                <td class="right" data-ratio="1 1">100%</td>
+            </tr>
+            <tr class="region">
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html#t1834">pySDC/helpers/spectral_helper.py</a></td>
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html#t1834"><data value='get_filter_matrix'>SpectralHelper.get_filter_matrix</data></a></td>
+                <td>5</td>
+                <td>3</td>
+                <td>1</td>
+                <td class="right" data-ratio="2 5">40%</td>
+            </tr>
+            <tr class="region">
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html#t1849">pySDC/helpers/spectral_helper.py</a></td>
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html#t1849"><data value='get_differentiation_matrix'>SpectralHelper.get_differentiation_matrix</data></a></td>
+                <td>18</td>
+                <td>0</td>
+                <td>1</td>
+                <td class="right" data-ratio="18 18">100%</td>
+            </tr>
+            <tr class="region">
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html#t1887">pySDC/helpers/spectral_helper.py</a></td>
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html#t1887"><data value='get_integration_matrix'>SpectralHelper.get_integration_matrix</data></a></td>
+                <td>18</td>
+                <td>0</td>
+                <td>1</td>
+                <td class="right" data-ratio="18 18">100%</td>
+            </tr>
+            <tr class="region">
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html#t1925">pySDC/helpers/spectral_helper.py</a></td>
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html#t1925"><data value='get_Id'>SpectralHelper.get_Id</data></a></td>
+                <td>15</td>
+                <td>0</td>
+                <td>1</td>
+                <td class="right" data-ratio="15 15">100%</td>
+            </tr>
+            <tr class="region">
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html#t1955">pySDC/helpers/spectral_helper.py</a></td>
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html#t1955"><data value='get_Dirichlet_recombination_matrix'>SpectralHelper.get_Dirichlet_recombination_matrix</data></a></td>
+                <td>13</td>
+                <td>13</td>
+                <td>1</td>
+                <td class="right" data-ratio="0 13">0%</td>
+            </tr>
+            <tr class="region">
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html#t1986">pySDC/helpers/spectral_helper.py</a></td>
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html#t1986"><data value='get_basis_change_matrix'>SpectralHelper.get_basis_change_matrix</data></a></td>
+                <td>19</td>
+                <td>0</td>
+                <td>2</td>
+                <td class="right" data-ratio="19 19">100%</td>
+            </tr>
+            <tr class="region">
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html">pySDC/helpers/spectral_helper.py</a></td>
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html"><data value=''><span class='no-noun'>(no function)</span></data></a></td>
+                <td>114</td>
+                <td>0</td>
+                <td>0</td>
+                <td class="right" data-ratio="114 114">100%</td>
+            </tr>
             <tr class="region">
                 <td class="name left"><a href="z_41564483689e14fd_stats_helper_py.html#t4">pySDC/helpers/stats_helper.py</a></td>
                 <td class="name left"><a href="z_41564483689e14fd_stats_helper_py.html#t4"><data value='filter_stats'>filter_stats</data></a></td>
@@ -3140,9 +3804,9 @@ <h2>
                 <td class="name left"><a href="z_3017878c2b4bc866_mesh_py.html#t65">pySDC/implementations/datatype_classes/mesh.py</a></td>
                 <td class="name left"><a href="z_3017878c2b4bc866_mesh_py.html#t65"><data value='abs__'>mesh.__abs__</data></a></td>
                 <td>7</td>
-                <td>1</td>
                 <td>0</td>
-                <td class="right" data-ratio="6 7">86%</td>
+                <td>0</td>
+                <td class="right" data-ratio="7 7">100%</td>
             </tr>
             <tr class="region">
                 <td class="name left"><a href="z_3017878c2b4bc866_mesh_py.html#t85">pySDC/implementations/datatype_classes/mesh.py</a></td>
@@ -4873,20 +5537,84 @@ <h2>
                 <td class="right" data-ratio="18 18">100%</td>
             </tr>
             <tr class="region">
-                <td class="name left"><a href="z_91faa57f8583c837_BuckConverter_py.html#t173">pySDC/implementations/problem_classes/BuckConverter.py</a></td>
-                <td class="name left"><a href="z_91faa57f8583c837_BuckConverter_py.html#t173"><data value='u_exact'>buck_converter.u_exact</data></a></td>
-                <td>6</td>
+                <td class="name left"><a href="z_91faa57f8583c837_BuckConverter_py.html#t173">pySDC/implementations/problem_classes/BuckConverter.py</a></td>
+                <td class="name left"><a href="z_91faa57f8583c837_BuckConverter_py.html#t173"><data value='u_exact'>buck_converter.u_exact</data></a></td>
+                <td>6</td>
+                <td>0</td>
+                <td>0</td>
+                <td class="right" data-ratio="6 6">100%</td>
+            </tr>
+            <tr class="region">
+                <td class="name left"><a href="z_91faa57f8583c837_BuckConverter_py.html">pySDC/implementations/problem_classes/BuckConverter.py</a></td>
+                <td class="name left"><a href="z_91faa57f8583c837_BuckConverter_py.html"><data value=''><span class='no-noun'>(no function)</span></data></a></td>
+                <td>10</td>
+                <td>0</td>
+                <td>0</td>
+                <td class="right" data-ratio="10 10">100%</td>
+            </tr>
+            <tr class="region">
+                <td class="name left"><a href="z_91faa57f8583c837_Burgers_py.html#t25">pySDC/implementations/problem_classes/Burgers.py</a></td>
+                <td class="name left"><a href="z_91faa57f8583c837_Burgers_py.html#t25"><data value='init__'>Burgers1D.__init__</data></a></td>
+                <td>16</td>
+                <td>0</td>
+                <td>0</td>
+                <td class="right" data-ratio="16 16">100%</td>
+            </tr>
+            <tr class="region">
+                <td class="name left"><a href="z_91faa57f8583c837_Burgers_py.html#t67">pySDC/implementations/problem_classes/Burgers.py</a></td>
+                <td class="name left"><a href="z_91faa57f8583c837_Burgers_py.html#t67"><data value='u_exact'>Burgers1D.u_exact</data></a></td>
+                <td>7</td>
+                <td>0</td>
+                <td>1</td>
+                <td class="right" data-ratio="7 7">100%</td>
+            </tr>
+            <tr class="region">
+                <td class="name left"><a href="z_91faa57f8583c837_Burgers_py.html#t95">pySDC/implementations/problem_classes/Burgers.py</a></td>
+                <td class="name left"><a href="z_91faa57f8583c837_Burgers_py.html#t95"><data value='eval_f'>Burgers1D.eval_f</data></a></td>
+                <td>8</td>
+                <td>0</td>
+                <td>0</td>
+                <td class="right" data-ratio="8 8">100%</td>
+            </tr>
+            <tr class="region">
+                <td class="name left"><a href="z_91faa57f8583c837_Burgers_py.html#t170">pySDC/implementations/problem_classes/Burgers.py</a></td>
+                <td class="name left"><a href="z_91faa57f8583c837_Burgers_py.html#t170"><data value='init__'>Burgers2D.__init__</data></a></td>
+                <td>23</td>
+                <td>0</td>
+                <td>0</td>
+                <td class="right" data-ratio="23 23">100%</td>
+            </tr>
+            <tr class="region">
+                <td class="name left"><a href="z_91faa57f8583c837_Burgers_py.html#t231">pySDC/implementations/problem_classes/Burgers.py</a></td>
+                <td class="name left"><a href="z_91faa57f8583c837_Burgers_py.html#t231"><data value='u_exact'>Burgers2D.u_exact</data></a></td>
+                <td>11</td>
+                <td>0</td>
+                <td>1</td>
+                <td class="right" data-ratio="11 11">100%</td>
+            </tr>
+            <tr class="region">
+                <td class="name left"><a href="z_91faa57f8583c837_Burgers_py.html#t252">pySDC/implementations/problem_classes/Burgers.py</a></td>
+                <td class="name left"><a href="z_91faa57f8583c837_Burgers_py.html#t252"><data value='eval_f'>Burgers2D.eval_f</data></a></td>
+                <td>10</td>
                 <td>0</td>
                 <td>0</td>
-                <td class="right" data-ratio="6 6">100%</td>
+                <td class="right" data-ratio="10 10">100%</td>
             </tr>
             <tr class="region">
-                <td class="name left"><a href="z_91faa57f8583c837_BuckConverter_py.html">pySDC/implementations/problem_classes/BuckConverter.py</a></td>
-                <td class="name left"><a href="z_91faa57f8583c837_BuckConverter_py.html"><data value=''><span class='no-noun'>(no function)</span></data></a></td>
-                <td>10</td>
+                <td class="name left"><a href="z_91faa57f8583c837_Burgers_py.html#t270">pySDC/implementations/problem_classes/Burgers.py</a></td>
+                <td class="name left"><a href="z_91faa57f8583c837_Burgers_py.html#t270"><data value='compute_vorticity'>Burgers2D.compute_vorticity</data></a></td>
+                <td>5</td>
+                <td>5</td>
                 <td>0</td>
+                <td class="right" data-ratio="0 5">0%</td>
+            </tr>
+            <tr class="region">
+                <td class="name left"><a href="z_91faa57f8583c837_Burgers_py.html">pySDC/implementations/problem_classes/Burgers.py</a></td>
+                <td class="name left"><a href="z_91faa57f8583c837_Burgers_py.html"><data value=''><span class='no-noun'>(no function)</span></data></a></td>
+                <td>16</td>
                 <td>0</td>
-                <td class="right" data-ratio="10 10">100%</td>
+                <td>4</td>
+                <td class="right" data-ratio="16 16">100%</td>
             </tr>
             <tr class="region">
                 <td class="name left"><a href="z_91faa57f8583c837_DiscontinuousTestODE_py.html#t44">pySDC/implementations/problem_classes/DiscontinuousTestODE.py</a></td>
@@ -5872,6 +6600,110 @@ <h2>
                 <td>0</td>
                 <td class="right" data-ratio="14 14">100%</td>
             </tr>
+            <tr class="region">
+                <td class="name left"><a href="z_91faa57f8583c837_HeatEquation_Chebychev_py.html#t17">pySDC/implementations/problem_classes/HeatEquation_Chebychev.py</a></td>
+                <td class="name left"><a href="z_91faa57f8583c837_HeatEquation_Chebychev_py.html#t17"><data value='init__'>Heat1DChebychev.__init__</data></a></td>
+                <td>16</td>
+                <td>0</td>
+                <td>0</td>
+                <td class="right" data-ratio="16 16">100%</td>
+            </tr>
+            <tr class="region">
+                <td class="name left"><a href="z_91faa57f8583c837_HeatEquation_Chebychev_py.html#t58">pySDC/implementations/problem_classes/HeatEquation_Chebychev.py</a></td>
+                <td class="name left"><a href="z_91faa57f8583c837_HeatEquation_Chebychev_py.html#t58"><data value='eval_f'>Heat1DChebychev.eval_f</data></a></td>
+                <td>11</td>
+                <td>0</td>
+                <td>0</td>
+                <td class="right" data-ratio="11 11">100%</td>
+            </tr>
+            <tr class="region">
+                <td class="name left"><a href="z_91faa57f8583c837_HeatEquation_Chebychev_py.html#t77">pySDC/implementations/problem_classes/HeatEquation_Chebychev.py</a></td>
+                <td class="name left"><a href="z_91faa57f8583c837_HeatEquation_Chebychev_py.html#t77"><data value='u_exact'>Heat1DChebychev.u_exact</data></a></td>
+                <td>19</td>
+                <td>0</td>
+                <td>2</td>
+                <td class="right" data-ratio="19 19">100%</td>
+            </tr>
+            <tr class="region">
+                <td class="name left"><a href="z_91faa57f8583c837_HeatEquation_Chebychev_py.html#t132">pySDC/implementations/problem_classes/HeatEquation_Chebychev.py</a></td>
+                <td class="name left"><a href="z_91faa57f8583c837_HeatEquation_Chebychev_py.html#t132"><data value='init__'>Heat1DUltraspherical.__init__</data></a></td>
+                <td>17</td>
+                <td>0</td>
+                <td>0</td>
+                <td class="right" data-ratio="17 17">100%</td>
+            </tr>
+            <tr class="region">
+                <td class="name left"><a href="z_91faa57f8583c837_HeatEquation_Chebychev_py.html#t172">pySDC/implementations/problem_classes/HeatEquation_Chebychev.py</a></td>
+                <td class="name left"><a href="z_91faa57f8583c837_HeatEquation_Chebychev_py.html#t172"><data value='eval_f'>Heat1DUltraspherical.eval_f</data></a></td>
+                <td>11</td>
+                <td>0</td>
+                <td>0</td>
+                <td class="right" data-ratio="11 11">100%</td>
+            </tr>
+            <tr class="region">
+                <td class="name left"><a href="z_91faa57f8583c837_HeatEquation_Chebychev_py.html#t191">pySDC/implementations/problem_classes/HeatEquation_Chebychev.py</a></td>
+                <td class="name left"><a href="z_91faa57f8583c837_HeatEquation_Chebychev_py.html#t191"><data value='u_exact'>Heat1DUltraspherical.u_exact</data></a></td>
+                <td>16</td>
+                <td>0</td>
+                <td>2</td>
+                <td class="right" data-ratio="16 16">100%</td>
+            </tr>
+            <tr class="region">
+                <td class="name left"><a href="z_91faa57f8583c837_HeatEquation_Chebychev_py.html#t240">pySDC/implementations/problem_classes/HeatEquation_Chebychev.py</a></td>
+                <td class="name left"><a href="z_91faa57f8583c837_HeatEquation_Chebychev_py.html#t240"><data value='init__'>Heat2DChebychev.__init__</data></a></td>
+                <td>31</td>
+                <td>0</td>
+                <td>0</td>
+                <td class="right" data-ratio="31 31">100%</td>
+            </tr>
+            <tr class="region">
+                <td class="name left"><a href="z_91faa57f8583c837_HeatEquation_Chebychev_py.html#t305">pySDC/implementations/problem_classes/HeatEquation_Chebychev.py</a></td>
+                <td class="name left"><a href="z_91faa57f8583c837_HeatEquation_Chebychev_py.html#t305"><data value='eval_f'>Heat2DChebychev.eval_f</data></a></td>
+                <td>8</td>
+                <td>0</td>
+                <td>0</td>
+                <td class="right" data-ratio="8 8">100%</td>
+            </tr>
+            <tr class="region">
+                <td class="name left"><a href="z_91faa57f8583c837_HeatEquation_Chebychev_py.html#t319">pySDC/implementations/problem_classes/HeatEquation_Chebychev.py</a></td>
+                <td class="name left"><a href="z_91faa57f8583c837_HeatEquation_Chebychev_py.html#t319"><data value='u_exact'>Heat2DChebychev.u_exact</data></a></td>
+                <td>13</td>
+                <td>0</td>
+                <td>0</td>
+                <td class="right" data-ratio="13 13">100%</td>
+            </tr>
+            <tr class="region">
+                <td class="name left"><a href="z_91faa57f8583c837_HeatEquation_Chebychev_py.html#t348">pySDC/implementations/problem_classes/HeatEquation_Chebychev.py</a></td>
+                <td class="name left"><a href="z_91faa57f8583c837_HeatEquation_Chebychev_py.html#t348"><data value='init__'>Heat2DUltraspherical.__init__</data></a></td>
+                <td>30</td>
+                <td>0</td>
+                <td>0</td>
+                <td class="right" data-ratio="30 30">100%</td>
+            </tr>
+            <tr class="region">
+                <td class="name left"><a href="z_91faa57f8583c837_HeatEquation_Chebychev_py.html#t416">pySDC/implementations/problem_classes/HeatEquation_Chebychev.py</a></td>
+                <td class="name left"><a href="z_91faa57f8583c837_HeatEquation_Chebychev_py.html#t416"><data value='eval_f'>Heat2DUltraspherical.eval_f</data></a></td>
+                <td>8</td>
+                <td>0</td>
+                <td>0</td>
+                <td class="right" data-ratio="8 8">100%</td>
+            </tr>
+            <tr class="region">
+                <td class="name left"><a href="z_91faa57f8583c837_HeatEquation_Chebychev_py.html#t428">pySDC/implementations/problem_classes/HeatEquation_Chebychev.py</a></td>
+                <td class="name left"><a href="z_91faa57f8583c837_HeatEquation_Chebychev_py.html#t428"><data value='u_exact'>Heat2DUltraspherical.u_exact</data></a></td>
+                <td>11</td>
+                <td>0</td>
+                <td>0</td>
+                <td class="right" data-ratio="11 11">100%</td>
+            </tr>
+            <tr class="region">
+                <td class="name left"><a href="z_91faa57f8583c837_HeatEquation_Chebychev_py.html">pySDC/implementations/problem_classes/HeatEquation_Chebychev.py</a></td>
+                <td class="name left"><a href="z_91faa57f8583c837_HeatEquation_Chebychev_py.html"><data value=''><span class='no-noun'>(no function)</span></data></a></td>
+                <td>29</td>
+                <td>0</td>
+                <td>0</td>
+                <td class="right" data-ratio="29 29">100%</td>
+            </tr>
             <tr class="region">
                 <td class="name left"><a href="z_91faa57f8583c837_HeatEquation_ND_FD_py.html#t64">pySDC/implementations/problem_classes/HeatEquation_ND_FD.py</a></td>
                 <td class="name left"><a href="z_91faa57f8583c837_HeatEquation_ND_FD_py.html#t64"><data value='init__'>heatNd_unforced.__init__</data></a></td>
@@ -6408,6 +7240,126 @@ <h2>
                 <td>0</td>
                 <td class="right" data-ratio="22 22">100%</td>
             </tr>
+            <tr class="region">
+                <td class="name left"><a href="z_91faa57f8583c837_RayleighBenard_py.html#t51">pySDC/implementations/problem_classes/RayleighBenard.py</a></td>
+                <td class="name left"><a href="z_91faa57f8583c837_RayleighBenard_py.html#t51"><data value='init__'>RayleighBenard.__init__</data></a></td>
+                <td>47</td>
+                <td>1</td>
+                <td>1</td>
+                <td class="right" data-ratio="46 47">98%</td>
+            </tr>
+            <tr class="region">
+                <td class="name left"><a href="z_91faa57f8583c837_RayleighBenard_py.html#t178">pySDC/implementations/problem_classes/RayleighBenard.py</a></td>
+                <td class="name left"><a href="z_91faa57f8583c837_RayleighBenard_py.html#t178"><data value='eval_f'>RayleighBenard.eval_f</data></a></td>
+                <td>30</td>
+                <td>0</td>
+                <td>0</td>
+                <td class="right" data-ratio="30 30">100%</td>
+            </tr>
+            <tr class="region">
+                <td class="name left"><a href="z_91faa57f8583c837_RayleighBenard_py.html#t226">pySDC/implementations/problem_classes/RayleighBenard.py</a></td>
+                <td class="name left"><a href="z_91faa57f8583c837_RayleighBenard_py.html#t226"><data value='u_exact'>RayleighBenard.u_exact</data></a></td>
+                <td>17</td>
+                <td>1</td>
+                <td>0</td>
+                <td class="right" data-ratio="16 17">94%</td>
+            </tr>
+            <tr class="region">
+                <td class="name left"><a href="z_91faa57f8583c837_RayleighBenard_py.html#t316">pySDC/implementations/problem_classes/RayleighBenard.py</a></td>
+                <td class="name left"><a href="z_91faa57f8583c837_RayleighBenard_py.html#t316"><data value='compute_vorticity'>RayleighBenard.compute_vorticity</data></a></td>
+                <td>9</td>
+                <td>1</td>
+                <td>0</td>
+                <td class="right" data-ratio="8 9">89%</td>
+            </tr>
+            <tr class="region">
+                <td class="name left"><a href="z_91faa57f8583c837_RayleighBenard_py.html#t329">pySDC/implementations/problem_classes/RayleighBenard.py</a></td>
+                <td class="name left"><a href="z_91faa57f8583c837_RayleighBenard_py.html#t329"><data value='compute_Nusselt_numbers'>RayleighBenard.compute_Nusselt_numbers</data></a></td>
+                <td>21</td>
+                <td>1</td>
+                <td>0</td>
+                <td class="right" data-ratio="20 21">95%</td>
+            </tr>
+            <tr class="region">
+                <td class="name left"><a href="z_91faa57f8583c837_RayleighBenard_py.html#t391">pySDC/implementations/problem_classes/RayleighBenard.py</a></td>
+                <td class="name left"><a href="z_91faa57f8583c837_RayleighBenard_py.html#t391"><data value='compute_viscous_dissipation'>RayleighBenard.compute_viscous_dissipation</data></a></td>
+                <td>9</td>
+                <td>9</td>
+                <td>0</td>
+                <td class="right" data-ratio="0 9">0%</td>
+            </tr>
+            <tr class="region">
+                <td class="name left"><a href="z_91faa57f8583c837_RayleighBenard_py.html#t406">pySDC/implementations/problem_classes/RayleighBenard.py</a></td>
+                <td class="name left"><a href="z_91faa57f8583c837_RayleighBenard_py.html#t406"><data value='compute_buoyancy_generation'>RayleighBenard.compute_buoyancy_generation</data></a></td>
+                <td>4</td>
+                <td>4</td>
+                <td>0</td>
+                <td class="right" data-ratio="0 4">0%</td>
+            </tr>
+            <tr class="region">
+                <td class="name left"><a href="z_91faa57f8583c837_RayleighBenard_py.html#t415">pySDC/implementations/problem_classes/RayleighBenard.py</a></td>
+                <td class="name left"><a href="z_91faa57f8583c837_RayleighBenard_py.html#t415"><data value='dependencies'>CFLLimit.dependencies</data></a></td>
+                <td>3</td>
+                <td>3</td>
+                <td>0</td>
+                <td class="right" data-ratio="0 3">0%</td>
+            </tr>
+            <tr class="region">
+                <td class="name left"><a href="z_91faa57f8583c837_RayleighBenard_py.html#t421">pySDC/implementations/problem_classes/RayleighBenard.py</a></td>
+                <td class="name left"><a href="z_91faa57f8583c837_RayleighBenard_py.html#t421"><data value='setup_status_variables'>CFLLimit.setup_status_variables</data></a></td>
+                <td>1</td>
+                <td>1</td>
+                <td>0</td>
+                <td class="right" data-ratio="0 1">0%</td>
+            </tr>
+            <tr class="region">
+                <td class="name left"><a href="z_91faa57f8583c837_RayleighBenard_py.html#t430">pySDC/implementations/problem_classes/RayleighBenard.py</a></td>
+                <td class="name left"><a href="z_91faa57f8583c837_RayleighBenard_py.html#t430"><data value='setup'>CFLLimit.setup</data></a></td>
+                <td>2</td>
+                <td>2</td>
+                <td>1</td>
+                <td class="right" data-ratio="0 2">0%</td>
+            </tr>
+            <tr class="region">
+                <td class="name left"><a href="z_91faa57f8583c837_RayleighBenard_py.html#t456">pySDC/implementations/problem_classes/RayleighBenard.py</a></td>
+                <td class="name left"><a href="z_91faa57f8583c837_RayleighBenard_py.html#t456"><data value='compute_max_step_size'>CFLLimit.compute_max_step_size</data></a></td>
+                <td>15</td>
+                <td>1</td>
+                <td>0</td>
+                <td class="right" data-ratio="14 15">93%</td>
+            </tr>
+            <tr class="region">
+                <td class="name left"><a href="z_91faa57f8583c837_RayleighBenard_py.html#t478">pySDC/implementations/problem_classes/RayleighBenard.py</a></td>
+                <td class="name left"><a href="z_91faa57f8583c837_RayleighBenard_py.html#t478"><data value='get_new_step_size'>CFLLimit.get_new_step_size</data></a></td>
+                <td>11</td>
+                <td>11</td>
+                <td>0</td>
+                <td class="right" data-ratio="0 11">0%</td>
+            </tr>
+            <tr class="region">
+                <td class="name left"><a href="z_91faa57f8583c837_RayleighBenard_py.html#t499">pySDC/implementations/problem_classes/RayleighBenard.py</a></td>
+                <td class="name left"><a href="z_91faa57f8583c837_RayleighBenard_py.html#t499"><data value='post_step'>LogCFL.post_step</data></a></td>
+                <td>3</td>
+                <td>3</td>
+                <td>0</td>
+                <td class="right" data-ratio="0 3">0%</td>
+            </tr>
+            <tr class="region">
+                <td class="name left"><a href="z_91faa57f8583c837_RayleighBenard_py.html#t527">pySDC/implementations/problem_classes/RayleighBenard.py</a></td>
+                <td class="name left"><a href="z_91faa57f8583c837_RayleighBenard_py.html#t527"><data value='post_step'>LogAnalysisVariables.post_step</data></a></td>
+                <td>9</td>
+                <td>9</td>
+                <td>0</td>
+                <td class="right" data-ratio="0 9">0%</td>
+            </tr>
+            <tr class="region">
+                <td class="name left"><a href="z_91faa57f8583c837_RayleighBenard_py.html">pySDC/implementations/problem_classes/RayleighBenard.py</a></td>
+                <td class="name left"><a href="z_91faa57f8583c837_RayleighBenard_py.html"><data value=''><span class='no-noun'>(no function)</span></data></a></td>
+                <td>28</td>
+                <td>0</td>
+                <td>2</td>
+                <td class="right" data-ratio="28 28">100%</td>
+            </tr>
             <tr class="region">
                 <td class="name left"><a href="z_91faa57f8583c837_TestEquation_0D_py.html#t36">pySDC/implementations/problem_classes/TestEquation_0D.py</a></td>
                 <td class="name left"><a href="z_91faa57f8583c837_TestEquation_0D_py.html#t36"><data value='setup_GPU'>testequation0d.setup_GPU</data></a></td>
@@ -6492,9 +7444,9 @@ <h2>
                 <td class="name left"><a href="z_91faa57f8583c837_Van_der_Pol_implicit_py.html#t130">pySDC/implementations/problem_classes/Van_der_Pol_implicit.py</a></td>
                 <td class="name left"><a href="z_91faa57f8583c837_Van_der_Pol_implicit_py.html#t130"><data value='solve_system'>vanderpol.solve_system</data></a></td>
                 <td>24</td>
-                <td>1</td>
                 <td>2</td>
-                <td class="right" data-ratio="23 24">96%</td>
+                <td>2</td>
+                <td class="right" data-ratio="22 24">92%</td>
             </tr>
             <tr class="region">
                 <td class="name left"><a href="z_91faa57f8583c837_Van_der_Pol_implicit_py.html">pySDC/implementations/problem_classes/Van_der_Pol_implicit.py</a></td>
@@ -7376,6 +8328,94 @@ <h2>
                 <td>0</td>
                 <td class="right" data-ratio="21 21">100%</td>
             </tr>
+            <tr class="region">
+                <td class="name left"><a href="z_91faa57f8583c837_generic_spectral_py.html#t32">pySDC/implementations/problem_classes/generic_spectral.py</a></td>
+                <td class="name left"><a href="z_91faa57f8583c837_generic_spectral_py.html#t32"><data value='setup_GPU'>GenericSpectralLinear.setup_GPU</data></a></td>
+                <td>6</td>
+                <td>6</td>
+                <td>0</td>
+                <td class="right" data-ratio="0 6">0%</td>
+            </tr>
+            <tr class="region">
+                <td class="name left"><a href="z_91faa57f8583c837_generic_spectral_py.html#t47">pySDC/implementations/problem_classes/generic_spectral.py</a></td>
+                <td class="name left"><a href="z_91faa57f8583c837_generic_spectral_py.html#t47"><data value='init__'>GenericSpectralLinear.__init__</data></a></td>
+                <td>14</td>
+                <td>1</td>
+                <td>0</td>
+                <td class="right" data-ratio="13 14">93%</td>
+            </tr>
+            <tr class="region">
+                <td class="name left"><a href="z_91faa57f8583c837_generic_spectral_py.html#t113">pySDC/implementations/problem_classes/generic_spectral.py</a></td>
+                <td class="name left"><a href="z_91faa57f8583c837_generic_spectral_py.html#t113"><data value='getattr__'>GenericSpectralLinear.__getattr__</data></a></td>
+                <td>1</td>
+                <td>0</td>
+                <td>0</td>
+                <td class="right" data-ratio="1 1">100%</td>
+            </tr>
+            <tr class="region">
+                <td class="name left"><a href="z_91faa57f8583c837_generic_spectral_py.html#t125">pySDC/implementations/problem_classes/generic_spectral.py</a></td>
+                <td class="name left"><a href="z_91faa57f8583c837_generic_spectral_py.html#t125"><data value='setup_operator'>GenericSpectralLinear._setup_operator</data></a></td>
+                <td>4</td>
+                <td>0</td>
+                <td>0</td>
+                <td class="right" data-ratio="4 4">100%</td>
+            </tr>
+            <tr class="region">
+                <td class="name left"><a href="z_91faa57f8583c837_generic_spectral_py.html#t140">pySDC/implementations/problem_classes/generic_spectral.py</a></td>
+                <td class="name left"><a href="z_91faa57f8583c837_generic_spectral_py.html#t140"><data value='setup_L'>GenericSpectralLinear.setup_L</data></a></td>
+                <td>1</td>
+                <td>0</td>
+                <td>0</td>
+                <td class="right" data-ratio="1 1">100%</td>
+            </tr>
+            <tr class="region">
+                <td class="name left"><a href="z_91faa57f8583c837_generic_spectral_py.html#t160">pySDC/implementations/problem_classes/generic_spectral.py</a></td>
+                <td class="name left"><a href="z_91faa57f8583c837_generic_spectral_py.html#t160"><data value='setup_M'>GenericSpectralLinear.setup_M</data></a></td>
+                <td>3</td>
+                <td>0</td>
+                <td>0</td>
+                <td class="right" data-ratio="3 3">100%</td>
+            </tr>
+            <tr class="region">
+                <td class="name left"><a href="z_91faa57f8583c837_generic_spectral_py.html#t168">pySDC/implementations/problem_classes/generic_spectral.py</a></td>
+                <td class="name left"><a href="z_91faa57f8583c837_generic_spectral_py.html#t168"><data value='setup_preconditioner'>GenericSpectralLinear.setup_preconditioner</data></a></td>
+                <td>18</td>
+                <td>2</td>
+                <td>0</td>
+                <td class="right" data-ratio="16 18">89%</td>
+            </tr>
+            <tr class="region">
+                <td class="name left"><a href="z_91faa57f8583c837_generic_spectral_py.html#t205">pySDC/implementations/problem_classes/generic_spectral.py</a></td>
+                <td class="name left"><a href="z_91faa57f8583c837_generic_spectral_py.html#t205"><data value='solve_system'>GenericSpectralLinear.solve_system</data></a></td>
+                <td>35</td>
+                <td>8</td>
+                <td>1</td>
+                <td class="right" data-ratio="27 35">77%</td>
+            </tr>
+            <tr class="region">
+                <td class="name left"><a href="z_91faa57f8583c837_generic_spectral_py.html#t289">pySDC/implementations/problem_classes/generic_spectral.py</a></td>
+                <td class="name left"><a href="z_91faa57f8583c837_generic_spectral_py.html#t289"><data value='compute_residual_DAE'>compute_residual_DAE</data></a></td>
+                <td>22</td>
+                <td>22</td>
+                <td>1</td>
+                <td class="right" data-ratio="0 22">0%</td>
+            </tr>
+            <tr class="region">
+                <td class="name left"><a href="z_91faa57f8583c837_generic_spectral_py.html#t345">pySDC/implementations/problem_classes/generic_spectral.py</a></td>
+                <td class="name left"><a href="z_91faa57f8583c837_generic_spectral_py.html#t345"><data value='compute_residual_DAE_MPI'>compute_residual_DAE_MPI</data></a></td>
+                <td>21</td>
+                <td>21</td>
+                <td>1</td>
+                <td class="right" data-ratio="0 21">0%</td>
+            </tr>
+            <tr class="region">
+                <td class="name left"><a href="z_91faa57f8583c837_generic_spectral_py.html">pySDC/implementations/problem_classes/generic_spectral.py</a></td>
+                <td class="name left"><a href="z_91faa57f8583c837_generic_spectral_py.html"><data value=''><span class='no-noun'>(no function)</span></data></a></td>
+                <td>16</td>
+                <td>0</td>
+                <td>1</td>
+                <td class="right" data-ratio="16 16">100%</td>
+            </tr>
             <tr class="region">
                 <td class="name left"><a href="z_91faa57f8583c837_nonlinear_ODE_1_py.html#t45">pySDC/implementations/problem_classes/nonlinear_ODE_1.py</a></td>
                 <td class="name left"><a href="z_91faa57f8583c837_nonlinear_ODE_1_py.html#t45"><data value='init__'>nonlinear_ODE_1.__init__</data></a></td>
@@ -15820,9 +16860,9 @@ <h2>
                 <td class="name left"><a href="z_064a9f2a35945611_generate_statistics_py.html#t251">pySDC/projects/soft_failure/generate_statistics.py</a></td>
                 <td class="name left"><a href="z_064a9f2a35945611_generate_statistics_py.html#t251"><data value='process_statistics'>process_statistics</data></a></td>
                 <td>68</td>
+                <td>1</td>
                 <td>0</td>
-                <td>0</td>
-                <td class="right" data-ratio="68 68">100%</td>
+                <td class="right" data-ratio="67 68">99%</td>
             </tr>
             <tr class="region">
                 <td class="name left"><a href="z_064a9f2a35945611_generate_statistics_py.html#t378">pySDC/projects/soft_failure/generate_statistics.py</a></td>
@@ -15900,9 +16940,9 @@ <h2>
                 <td class="name left"><a href="z_064a9f2a35945611_implicit_sweeper_faults_py.html#t174">pySDC/projects/soft_failure/implicit_sweeper_faults.py</a></td>
                 <td class="name left"><a href="z_064a9f2a35945611_implicit_sweeper_faults_py.html#t174"><data value='detect_fault'>implicit_sweeper_faults.detect_fault</data></a></td>
                 <td>15</td>
+                <td>3</td>
                 <td>0</td>
-                <td>0</td>
-                <td class="right" data-ratio="15 15">100%</td>
+                <td class="right" data-ratio="12 15">80%</td>
             </tr>
             <tr class="region">
                 <td class="name left"><a href="z_064a9f2a35945611_implicit_sweeper_faults_py.html#t212">pySDC/projects/soft_failure/implicit_sweeper_faults.py</a></td>
@@ -16661,23 +17701,23 @@ <h2>
             <tr class="total">
                 <td class="name left">Total</td>
                 <td class="name left">&nbsp;</td>
-                <td>26137</td>
-                <td>5718</td>
-                <td>4998</td>
-                <td class="right" data-ratio="20419 26137">78%</td>
+                <td>27567</td>
+                <td>5896</td>
+                <td>5197</td>
+                <td class="right" data-ratio="21671 27567">79%</td>
             </tr>
         </tfoot>
     </table>
     <p id="no_rows">
         No items found using the specified filter.
     </p>
-        <p>137 empty functions skipped.</p>
+        <p>148 empty functions skipped.</p>
 </main>
 <footer>
     <div class="content">
         <p>
             <a class="nav" href="https://coverage.readthedocs.io/en/7.6.1">coverage.py v7.6.1</a>,
-            created at 2024-09-19 09:13 +0000
+            created at 2024-09-20 16:55 +0000
         </p>
     </div>
     <aside class="hidden">
diff --git a/coverage/index.html b/coverage/index.html
index 48c13a41b6..905cc7684f 100644
--- a/coverage/index.html
+++ b/coverage/index.html
@@ -11,7 +11,7 @@
 <header>
     <div class="content">
         <h1>Coverage report:
-            <span class="pc_cov">78%</span>
+            <span class="pc_cov">79%</span>
         </h1>
         <aside id="help_panel_wrapper">
             <input id="help_panel_state" type="checkbox">
@@ -54,7 +54,7 @@ <h2>
         </h2>
         <p class="text">
             <a class="nav" href="https://coverage.readthedocs.io/en/7.6.1">coverage.py v7.6.1</a>,
-            created at 2024-09-19 09:13 +0000
+            created at 2024-09-20 16:55 +0000
         </p>
     </div>
 </header>
@@ -189,6 +189,13 @@ <h2>
                 <td>2</td>
                 <td class="right" data-ratio="19 19">100%</td>
             </tr>
+            <tr class="region">
+                <td class="name left"><a href="z_41564483689e14fd_spectral_helper_py.html">pySDC/helpers/spectral_helper.py</a></td>
+                <td>764</td>
+                <td>62</td>
+                <td>31</td>
+                <td class="right" data-ratio="702 764">92%</td>
+            </tr>
             <tr class="region">
                 <td class="name left"><a href="z_41564483689e14fd_stats_helper_py.html">pySDC/helpers/stats_helper.py</a></td>
                 <td>36</td>
@@ -360,9 +367,9 @@ <h2>
             <tr class="region">
                 <td class="name left"><a href="z_3017878c2b4bc866_mesh_py.html">pySDC/implementations/datatype_classes/mesh.py</a></td>
                 <td>56</td>
-                <td>1</td>
+                <td>0</td>
                 <td>3</td>
-                <td class="right" data-ratio="55 56">98%</td>
+                <td class="right" data-ratio="56 56">100%</td>
             </tr>
             <tr class="region">
                 <td class="name left"><a href="z_3017878c2b4bc866_particles_py.html">pySDC/implementations/datatype_classes/particles.py</a></td>
@@ -539,6 +546,13 @@ <h2>
                 <td>1</td>
                 <td class="right" data-ratio="47 47">100%</td>
             </tr>
+            <tr class="region">
+                <td class="name left"><a href="z_91faa57f8583c837_Burgers_py.html">pySDC/implementations/problem_classes/Burgers.py</a></td>
+                <td>96</td>
+                <td>5</td>
+                <td>100</td>
+                <td class="right" data-ratio="91 96">95%</td>
+            </tr>
             <tr class="region">
                 <td class="name left"><a href="z_91faa57f8583c837_DiscontinuousTestODE_py.html">pySDC/implementations/problem_classes/DiscontinuousTestODE.py</a></td>
                 <td>87</td>
@@ -623,6 +637,13 @@ <h2>
                 <td>2</td>
                 <td class="right" data-ratio="90 90">100%</td>
             </tr>
+            <tr class="region">
+                <td class="name left"><a href="z_91faa57f8583c837_HeatEquation_Chebychev_py.html">pySDC/implementations/problem_classes/HeatEquation_Chebychev.py</a></td>
+                <td>220</td>
+                <td>0</td>
+                <td>4</td>
+                <td class="right" data-ratio="220 220">100%</td>
+            </tr>
             <tr class="region">
                 <td class="name left"><a href="z_91faa57f8583c837_HeatEquation_ND_FD_py.html">pySDC/implementations/problem_classes/HeatEquation_ND_FD.py</a></td>
                 <td>58</td>
@@ -693,6 +714,13 @@ <h2>
                 <td>8</td>
                 <td class="right" data-ratio="117 152">77%</td>
             </tr>
+            <tr class="region">
+                <td class="name left"><a href="z_91faa57f8583c837_RayleighBenard_py.html">pySDC/implementations/problem_classes/RayleighBenard.py</a></td>
+                <td>209</td>
+                <td>47</td>
+                <td>60</td>
+                <td class="right" data-ratio="162 209">78%</td>
+            </tr>
             <tr class="region">
                 <td class="name left"><a href="z_91faa57f8583c837_TestEquation_0D_py.html">pySDC/implementations/problem_classes/TestEquation_0D.py</a></td>
                 <td>53</td>
@@ -703,9 +731,9 @@ <h2>
             <tr class="region">
                 <td class="name left"><a href="z_91faa57f8583c837_Van_der_Pol_implicit_py.html">pySDC/implementations/problem_classes/Van_der_Pol_implicit.py</a></td>
                 <td>57</td>
+                <td>3</td>
                 <td>2</td>
-                <td>2</td>
-                <td class="right" data-ratio="55 57">96%</td>
+                <td class="right" data-ratio="54 57">95%</td>
             </tr>
             <tr class="region">
                 <td class="name left"><a href="z_91faa57f8583c837_VorticityVelocity_2D_FEniCS_periodic_py.html">pySDC/implementations/problem_classes/VorticityVelocity_2D_FEniCS_periodic.py</a></td>
@@ -791,6 +819,13 @@ <h2>
                 <td>10</td>
                 <td class="right" data-ratio="67 71">94%</td>
             </tr>
+            <tr class="region">
+                <td class="name left"><a href="z_91faa57f8583c837_generic_spectral_py.html">pySDC/implementations/problem_classes/generic_spectral.py</a></td>
+                <td>141</td>
+                <td>60</td>
+                <td>4</td>
+                <td class="right" data-ratio="81 141">57%</td>
+            </tr>
             <tr class="region">
                 <td class="name left"><a href="z_91faa57f8583c837_nonlinear_ODE_1_py.html">pySDC/implementations/problem_classes/nonlinear_ODE_1.py</a></td>
                 <td>41</td>
@@ -2173,16 +2208,16 @@ <h2>
             <tr class="region">
                 <td class="name left"><a href="z_064a9f2a35945611_generate_statistics_py.html">pySDC/projects/soft_failure/generate_statistics.py</a></td>
                 <td>199</td>
-                <td>49</td>
+                <td>50</td>
                 <td>16</td>
-                <td class="right" data-ratio="150 199">75%</td>
+                <td class="right" data-ratio="149 199">75%</td>
             </tr>
             <tr class="region">
                 <td class="name left"><a href="z_064a9f2a35945611_implicit_sweeper_faults_py.html">pySDC/projects/soft_failure/implicit_sweeper_faults.py</a></td>
                 <td>144</td>
-                <td>5</td>
+                <td>8</td>
                 <td>0</td>
-                <td class="right" data-ratio="139 144">97%</td>
+                <td class="right" data-ratio="136 144">94%</td>
             </tr>
             <tr class="region">
                 <td class="name left"><a href="z_064a9f2a35945611_visualization_helper_py.html">pySDC/projects/soft_failure/visualization_helper.py</a></td>
@@ -2412,10 +2447,10 @@ <h2>
         <tfoot>
             <tr class="total">
                 <td class="name left">Total</td>
-                <td>26137</td>
-                <td>5718</td>
-                <td>5001</td>
-                <td class="right" data-ratio="20419 26137">78%</td>
+                <td>27567</td>
+                <td>5896</td>
+                <td>5200</td>
+                <td class="right" data-ratio="21671 27567">79%</td>
             </tr>
         </tfoot>
     </table>
@@ -2428,7 +2463,7 @@ <h2>
     <div class="content">
         <p>
             <a class="nav" href="https://coverage.readthedocs.io/en/7.6.1">coverage.py v7.6.1</a>,
-            created at 2024-09-19 09:13 +0000
+            created at 2024-09-20 16:55 +0000
         </p>
     </div>
     <aside class="hidden">
diff --git a/coverage/status.json b/coverage/status.json
index f824471ee1..c371dd1af1 100644
--- a/coverage/status.json
+++ b/coverage/status.json
@@ -1 +1 @@
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diff --git a/coverage/z_020efe120a771d8a_hamiltonian_output_py.html b/coverage/z_020efe120a771d8a_hamiltonian_output_py.html
index 2c7a57dd9c..113665247f 100644
--- a/coverage/z_020efe120a771d8a_hamiltonian_output_py.html
+++ b/coverage/z_020efe120a771d8a_hamiltonian_output_py.html
@@ -65,7 +65,7 @@ <h2>
             <a id="nextFileLink" class="nav" href="z_020efe120a771d8a_harmonic_oscillator_py.html">&#xbb; next</a>
             &nbsp; &nbsp; &nbsp;
             <a class="nav" href="https://coverage.readthedocs.io/en/7.6.1">coverage.py v7.6.1</a>,
-            created at 2024-09-19 09:13 +0000
+            created at 2024-09-20 16:55 +0000
         </p>
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@@ -170,7 +170,7 @@ <h2>
             <a class="nav" href="z_020efe120a771d8a_harmonic_oscillator_py.html">&#xbb; next</a>
             &nbsp; &nbsp; &nbsp;
             <a class="nav" href="https://coverage.readthedocs.io/en/7.6.1">coverage.py v7.6.1</a>,
-            created at 2024-09-19 09:13 +0000
+            created at 2024-09-20 16:55 +0000
         </p>
     </div>
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diff --git a/coverage/z_020efe120a771d8a_harmonic_oscillator_py.html b/coverage/z_020efe120a771d8a_harmonic_oscillator_py.html
index 3689432b4c..fee2a3aa22 100644
--- a/coverage/z_020efe120a771d8a_harmonic_oscillator_py.html
+++ b/coverage/z_020efe120a771d8a_harmonic_oscillator_py.html
@@ -65,7 +65,7 @@ <h2>
             <a id="nextFileLink" class="nav" href="z_020efe120a771d8a_simple_problems_py.html">&#xbb; next</a>
             &nbsp; &nbsp; &nbsp;
             <a class="nav" href="https://coverage.readthedocs.io/en/7.6.1">coverage.py v7.6.1</a>,
-            created at 2024-09-19 09:13 +0000
+            created at 2024-09-20 16:55 +0000
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@@ -254,7 +254,7 @@ <h2>
             <a class="nav" href="z_020efe120a771d8a_simple_problems_py.html">&#xbb; next</a>
             &nbsp; &nbsp; &nbsp;
             <a class="nav" href="https://coverage.readthedocs.io/en/7.6.1">coverage.py v7.6.1</a>,
-            created at 2024-09-19 09:13 +0000
+            created at 2024-09-20 16:55 +0000
         </p>
     </div>
 </footer>
diff --git a/coverage/z_020efe120a771d8a_simple_problems_py.html b/coverage/z_020efe120a771d8a_simple_problems_py.html
index df8ee5054e..c096234557 100644
--- a/coverage/z_020efe120a771d8a_simple_problems_py.html
+++ b/coverage/z_020efe120a771d8a_simple_problems_py.html
@@ -65,7 +65,7 @@ <h2>
             <a id="nextFileLink" class="nav" href="z_020efe120a771d8a_solar_system_py.html">&#xbb; next</a>
             &nbsp; &nbsp; &nbsp;
             <a class="nav" href="https://coverage.readthedocs.io/en/7.6.1">coverage.py v7.6.1</a>,
-            created at 2024-09-19 09:13 +0000
+            created at 2024-09-20 16:55 +0000
         </p>
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             <button type="button" class="button_next_chunk" data-shortcut="j"></button>
@@ -323,7 +323,7 @@ <h2>
             <a class="nav" href="z_020efe120a771d8a_solar_system_py.html">&#xbb; next</a>
             &nbsp; &nbsp; &nbsp;
             <a class="nav" href="https://coverage.readthedocs.io/en/7.6.1">coverage.py v7.6.1</a>,
-            created at 2024-09-19 09:13 +0000
+            created at 2024-09-20 16:55 +0000
         </p>
     </div>
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diff --git a/coverage/z_020efe120a771d8a_solar_system_py.html b/coverage/z_020efe120a771d8a_solar_system_py.html
index 8cf2d17512..3049fb9b39 100644
--- a/coverage/z_020efe120a771d8a_solar_system_py.html
+++ b/coverage/z_020efe120a771d8a_solar_system_py.html
@@ -65,7 +65,7 @@ <h2>
             <a id="nextFileLink" class="nav" href="z_020efe120a771d8a_stop_at_error_hook_py.html">&#xbb; next</a>
             &nbsp; &nbsp; &nbsp;
             <a class="nav" href="https://coverage.readthedocs.io/en/7.6.1">coverage.py v7.6.1</a>,
-            created at 2024-09-19 09:13 +0000
+            created at 2024-09-20 16:55 +0000
         </p>
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             <button type="button" class="button_next_chunk" data-shortcut="j"></button>
@@ -365,7 +365,7 @@ <h2>
             <a class="nav" href="z_020efe120a771d8a_stop_at_error_hook_py.html">&#xbb; next</a>
             &nbsp; &nbsp; &nbsp;
             <a class="nav" href="https://coverage.readthedocs.io/en/7.6.1">coverage.py v7.6.1</a>,
-            created at 2024-09-19 09:13 +0000
+            created at 2024-09-20 16:55 +0000
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     </div>
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diff --git a/coverage/z_020efe120a771d8a_stop_at_error_hook_py.html b/coverage/z_020efe120a771d8a_stop_at_error_hook_py.html
index 0b2469ed04..a977d78d2f 100644
--- a/coverage/z_020efe120a771d8a_stop_at_error_hook_py.html
+++ b/coverage/z_020efe120a771d8a_stop_at_error_hook_py.html
@@ -65,7 +65,7 @@ <h2>
             <a id="nextFileLink" class="nav" href="z_6211a37896fa1d13_my_mesh_py.html">&#xbb; next</a>
             &nbsp; &nbsp; &nbsp;
             <a class="nav" href="https://coverage.readthedocs.io/en/7.6.1">coverage.py v7.6.1</a>,
-            created at 2024-09-19 09:13 +0000
+            created at 2024-09-20 16:55 +0000
         </p>
         <aside class="hidden">
             <button type="button" class="button_next_chunk" data-shortcut="j"></button>
@@ -117,7 +117,7 @@ <h2>
             <a class="nav" href="z_6211a37896fa1d13_my_mesh_py.html">&#xbb; next</a>
             &nbsp; &nbsp; &nbsp;
             <a class="nav" href="https://coverage.readthedocs.io/en/7.6.1">coverage.py v7.6.1</a>,
-            created at 2024-09-19 09:13 +0000
+            created at 2024-09-20 16:55 +0000
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diff --git a/coverage/z_04cb75334c2a5eef_data_management_py.html b/coverage/z_04cb75334c2a5eef_data_management_py.html
index ab3e668a37..eb14738c8e 100644
--- a/coverage/z_04cb75334c2a5eef_data_management_py.html
+++ b/coverage/z_04cb75334c2a5eef_data_management_py.html
@@ -65,7 +65,7 @@ <h2>
             <a id="nextFileLink" class="nav" href="z_74ce84641ea8e5a0_controller_MPI_scorep_py.html">&#xbb; next</a>
             &nbsp; &nbsp; &nbsp;
             <a class="nav" href="https://coverage.readthedocs.io/en/7.6.1">coverage.py v7.6.1</a>,
-            created at 2024-09-19 09:13 +0000
+            created at 2024-09-20 16:55 +0000
         </p>
         <aside class="hidden">
             <button type="button" class="button_next_chunk" data-shortcut="j"></button>
@@ -196,7 +196,7 @@ <h2>
             <a class="nav" href="z_74ce84641ea8e5a0_controller_MPI_scorep_py.html">&#xbb; next</a>
             &nbsp; &nbsp; &nbsp;
             <a class="nav" href="https://coverage.readthedocs.io/en/7.6.1">coverage.py v7.6.1</a>,
-            created at 2024-09-19 09:13 +0000
+            created at 2024-09-20 16:55 +0000
         </p>
     </div>
 </footer>
diff --git a/coverage/z_064a9f2a35945611_FaultHooks_py.html b/coverage/z_064a9f2a35945611_FaultHooks_py.html
index f947d22da1..d645e45bd6 100644
--- a/coverage/z_064a9f2a35945611_FaultHooks_py.html
+++ b/coverage/z_064a9f2a35945611_FaultHooks_py.html
@@ -65,7 +65,7 @@ <h2>
             <a id="nextFileLink" class="nav" href="z_064a9f2a35945611_generate_statistics_py.html">&#xbb; next</a>
             &nbsp; &nbsp; &nbsp;
             <a class="nav" href="https://coverage.readthedocs.io/en/7.6.1">coverage.py v7.6.1</a>,
-            created at 2024-09-19 09:13 +0000
+            created at 2024-09-20 16:55 +0000
         </p>
         <aside class="hidden">
             <button type="button" class="button_next_chunk" data-shortcut="j"></button>
@@ -135,7 +135,7 @@ <h2>
             <a class="nav" href="z_064a9f2a35945611_generate_statistics_py.html">&#xbb; next</a>
             &nbsp; &nbsp; &nbsp;
             <a class="nav" href="https://coverage.readthedocs.io/en/7.6.1">coverage.py v7.6.1</a>,
-            created at 2024-09-19 09:13 +0000
+            created at 2024-09-20 16:55 +0000
         </p>
     </div>
 </footer>
diff --git a/coverage/z_064a9f2a35945611_generate_statistics_py.html b/coverage/z_064a9f2a35945611_generate_statistics_py.html
index fc861f2ef9..01fda68405 100644
--- a/coverage/z_064a9f2a35945611_generate_statistics_py.html
+++ b/coverage/z_064a9f2a35945611_generate_statistics_py.html
@@ -55,8 +55,8 @@ <h1>
         </aside>
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-            <button type="button" class="run button_toggle_run" value="run" data-shortcut="r" title="Toggle lines run">150<span class="text"> run</span></button>
-            <button type="button" class="mis show_mis button_toggle_mis" value="mis" data-shortcut="m" title="Toggle lines missing">49<span class="text"> missing</span></button>
+            <button type="button" class="run button_toggle_run" value="run" data-shortcut="r" title="Toggle lines run">149<span class="text"> run</span></button>
+            <button type="button" class="mis show_mis button_toggle_mis" value="mis" data-shortcut="m" title="Toggle lines missing">50<span class="text"> missing</span></button>
             <button type="button" class="exc show_exc button_toggle_exc" value="exc" data-shortcut="x" title="Toggle lines excluded">16<span class="text"> excluded</span></button>
         </h2>
         <p class="text">
@@ -65,7 +65,7 @@ <h2>
             <a id="nextFileLink" class="nav" href="z_064a9f2a35945611_implicit_sweeper_faults_py.html">&#xbb; next</a>
             &nbsp; &nbsp; &nbsp;
             <a class="nav" href="https://coverage.readthedocs.io/en/7.6.1">coverage.py v7.6.1</a>,
-            created at 2024-09-19 09:13 +0000
+            created at 2024-09-20 16:55 +0000
         </p>
         <aside class="hidden">
             <button type="button" class="button_next_chunk" data-shortcut="j"></button>
@@ -364,7 +364,7 @@ <h2>
     <p class="pln"><span class="n"><a id="t282" href="#t282">282</a></span><span class="t">        <span class="com"># calculate minimal, maximal, mean residual vectors</span>&nbsp;</span><span class="r"></span></p>
     <p class="run"><span class="n"><a id="t283" href="#t283">283</a></span><span class="t">        <span class="key">for</span> <span class="nam">i</span> <span class="key">in</span> <span class="nam">range</span><span class="op">(</span><span class="nam">minlen</span><span class="op">)</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
     <p class="run"><span class="n"><a id="t284" href="#t284">284</a></span><span class="t">            <span class="key">if</span> <span class="nam">np</span><span class="op">.</span><span class="nam">isnan</span><span class="op">(</span><span class="nam">residuals</span><span class="op">[</span><span class="nam">i</span><span class="op">]</span><span class="op">)</span> <span class="key">or</span> <span class="nam">np</span><span class="op">.</span><span class="nam">isinf</span><span class="op">(</span><span class="nam">residuals</span><span class="op">[</span><span class="nam">i</span><span class="op">]</span><span class="op">)</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
-    <p class="run"><span class="n"><a id="t285" href="#t285">285</a></span><span class="t">                <span class="nam">residuals</span><span class="op">[</span><span class="nam">i</span><span class="op">]</span> <span class="op">=</span> <span class="num">1000</span>&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t285" href="#t285">285</a></span><span class="t">                <span class="nam">residuals</span><span class="op">[</span><span class="nam">i</span><span class="op">]</span> <span class="op">=</span> <span class="num">1000</span>&nbsp;</span><span class="r"></span></p>
     <p class="run"><span class="n"><a id="t286" href="#t286">286</a></span><span class="t">            <span class="nam">minres</span><span class="op">[</span><span class="nam">i</span><span class="op">]</span> <span class="op">=</span> <span class="nam">min</span><span class="op">(</span><span class="nam">minres</span><span class="op">[</span><span class="nam">i</span><span class="op">]</span><span class="op">,</span> <span class="nam">residuals</span><span class="op">[</span><span class="nam">i</span><span class="op">]</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
     <p class="run"><span class="n"><a id="t287" href="#t287">287</a></span><span class="t">            <span class="nam">maxres</span><span class="op">[</span><span class="nam">i</span><span class="op">]</span> <span class="op">=</span> <span class="nam">max</span><span class="op">(</span><span class="nam">maxres</span><span class="op">[</span><span class="nam">i</span><span class="op">]</span><span class="op">,</span> <span class="nam">residuals</span><span class="op">[</span><span class="nam">i</span><span class="op">]</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
     <p class="run"><span class="n"><a id="t288" href="#t288">288</a></span><span class="t">            <span class="nam">meanres</span><span class="op">[</span><span class="nam">i</span><span class="op">]</span> <span class="op">+=</span> <span class="nam">residuals</span><span class="op">[</span><span class="nam">i</span><span class="op">]</span>&nbsp;</span><span class="r"></span></p>
@@ -485,7 +485,7 @@ <h2>
             <a class="nav" href="z_064a9f2a35945611_implicit_sweeper_faults_py.html">&#xbb; next</a>
             &nbsp; &nbsp; &nbsp;
             <a class="nav" href="https://coverage.readthedocs.io/en/7.6.1">coverage.py v7.6.1</a>,
-            created at 2024-09-19 09:13 +0000
+            created at 2024-09-20 16:55 +0000
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     </div>
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diff --git a/coverage/z_064a9f2a35945611_implicit_sweeper_faults_py.html b/coverage/z_064a9f2a35945611_implicit_sweeper_faults_py.html
index bcfe0008df..88293531d1 100644
--- a/coverage/z_064a9f2a35945611_implicit_sweeper_faults_py.html
+++ b/coverage/z_064a9f2a35945611_implicit_sweeper_faults_py.html
@@ -2,7 +2,7 @@
 <html lang="en">
 <head>
     <meta http-equiv="Content-Type" content="text/html; charset=utf-8">
-    <title>Coverage for pySDC/projects/soft_failure/implicit_sweeper_faults.py: 97%</title>
+    <title>Coverage for pySDC/projects/soft_failure/implicit_sweeper_faults.py: 94%</title>
     <link rel="icon" sizes="32x32" href="favicon_32_cb_58284776.png">
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         <h1>
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-            <span class="pc_cov">97%</span>
+            <span class="pc_cov">94%</span>
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@@ -55,8 +55,8 @@ <h1>
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-            <button type="button" class="mis show_mis button_toggle_mis" value="mis" data-shortcut="m" title="Toggle lines missing">5<span class="text"> missing</span></button>
+            <button type="button" class="run button_toggle_run" value="run" data-shortcut="r" title="Toggle lines run">136<span class="text"> run</span></button>
+            <button type="button" class="mis show_mis button_toggle_mis" value="mis" data-shortcut="m" title="Toggle lines missing">8<span class="text"> missing</span></button>
             <button type="button" class="exc show_exc button_toggle_exc" value="exc" data-shortcut="x" title="Toggle lines excluded">0<span class="text"> excluded</span></button>
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@@ -65,7 +65,7 @@ <h2>
             <a id="nextFileLink" class="nav" href="z_064a9f2a35945611_visualization_helper_py.html">&#xbb; next</a>
             &nbsp; &nbsp; &nbsp;
             <a class="nav" href="https://coverage.readthedocs.io/en/7.6.1">coverage.py v7.6.1</a>,
-            created at 2024-09-19 09:13 +0000
+            created at 2024-09-20 16:55 +0000
         </p>
         <aside class="hidden">
             <button type="button" class="button_next_chunk" data-shortcut="j"></button>
@@ -280,10 +280,10 @@ <h2>
     <p class="run"><span class="n"><a id="t198" href="#t198">198</a></span><span class="t">            <span class="nam">self</span><span class="op">.</span><span class="nam">fault_stats</span><span class="op">.</span><span class="nam">nfaults_detected</span> <span class="op">+=</span> <span class="num">1</span>&nbsp;</span><span class="r"></span></p>
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+    <p class="mis show_mis"><span class="n"><a id="t201" href="#t201">201</a></span><span class="t">            <span class="nam">self</span><span class="op">.</span><span class="nam">fault_stats</span><span class="op">.</span><span class="nam">nfalse_positives</span> <span class="op">+=</span> <span class="num">1</span>&nbsp;</span><span class="r"></span></p>
     <p class="pln"><span class="n"><a id="t202" href="#t202">202</a></span><span class="t">            <span class="com"># in correction mode and fault detected -> meeeh!</span>&nbsp;</span><span class="r"></span></p>
-    <p class="run"><span class="n"><a id="t203" href="#t203">203</a></span><span class="t">            <span class="key">if</span> <span class="nam">self</span><span class="op">.</span><span class="nam">in_correction</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
-    <p class="run"><span class="n"><a id="t204" href="#t204">204</a></span><span class="t">                <span class="nam">self</span><span class="op">.</span><span class="nam">fault_stats</span><span class="op">.</span><span class="nam">nfalse_positives_in_correction</span> <span class="op">+=</span> <span class="num">1</span>&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t203" href="#t203">203</a></span><span class="t">            <span class="key">if</span> <span class="nam">self</span><span class="op">.</span><span class="nam">in_correction</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t204" href="#t204">204</a></span><span class="t">                <span class="nam">self</span><span class="op">.</span><span class="nam">fault_stats</span><span class="op">.</span><span class="nam">nfalse_positives_in_correction</span> <span class="op">+=</span> <span class="num">1</span>&nbsp;</span><span class="r"></span></p>
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@@ -416,7 +416,7 @@ <h2>
             <a class="nav" href="z_064a9f2a35945611_visualization_helper_py.html">&#xbb; next</a>
             &nbsp; &nbsp; &nbsp;
             <a class="nav" href="https://coverage.readthedocs.io/en/7.6.1">coverage.py v7.6.1</a>,
-            created at 2024-09-19 09:13 +0000
+            created at 2024-09-20 16:55 +0000
         </p>
     </div>
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             <a class="nav" href="https://coverage.readthedocs.io/en/7.6.1">coverage.py v7.6.1</a>,
-            created at 2024-09-19 09:13 +0000
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             <a class="nav" href="https://coverage.readthedocs.io/en/7.6.1">coverage.py v7.6.1</a>,
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-            created at 2024-09-19 09:13 +0000
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             <a class="nav" href="https://coverage.readthedocs.io/en/7.6.1">coverage.py v7.6.1</a>,
-            created at 2024-09-19 09:13 +0000
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-            created at 2024-09-19 09:13 +0000
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@@ -253,7 +253,7 @@ <h2>
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+            created at 2024-09-20 16:55 +0000
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index 45dfd14c32..2959568ef1 100644
--- a/coverage/z_41564483689e14fd_NCCL_communicator_py.html
+++ b/coverage/z_41564483689e14fd_NCCL_communicator_py.html
@@ -65,7 +65,7 @@ <h2>
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+            created at 2024-09-20 16:55 +0000
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+            created at 2024-09-20 16:55 +0000
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index 42ed8f175c..8c6a31d4b2 100644
--- a/coverage/z_41564483689e14fd_plot_helper_py.html
+++ b/coverage/z_41564483689e14fd_plot_helper_py.html
@@ -65,7 +65,7 @@ <h2>
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+            created at 2024-09-20 16:55 +0000
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index cbcfec9fb8..24ba50c1d3 100644
--- a/coverage/z_41564483689e14fd_problem_helper_py.html
+++ b/coverage/z_41564483689e14fd_problem_helper_py.html
@@ -65,7 +65,7 @@ <h2>
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+            created at 2024-09-20 16:55 +0000
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index 8f5c34d211..9231eabe14 100644
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@@ -65,7 +65,7 @@ <h2>
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+            created at 2024-09-20 16:55 +0000
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index ddf40771ce..282630c4c7 100644
--- a/coverage/z_41564483689e14fd_setup_helper_py.html
+++ b/coverage/z_41564483689e14fd_setup_helper_py.html
@@ -62,10 +62,10 @@ <h2>
         <p class="text">
             <a id="prevFileLink" class="nav" href="z_41564483689e14fd_pysdc_helper_py.html">&#xab; prev</a> &nbsp; &nbsp;
             <a id="indexLink" class="nav" href="index.html">&Hat; index</a> &nbsp; &nbsp;
-            <a id="nextFileLink" class="nav" href="z_41564483689e14fd_stats_helper_py.html">&#xbb; next</a>
+            <a id="nextFileLink" class="nav" href="z_41564483689e14fd_spectral_helper_py.html">&#xbb; next</a>
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+            created at 2024-09-20 16:55 +0000
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diff --git a/coverage/z_41564483689e14fd_spectral_helper_py.html b/coverage/z_41564483689e14fd_spectral_helper_py.html
new file mode 100644
index 0000000000..268bd4c276
--- /dev/null
+++ b/coverage/z_41564483689e14fd_spectral_helper_py.html
@@ -0,0 +1,2122 @@
+<!DOCTYPE html>
+<html lang="en">
+<head>
+    <meta http-equiv="Content-Type" content="text/html; charset=utf-8">
+    <title>Coverage for pySDC/helpers/spectral_helper.py: 92%</title>
+    <link rel="icon" sizes="32x32" href="favicon_32_cb_58284776.png">
+    <link rel="stylesheet" href="style_cb_8e611ae1.css" type="text/css">
+    <script src="coverage_html_cb_6fb7b396.js" defer></script>
+</head>
+<body class="pyfile">
+<header>
+    <div class="content">
+        <h1>
+            <span class="text">Coverage for </span><b>pySDC/helpers/spectral_helper.py</b>:
+            <span class="pc_cov">92%</span>
+        </h1>
+        <aside id="help_panel_wrapper">
+            <input id="help_panel_state" type="checkbox">
+            <label for="help_panel_state">
+                <img id="keyboard_icon" src="keybd_closed_cb_ce680311.png" alt="Show/hide keyboard shortcuts">
+            </label>
+            <div id="help_panel">
+                <p class="legend">Shortcuts on this page</p>
+                <div class="keyhelp">
+                    <p>
+                        <kbd>r</kbd>
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+                        <kbd>x</kbd>
+                        &nbsp; toggle line displays
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+                        <kbd>j</kbd>
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+                    <p>
+                        <kbd>0</kbd> &nbsp; (zero) top of page
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+                        <kbd>1</kbd> &nbsp; (one) first highlighted chunk
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+                </div>
+            </div>
+        </aside>
+        <h2>
+            <span class="text">764 statements &nbsp;</span>
+            <button type="button" class="run button_toggle_run" value="run" data-shortcut="r" title="Toggle lines run">702<span class="text"> run</span></button>
+            <button type="button" class="mis show_mis button_toggle_mis" value="mis" data-shortcut="m" title="Toggle lines missing">62<span class="text"> missing</span></button>
+            <button type="button" class="exc show_exc button_toggle_exc" value="exc" data-shortcut="x" title="Toggle lines excluded">31<span class="text"> excluded</span></button>
+        </h2>
+        <p class="text">
+            <a id="prevFileLink" class="nav" href="z_41564483689e14fd_setup_helper_py.html">&#xab; prev</a> &nbsp; &nbsp;
+            <a id="indexLink" class="nav" href="index.html">&Hat; index</a> &nbsp; &nbsp;
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+            &nbsp; &nbsp; &nbsp;
+            <a class="nav" href="https://coverage.readthedocs.io/en/7.6.1">coverage.py v7.6.1</a>,
+            created at 2024-09-20 16:55 +0000
+        </p>
+        <aside class="hidden">
+            <button type="button" class="button_next_chunk" data-shortcut="j"></button>
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+            <button type="button" class="button_to_index" data-shortcut="u"></button>
+            <button type="button" class="button_show_hide_help" data-shortcut="?"></button>
+        </aside>
+    </div>
+</header>
+<main id="source">
+    <p class="run"><span class="n"><a id="t1" href="#t1">1</a></span><span class="t"><span class="key">import</span> <span class="nam">numpy</span> <span class="key">as</span> <span class="nam">np</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t2" href="#t2">2</a></span><span class="t"><span class="key">import</span> <span class="nam">scipy</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t3" href="#t3">3</a></span><span class="t"><span class="key">from</span> <span class="nam">pySDC</span><span class="op">.</span><span class="nam">implementations</span><span class="op">.</span><span class="nam">datatype_classes</span><span class="op">.</span><span class="nam">mesh</span> <span class="key">import</span> <span class="nam">mesh</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t4" href="#t4">4</a></span><span class="t"><span class="key">from</span> <span class="nam">scipy</span><span class="op">.</span><span class="nam">special</span> <span class="key">import</span> <span class="nam">factorial</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t5" href="#t5">5</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t6" href="#t6">6</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t7" href="#t7">7</a></span><span class="t"><span class="key">class</span> <span class="nam">SpectralHelper1D</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t8" href="#t8">8</a></span><span class="t">    <span class="str">"""</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t9" href="#t9">9</a></span><span class="t"><span class="str">    Abstract base class for 1D spectral discretizations. Defines a common interface with parameters and functions that</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t10" href="#t10">10</a></span><span class="t"><span class="str">    all bases need to have.</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t11" href="#t11">11</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t12" href="#t12">12</a></span><span class="t"><span class="str">    When implementing new bases, please take care to use the modules that are supplied as class attributes to enable</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t13" href="#t13">13</a></span><span class="t"><span class="str">    the code for GPUs.</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t14" href="#t14">14</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t15" href="#t15">15</a></span><span class="t"><span class="str">    Attributes:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t16" href="#t16">16</a></span><span class="t"><span class="str">        N (int): Resolution</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t17" href="#t17">17</a></span><span class="t"><span class="str">        x0 (float): Coordinate of left boundary</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t18" href="#t18">18</a></span><span class="t"><span class="str">        x1 (float): Coordinate of right boundary</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t19" href="#t19">19</a></span><span class="t"><span class="str">        L (float): Length of the domain</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t20" href="#t20">20</a></span><span class="t"><span class="str">        useGPU (bool): Whether to use GPUs</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t21" href="#t21">21</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t22" href="#t22">22</a></span><span class="t"><span class="str">    """</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t23" href="#t23">23</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t24" href="#t24">24</a></span><span class="t">    <span class="nam">fft_lib</span> <span class="op">=</span> <span class="nam">scipy</span><span class="op">.</span><span class="nam">fft</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t25" href="#t25">25</a></span><span class="t">    <span class="nam">sparse_lib</span> <span class="op">=</span> <span class="nam">scipy</span><span class="op">.</span><span class="nam">sparse</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t26" href="#t26">26</a></span><span class="t">    <span class="nam">linalg</span> <span class="op">=</span> <span class="nam">scipy</span><span class="op">.</span><span class="nam">sparse</span><span class="op">.</span><span class="nam">linalg</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t27" href="#t27">27</a></span><span class="t">    <span class="nam">xp</span> <span class="op">=</span> <span class="nam">np</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t28" href="#t28">28</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t29" href="#t29">29</a></span><span class="t">    <span class="key">def</span> <span class="nam">__init__</span><span class="op">(</span><span class="nam">self</span><span class="op">,</span> <span class="nam">N</span><span class="op">,</span> <span class="nam">x0</span><span class="op">=</span><span class="key">None</span><span class="op">,</span> <span class="nam">x1</span><span class="op">=</span><span class="key">None</span><span class="op">,</span> <span class="nam">useGPU</span><span class="op">=</span><span class="key">False</span><span class="op">)</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t30" href="#t30">30</a></span><span class="t">        <span class="str">"""</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t31" href="#t31">31</a></span><span class="t"><span class="str">        Constructor</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t32" href="#t32">32</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t33" href="#t33">33</a></span><span class="t"><span class="str">        Args:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t34" href="#t34">34</a></span><span class="t"><span class="str">            N (int): Resolution</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t35" href="#t35">35</a></span><span class="t"><span class="str">            x0 (float): Coordinate of left boundary</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t36" href="#t36">36</a></span><span class="t"><span class="str">            x1 (float): Coordinate of right boundary</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t37" href="#t37">37</a></span><span class="t"><span class="str">            useGPU (bool): Whether to use GPUs</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t38" href="#t38">38</a></span><span class="t"><span class="str">        """</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t39" href="#t39">39</a></span><span class="t">        <span class="nam">self</span><span class="op">.</span><span class="nam">N</span> <span class="op">=</span> <span class="nam">N</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t40" href="#t40">40</a></span><span class="t">        <span class="nam">self</span><span class="op">.</span><span class="nam">x0</span> <span class="op">=</span> <span class="nam">x0</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t41" href="#t41">41</a></span><span class="t">        <span class="nam">self</span><span class="op">.</span><span class="nam">x1</span> <span class="op">=</span> <span class="nam">x1</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t42" href="#t42">42</a></span><span class="t">        <span class="nam">self</span><span class="op">.</span><span class="nam">L</span> <span class="op">=</span> <span class="nam">x1</span> <span class="op">-</span> <span class="nam">x0</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t43" href="#t43">43</a></span><span class="t">        <span class="nam">self</span><span class="op">.</span><span class="nam">useGPU</span> <span class="op">=</span> <span class="nam">useGPU</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t44" href="#t44">44</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t45" href="#t45">45</a></span><span class="t">        <span class="key">if</span> <span class="nam">useGPU</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t46" href="#t46">46</a></span><span class="t">            <span class="nam">self</span><span class="op">.</span><span class="nam">setup_GPU</span><span class="op">(</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t47" href="#t47">47</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t48" href="#t48">48</a></span><span class="t">    <span class="op">@</span><span class="nam">classmethod</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t49" href="#t49">49</a></span><span class="t">    <span class="key">def</span> <span class="nam">setup_GPU</span><span class="op">(</span><span class="nam">cls</span><span class="op">)</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t50" href="#t50">50</a></span><span class="t">        <span class="str">"""switch to GPU modules"""</span>&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t51" href="#t51">51</a></span><span class="t">        <span class="key">import</span> <span class="nam">cupy</span> <span class="key">as</span> <span class="nam">cp</span>&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t52" href="#t52">52</a></span><span class="t">        <span class="key">import</span> <span class="nam">cupyx</span><span class="op">.</span><span class="nam">scipy</span><span class="op">.</span><span class="nam">sparse</span> <span class="key">as</span> <span class="nam">sparse_lib</span>&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t53" href="#t53">53</a></span><span class="t">        <span class="key">import</span> <span class="nam">cupyx</span><span class="op">.</span><span class="nam">scipy</span><span class="op">.</span><span class="nam">sparse</span><span class="op">.</span><span class="nam">linalg</span> <span class="key">as</span> <span class="nam">linalg</span>&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t54" href="#t54">54</a></span><span class="t">        <span class="key">import</span> <span class="nam">cupyx</span><span class="op">.</span><span class="nam">scipy</span><span class="op">.</span><span class="nam">fft</span> <span class="key">as</span> <span class="nam">fft_lib</span>&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t55" href="#t55">55</a></span><span class="t">        <span class="key">from</span> <span class="nam">pySDC</span><span class="op">.</span><span class="nam">implementations</span><span class="op">.</span><span class="nam">datatype_classes</span><span class="op">.</span><span class="nam">cupy_mesh</span> <span class="key">import</span> <span class="nam">cupy_mesh</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t56" href="#t56">56</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t57" href="#t57">57</a></span><span class="t">        <span class="nam">cls</span><span class="op">.</span><span class="nam">xp</span> <span class="op">=</span> <span class="nam">cp</span>&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t58" href="#t58">58</a></span><span class="t">        <span class="nam">cls</span><span class="op">.</span><span class="nam">sparse_lib</span> <span class="op">=</span> <span class="nam">sparse_lib</span>&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t59" href="#t59">59</a></span><span class="t">        <span class="nam">cls</span><span class="op">.</span><span class="nam">linalg</span> <span class="op">=</span> <span class="nam">linalg</span>&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t60" href="#t60">60</a></span><span class="t">        <span class="nam">cls</span><span class="op">.</span><span class="nam">fft_lib</span> <span class="op">=</span> <span class="nam">fft_lib</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t61" href="#t61">61</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t62" href="#t62">62</a></span><span class="t">    <span class="key">def</span> <span class="nam">get_Id</span><span class="op">(</span><span class="nam">self</span><span class="op">)</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t63" href="#t63">63</a></span><span class="t">        <span class="str">"""</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t64" href="#t64">64</a></span><span class="t"><span class="str">        Get identity matrix</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t65" href="#t65">65</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t66" href="#t66">66</a></span><span class="t"><span class="str">        Returns:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t67" href="#t67">67</a></span><span class="t"><span class="str">            sparse diagonal identity matrix</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t68" href="#t68">68</a></span><span class="t"><span class="str">        """</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t69" href="#t69">69</a></span><span class="t">        <span class="key">return</span> <span class="nam">self</span><span class="op">.</span><span class="nam">sparse_lib</span><span class="op">.</span><span class="nam">eye</span><span class="op">(</span><span class="nam">self</span><span class="op">.</span><span class="nam">N</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t70" href="#t70">70</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t71" href="#t71">71</a></span><span class="t">    <span class="key">def</span> <span class="nam">get_zero</span><span class="op">(</span><span class="nam">self</span><span class="op">)</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t72" href="#t72">72</a></span><span class="t">        <span class="str">"""</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t73" href="#t73">73</a></span><span class="t"><span class="str">        Get a matrix with all zeros of the correct size.</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t74" href="#t74">74</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t75" href="#t75">75</a></span><span class="t"><span class="str">        Returns:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t76" href="#t76">76</a></span><span class="t"><span class="str">            sparse matrix with zeros everywhere</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t77" href="#t77">77</a></span><span class="t"><span class="str">        """</span>&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t78" href="#t78">78</a></span><span class="t">        <span class="key">return</span> <span class="num">0</span> <span class="op">*</span> <span class="nam">self</span><span class="op">.</span><span class="nam">get_Id</span><span class="op">(</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t79" href="#t79">79</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t80" href="#t80">80</a></span><span class="t">    <span class="key">def</span> <span class="nam">get_differentiation_matrix</span><span class="op">(</span><span class="nam">self</span><span class="op">)</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t81" href="#t81">81</a></span><span class="t">        <span class="key">raise</span> <span class="nam">NotImplementedError</span><span class="op">(</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t82" href="#t82">82</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t83" href="#t83">83</a></span><span class="t">    <span class="key">def</span> <span class="nam">get_integration_matrix</span><span class="op">(</span><span class="nam">self</span><span class="op">)</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t84" href="#t84">84</a></span><span class="t">        <span class="key">raise</span> <span class="nam">NotImplementedError</span><span class="op">(</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t85" href="#t85">85</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t86" href="#t86">86</a></span><span class="t">    <span class="key">def</span> <span class="nam">get_wavenumbers</span><span class="op">(</span><span class="nam">self</span><span class="op">)</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t87" href="#t87">87</a></span><span class="t">        <span class="str">"""</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t88" href="#t88">88</a></span><span class="t"><span class="str">        Get the grid in spectral space</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t89" href="#t89">89</a></span><span class="t"><span class="str">        """</span>&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t90" href="#t90">90</a></span><span class="t">        <span class="key">raise</span> <span class="nam">NotImplementedError</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t91" href="#t91">91</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t92" href="#t92">92</a></span><span class="t">    <span class="key">def</span> <span class="nam">get_empty_operator_matrix</span><span class="op">(</span><span class="nam">self</span><span class="op">,</span> <span class="nam">S</span><span class="op">,</span> <span class="nam">O</span><span class="op">)</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t93" href="#t93">93</a></span><span class="t">        <span class="str">"""</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t94" href="#t94">94</a></span><span class="t"><span class="str">        Return a matrix of operators to be filled with the connections between the solution components.</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t95" href="#t95">95</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t96" href="#t96">96</a></span><span class="t"><span class="str">        Args:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t97" href="#t97">97</a></span><span class="t"><span class="str">            S (int): Number of components in the solution</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t98" href="#t98">98</a></span><span class="t"><span class="str">            O (sparse matrix): Zero matrix used for initialization</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t99" href="#t99">99</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t100" href="#t100">100</a></span><span class="t"><span class="str">        Returns:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t101" href="#t101">101</a></span><span class="t"><span class="str">            list of lists containing sparse zeros</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t102" href="#t102">102</a></span><span class="t"><span class="str">        """</span>&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t103" href="#t103">103</a></span><span class="t">        <span class="key">return</span> <span class="op">[</span><span class="op">[</span><span class="nam">O</span> <span class="key">for</span> <span class="nam">_</span> <span class="key">in</span> <span class="nam">range</span><span class="op">(</span><span class="nam">S</span><span class="op">)</span><span class="op">]</span> <span class="key">for</span> <span class="nam">_</span> <span class="key">in</span> <span class="nam">range</span><span class="op">(</span><span class="nam">S</span><span class="op">)</span><span class="op">]</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t104" href="#t104">104</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t105" href="#t105">105</a></span><span class="t">    <span class="key">def</span> <span class="nam">get_basis_change_matrix</span><span class="op">(</span><span class="nam">self</span><span class="op">,</span> <span class="op">*</span><span class="nam">args</span><span class="op">,</span> <span class="op">**</span><span class="nam">kwargs</span><span class="op">)</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t106" href="#t106">106</a></span><span class="t">        <span class="str">"""</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t107" href="#t107">107</a></span><span class="t"><span class="str">        Some spectral discretization change the basis during differentiation. This method can be used to transfer</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t108" href="#t108">108</a></span><span class="t"><span class="str">        between the various bases.</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t109" href="#t109">109</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t110" href="#t110">110</a></span><span class="t"><span class="str">        This method accepts arbitrary arguments that may not be used in order to provide an easy interface for multi-</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t111" href="#t111">111</a></span><span class="t"><span class="str">        dimensional bases. For instance, you may combine an FFT discretization with an ultraspherical discretization.</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t112" href="#t112">112</a></span><span class="t"><span class="str">        The FFT discretization will always be in the same base, but the ultraspherical discretization uses a different</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t113" href="#t113">113</a></span><span class="t"><span class="str">        base for every derivative. You can then ask all bases for transfer matrices from one ultraspherical derivative</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t114" href="#t114">114</a></span><span class="t"><span class="str">        base to the next. The FFT discretization will ignore this and return an identity while the ultraspherical</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t115" href="#t115">115</a></span><span class="t"><span class="str">        discretization will return the desired matrix. After a Kronecker product, you get the 2D version of the matrix</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t116" href="#t116">116</a></span><span class="t"><span class="str">        you want. This is what the `SpectralHelper` does when you call the method of the same name on it.</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t117" href="#t117">117</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t118" href="#t118">118</a></span><span class="t"><span class="str">        Returns:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t119" href="#t119">119</a></span><span class="t"><span class="str">            sparse bases change matrix</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t120" href="#t120">120</a></span><span class="t"><span class="str">        """</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t121" href="#t121">121</a></span><span class="t">        <span class="key">return</span> <span class="nam">self</span><span class="op">.</span><span class="nam">sparse_lib</span><span class="op">.</span><span class="nam">eye</span><span class="op">(</span><span class="nam">self</span><span class="op">.</span><span class="nam">N</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t122" href="#t122">122</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t123" href="#t123">123</a></span><span class="t">    <span class="key">def</span> <span class="nam">get_BC</span><span class="op">(</span><span class="nam">self</span><span class="op">,</span> <span class="nam">kind</span><span class="op">)</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t124" href="#t124">124</a></span><span class="t">        <span class="str">"""</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t125" href="#t125">125</a></span><span class="t"><span class="str">        To facilitate boundary conditions (BCs) we use either a basis where all functions satisfy the BCs automatically,</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t126" href="#t126">126</a></span><span class="t"><span class="str">        as is the case in FFT basis for periodic BCs, or boundary bordering. In boundary bordering, specific lines in</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t127" href="#t127">127</a></span><span class="t"><span class="str">        the matrix are replaced by the boundary conditions as obtained by this method.</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t128" href="#t128">128</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t129" href="#t129">129</a></span><span class="t"><span class="str">        Args:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t130" href="#t130">130</a></span><span class="t"><span class="str">            kind (str): The type of BC you want to implement please refer to the implementations of this method in the</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t131" href="#t131">131</a></span><span class="t"><span class="str">            individual 1D bases for what is implemented</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t132" href="#t132">132</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t133" href="#t133">133</a></span><span class="t"><span class="str">        Returns:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t134" href="#t134">134</a></span><span class="t"><span class="str">            self.xp.array: Boundary condition</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t135" href="#t135">135</a></span><span class="t"><span class="str">        """</span>&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t136" href="#t136">136</a></span><span class="t">        <span class="key">raise</span> <span class="nam">NotImplementedError</span><span class="op">(</span><span class="fst">f'</span><span class="fst">No boundary conditions of </span><span class="op">{</span><span class="nam">kind</span><span class="op">=</span><span class="op">!</span><span class="nam">r</span><span class="op">}</span><span class="fst"> implemented!</span><span class="fst">'</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t137" href="#t137">137</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t138" href="#t138">138</a></span><span class="t">    <span class="key">def</span> <span class="nam">get_filter_matrix</span><span class="op">(</span><span class="nam">self</span><span class="op">,</span> <span class="nam">kmin</span><span class="op">=</span><span class="num">0</span><span class="op">,</span> <span class="nam">kmax</span><span class="op">=</span><span class="key">None</span><span class="op">)</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t139" href="#t139">139</a></span><span class="t">        <span class="str">"""</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t140" href="#t140">140</a></span><span class="t"><span class="str">        Get a bandpass filter.</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t141" href="#t141">141</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t142" href="#t142">142</a></span><span class="t"><span class="str">        Args:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t143" href="#t143">143</a></span><span class="t"><span class="str">            kmin (int): Lower limit of the bandpass filter</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t144" href="#t144">144</a></span><span class="t"><span class="str">            kmax (int): Upper limit of the bandpass filter</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t145" href="#t145">145</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t146" href="#t146">146</a></span><span class="t"><span class="str">        Returns:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t147" href="#t147">147</a></span><span class="t"><span class="str">            sparse matrix</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t148" href="#t148">148</a></span><span class="t"><span class="str">        """</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t149" href="#t149">149</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t150" href="#t150">150</a></span><span class="t">        <span class="nam">k</span> <span class="op">=</span> <span class="nam">abs</span><span class="op">(</span><span class="nam">self</span><span class="op">.</span><span class="nam">get_wavenumbers</span><span class="op">(</span><span class="op">)</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t151" href="#t151">151</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t152" href="#t152">152</a></span><span class="t">        <span class="nam">kmax</span> <span class="op">=</span> <span class="nam">max</span><span class="op">(</span><span class="nam">k</span><span class="op">)</span> <span class="key">if</span> <span class="nam">kmax</span> <span class="key">is</span> <span class="key">None</span> <span class="key">else</span> <span class="nam">kmax</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t153" href="#t153">153</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t154" href="#t154">154</a></span><span class="t">        <span class="nam">mask</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">xp</span><span class="op">.</span><span class="nam">logical_or</span><span class="op">(</span><span class="nam">k</span> <span class="op">>=</span> <span class="nam">kmax</span><span class="op">,</span> <span class="nam">k</span> <span class="op">&lt;</span> <span class="nam">kmin</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t155" href="#t155">155</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t156" href="#t156">156</a></span><span class="t">        <span class="key">if</span> <span class="nam">self</span><span class="op">.</span><span class="nam">useGPU</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t157" href="#t157">157</a></span><span class="t">            <span class="nam">Id</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">get_Id</span><span class="op">(</span><span class="op">)</span><span class="op">.</span><span class="nam">get</span><span class="op">(</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t158" href="#t158">158</a></span><span class="t">        <span class="key">else</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t159" href="#t159">159</a></span><span class="t">            <span class="nam">Id</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">get_Id</span><span class="op">(</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t160" href="#t160">160</a></span><span class="t">        <span class="nam">F</span> <span class="op">=</span> <span class="nam">Id</span><span class="op">.</span><span class="nam">tolil</span><span class="op">(</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t161" href="#t161">161</a></span><span class="t">        <span class="nam">F</span><span class="op">[</span><span class="op">:</span><span class="op">,</span> <span class="nam">mask</span><span class="op">]</span> <span class="op">=</span> <span class="num">0</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t162" href="#t162">162</a></span><span class="t">        <span class="key">return</span> <span class="nam">F</span><span class="op">.</span><span class="nam">tocsc</span><span class="op">(</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t163" href="#t163">163</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t164" href="#t164">164</a></span><span class="t">    <span class="key">def</span> <span class="nam">get_1dgrid</span><span class="op">(</span><span class="nam">self</span><span class="op">)</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t165" href="#t165">165</a></span><span class="t">        <span class="str">"""</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t166" href="#t166">166</a></span><span class="t"><span class="str">        Get the grid in physical space</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t167" href="#t167">167</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t168" href="#t168">168</a></span><span class="t"><span class="str">        Returns:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t169" href="#t169">169</a></span><span class="t"><span class="str">            self.xp.array: Grid</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t170" href="#t170">170</a></span><span class="t"><span class="str">        """</span>&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t171" href="#t171">171</a></span><span class="t">        <span class="key">raise</span> <span class="nam">NotImplementedError</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t172" href="#t172">172</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t173" href="#t173">173</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t174" href="#t174">174</a></span><span class="t"><span class="key">class</span> <span class="nam">ChebychevHelper</span><span class="op">(</span><span class="nam">SpectralHelper1D</span><span class="op">)</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t175" href="#t175">175</a></span><span class="t">    <span class="str">"""</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t176" href="#t176">176</a></span><span class="t"><span class="str">    The Chebychev base consists of special kinds of polynomials, with the main advantage that you can easily transform</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t177" href="#t177">177</a></span><span class="t"><span class="str">    between physical and spectral space by discrete cosine transform.</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t178" href="#t178">178</a></span><span class="t"><span class="str">    The differentiation in the Chebychev T base is dense, but can be preconditioned to yield a differentiation operator</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t179" href="#t179">179</a></span><span class="t"><span class="str">    that moves to Chebychev U basis during differentiation, which is sparse. When using this technique, problems need to</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t180" href="#t180">180</a></span><span class="t"><span class="str">    be formulated in first order formulation.</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t181" href="#t181">181</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t182" href="#t182">182</a></span><span class="t"><span class="str">    This implementation is largely based on the Dedalus paper (arXiv:1905.10388).</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t183" href="#t183">183</a></span><span class="t"><span class="str">    """</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t184" href="#t184">184</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t185" href="#t185">185</a></span><span class="t">    <span class="key">def</span> <span class="nam">__init__</span><span class="op">(</span><span class="nam">self</span><span class="op">,</span> <span class="op">*</span><span class="nam">args</span><span class="op">,</span> <span class="nam">transform_type</span><span class="op">=</span><span class="str">'fft'</span><span class="op">,</span> <span class="nam">x0</span><span class="op">=</span><span class="op">-</span><span class="num">1</span><span class="op">,</span> <span class="nam">x1</span><span class="op">=</span><span class="num">1</span><span class="op">,</span> <span class="op">**</span><span class="nam">kwargs</span><span class="op">)</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t186" href="#t186">186</a></span><span class="t">        <span class="str">"""</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t187" href="#t187">187</a></span><span class="t"><span class="str">        Constructor.</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t188" href="#t188">188</a></span><span class="t"><span class="str">        Please refer to the parent class for additional arguments. Notably, you have to supply a resolution `N` and you</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t189" href="#t189">189</a></span><span class="t"><span class="str">        may choose to run on GPUs via the `useGPU` argument.</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t190" href="#t190">190</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t191" href="#t191">191</a></span><span class="t"><span class="str">        Args:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t192" href="#t192">192</a></span><span class="t"><span class="str">            transform_type ('fft' or 'dct'): Either use DCT functions directly implemented in the transform library or</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t193" href="#t193">193</a></span><span class="t"><span class="str">                                             use the FFT from the library to compute the DCT</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t194" href="#t194">194</a></span><span class="t"><span class="str">            x0 (float): Coordinate of left boundary. Note that only -1 is currently implented</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t195" href="#t195">195</a></span><span class="t"><span class="str">            x1 (float): Coordinate of right boundary. Note that only +1 is currently implented</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t196" href="#t196">196</a></span><span class="t"><span class="str">        """</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t197" href="#t197">197</a></span><span class="t">        <span class="key">assert</span> <span class="nam">x0</span> <span class="op">==</span> <span class="op">-</span><span class="num">1</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t198" href="#t198">198</a></span><span class="t">        <span class="key">assert</span> <span class="nam">x1</span> <span class="op">==</span> <span class="num">1</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t199" href="#t199">199</a></span><span class="t">        <span class="nam">super</span><span class="op">(</span><span class="op">)</span><span class="op">.</span><span class="nam">__init__</span><span class="op">(</span><span class="op">*</span><span class="nam">args</span><span class="op">,</span> <span class="nam">x0</span><span class="op">=</span><span class="nam">x0</span><span class="op">,</span> <span class="nam">x1</span><span class="op">=</span><span class="nam">x1</span><span class="op">,</span> <span class="op">**</span><span class="nam">kwargs</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t200" href="#t200">200</a></span><span class="t">        <span class="nam">self</span><span class="op">.</span><span class="nam">transform_type</span> <span class="op">=</span> <span class="nam">transform_type</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t201" href="#t201">201</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t202" href="#t202">202</a></span><span class="t">        <span class="key">if</span> <span class="nam">self</span><span class="op">.</span><span class="nam">transform_type</span> <span class="op">==</span> <span class="str">'fft'</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t203" href="#t203">203</a></span><span class="t">            <span class="nam">self</span><span class="op">.</span><span class="nam">get_fft_utils</span><span class="op">(</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t204" href="#t204">204</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t205" href="#t205">205</a></span><span class="t">        <span class="nam">self</span><span class="op">.</span><span class="nam">cache</span> <span class="op">=</span> <span class="op">{</span><span class="op">}</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t206" href="#t206">206</a></span><span class="t">        <span class="nam">self</span><span class="op">.</span><span class="nam">norm</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">get_norm</span><span class="op">(</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t207" href="#t207">207</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t208" href="#t208">208</a></span><span class="t">    <span class="key">def</span> <span class="nam">get_1dgrid</span><span class="op">(</span><span class="nam">self</span><span class="op">)</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t209" href="#t209">209</a></span><span class="t">        <span class="str">'''</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t210" href="#t210">210</a></span><span class="t"><span class="str">        Generates a 1D grid with Chebychev points. These are clustered at the boundary. You need this kind of grid to</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t211" href="#t211">211</a></span><span class="t"><span class="str">        use discrete cosine transformation (DCT) to get the Chebychev representation. If you want a different grid, you</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t212" href="#t212">212</a></span><span class="t"><span class="str">        need to do an affine transformation before any Chebychev business.</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t213" href="#t213">213</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t214" href="#t214">214</a></span><span class="t"><span class="str">        Returns:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t215" href="#t215">215</a></span><span class="t"><span class="str">            numpy.ndarray: 1D grid</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t216" href="#t216">216</a></span><span class="t"><span class="str">        '''</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t217" href="#t217">217</a></span><span class="t">        <span class="key">return</span> <span class="nam">self</span><span class="op">.</span><span class="nam">xp</span><span class="op">.</span><span class="nam">cos</span><span class="op">(</span><span class="nam">np</span><span class="op">.</span><span class="nam">pi</span> <span class="op">/</span> <span class="nam">self</span><span class="op">.</span><span class="nam">N</span> <span class="op">*</span> <span class="op">(</span><span class="nam">self</span><span class="op">.</span><span class="nam">xp</span><span class="op">.</span><span class="nam">arange</span><span class="op">(</span><span class="nam">self</span><span class="op">.</span><span class="nam">N</span><span class="op">)</span> <span class="op">+</span> <span class="num">0.5</span><span class="op">)</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t218" href="#t218">218</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t219" href="#t219">219</a></span><span class="t">    <span class="key">def</span> <span class="nam">get_wavenumbers</span><span class="op">(</span><span class="nam">self</span><span class="op">)</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t220" href="#t220">220</a></span><span class="t">        <span class="str">"""Get the domain in spectral space"""</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t221" href="#t221">221</a></span><span class="t">        <span class="key">return</span> <span class="nam">self</span><span class="op">.</span><span class="nam">xp</span><span class="op">.</span><span class="nam">arange</span><span class="op">(</span><span class="nam">self</span><span class="op">.</span><span class="nam">N</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t222" href="#t222">222</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t223" href="#t223">223</a></span><span class="t">    <span class="key">def</span> <span class="nam">get_conv</span><span class="op">(</span><span class="nam">self</span><span class="op">,</span> <span class="nam">name</span><span class="op">,</span> <span class="nam">N</span><span class="op">=</span><span class="key">None</span><span class="op">)</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t224" href="#t224">224</a></span><span class="t">        <span class="str">'''</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t225" href="#t225">225</a></span><span class="t"><span class="str">        Get conversion matrix between different kinds of polynomials. The supported kinds are</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t226" href="#t226">226</a></span><span class="t"><span class="str">         - T: Chebychev polynomials of first kind</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t227" href="#t227">227</a></span><span class="t"><span class="str">         - U: Chebychev polynomials of second kind</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t228" href="#t228">228</a></span><span class="t"><span class="str">         - D: Dirichlet recombination.</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t229" href="#t229">229</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t230" href="#t230">230</a></span><span class="t"><span class="str">        You get the desired matrix by choosing a name as ``A2B``. I.e. ``T2U`` for the conversion matrix from T to U.</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t231" href="#t231">231</a></span><span class="t"><span class="str">        Once generates matrices are cached. So feel free to call the method as often as you like.</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t232" href="#t232">232</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t233" href="#t233">233</a></span><span class="t"><span class="str">        Args:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t234" href="#t234">234</a></span><span class="t"><span class="str">         name (str): Conversion code, e.g. 'T2U'</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t235" href="#t235">235</a></span><span class="t"><span class="str">         N (int): Size of the matrix (optional)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t236" href="#t236">236</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t237" href="#t237">237</a></span><span class="t"><span class="str">        Returns:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t238" href="#t238">238</a></span><span class="t"><span class="str">            scipy.sparse: Sparse conversion matrix</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t239" href="#t239">239</a></span><span class="t"><span class="str">        '''</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t240" href="#t240">240</a></span><span class="t">        <span class="key">if</span> <span class="nam">name</span> <span class="key">in</span> <span class="nam">self</span><span class="op">.</span><span class="nam">cache</span><span class="op">.</span><span class="nam">keys</span><span class="op">(</span><span class="op">)</span> <span class="key">and</span> <span class="key">not</span> <span class="nam">N</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t241" href="#t241">241</a></span><span class="t">            <span class="key">return</span> <span class="nam">self</span><span class="op">.</span><span class="nam">cache</span><span class="op">[</span><span class="nam">name</span><span class="op">]</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t242" href="#t242">242</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t243" href="#t243">243</a></span><span class="t">        <span class="nam">N</span> <span class="op">=</span> <span class="nam">N</span> <span class="key">if</span> <span class="nam">N</span> <span class="key">else</span> <span class="nam">self</span><span class="op">.</span><span class="nam">N</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t244" href="#t244">244</a></span><span class="t">        <span class="nam">sp</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">sparse_lib</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t245" href="#t245">245</a></span><span class="t">        <span class="nam">xp</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">xp</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t246" href="#t246">246</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t247" href="#t247">247</a></span><span class="t">        <span class="key">def</span> <span class="nam">get_forward_conv</span><span class="op">(</span><span class="nam">name</span><span class="op">)</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t248" href="#t248">248</a></span><span class="t">            <span class="key">if</span> <span class="nam">name</span> <span class="op">==</span> <span class="str">'T2U'</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t249" href="#t249">249</a></span><span class="t">                <span class="nam">mat</span> <span class="op">=</span> <span class="op">(</span><span class="nam">sp</span><span class="op">.</span><span class="nam">eye</span><span class="op">(</span><span class="nam">N</span><span class="op">)</span> <span class="op">-</span> <span class="nam">sp</span><span class="op">.</span><span class="nam">diags</span><span class="op">(</span><span class="nam">xp</span><span class="op">.</span><span class="nam">ones</span><span class="op">(</span><span class="nam">N</span> <span class="op">-</span> <span class="num">2</span><span class="op">)</span><span class="op">,</span> <span class="nam">offsets</span><span class="op">=</span><span class="op">+</span><span class="num">2</span><span class="op">)</span><span class="op">)</span> <span class="op">/</span> <span class="num">2.0</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t250" href="#t250">250</a></span><span class="t">                <span class="nam">mat</span><span class="op">[</span><span class="op">:</span><span class="op">,</span> <span class="num">0</span><span class="op">]</span> <span class="op">*=</span> <span class="num">2</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t251" href="#t251">251</a></span><span class="t">            <span class="key">elif</span> <span class="nam">name</span> <span class="op">==</span> <span class="str">'D2T'</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t252" href="#t252">252</a></span><span class="t">                <span class="nam">mat</span> <span class="op">=</span> <span class="nam">sp</span><span class="op">.</span><span class="nam">eye</span><span class="op">(</span><span class="nam">N</span><span class="op">)</span> <span class="op">-</span> <span class="nam">sp</span><span class="op">.</span><span class="nam">diags</span><span class="op">(</span><span class="nam">xp</span><span class="op">.</span><span class="nam">ones</span><span class="op">(</span><span class="nam">N</span> <span class="op">-</span> <span class="num">2</span><span class="op">)</span><span class="op">,</span> <span class="nam">offsets</span><span class="op">=</span><span class="op">+</span><span class="num">2</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t253" href="#t253">253</a></span><span class="t">            <span class="key">elif</span> <span class="nam">name</span><span class="op">[</span><span class="num">0</span><span class="op">]</span> <span class="op">==</span> <span class="nam">name</span><span class="op">[</span><span class="op">-</span><span class="num">1</span><span class="op">]</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t254" href="#t254">254</a></span><span class="t">                <span class="nam">mat</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">sparse_lib</span><span class="op">.</span><span class="nam">eye</span><span class="op">(</span><span class="nam">self</span><span class="op">.</span><span class="nam">N</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t255" href="#t255">255</a></span><span class="t">            <span class="key">else</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t256" href="#t256">256</a></span><span class="t">                <span class="key">raise</span> <span class="nam">NotImplementedError</span><span class="op">(</span><span class="fst">f'</span><span class="fst">Don\'t have conversion matrix </span><span class="op">{</span><span class="nam">name</span><span class="op">!</span><span class="nam">r</span><span class="op">}</span><span class="fst">'</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t257" href="#t257">257</a></span><span class="t">            <span class="key">return</span> <span class="nam">mat</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t258" href="#t258">258</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t259" href="#t259">259</a></span><span class="t">        <span class="key">try</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t260" href="#t260">260</a></span><span class="t">            <span class="nam">mat</span> <span class="op">=</span> <span class="nam">get_forward_conv</span><span class="op">(</span><span class="nam">name</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t261" href="#t261">261</a></span><span class="t">        <span class="key">except</span> <span class="nam">NotImplementedError</span> <span class="key">as</span> <span class="nam">E</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t262" href="#t262">262</a></span><span class="t">            <span class="key">try</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t263" href="#t263">263</a></span><span class="t">                <span class="nam">fwd</span> <span class="op">=</span> <span class="nam">get_forward_conv</span><span class="op">(</span><span class="nam">name</span><span class="op">[</span><span class="op">:</span><span class="op">:</span><span class="op">-</span><span class="num">1</span><span class="op">]</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t264" href="#t264">264</a></span><span class="t">                <span class="key">import</span> <span class="nam">scipy</span><span class="op">.</span><span class="nam">sparse</span> <span class="key">as</span> <span class="nam">sp</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t265" href="#t265">265</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t266" href="#t266">266</a></span><span class="t">                <span class="key">if</span> <span class="nam">self</span><span class="op">.</span><span class="nam">sparse_lib</span> <span class="op">==</span> <span class="nam">sp</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t267" href="#t267">267</a></span><span class="t">                    <span class="nam">mat</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">sparse_lib</span><span class="op">.</span><span class="nam">linalg</span><span class="op">.</span><span class="nam">inv</span><span class="op">(</span><span class="nam">fwd</span><span class="op">.</span><span class="nam">tocsc</span><span class="op">(</span><span class="op">)</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t268" href="#t268">268</a></span><span class="t">                <span class="key">else</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t269" href="#t269">269</a></span><span class="t">                    <span class="nam">mat</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">sparse_lib</span><span class="op">.</span><span class="nam">csc_matrix</span><span class="op">(</span><span class="nam">sp</span><span class="op">.</span><span class="nam">linalg</span><span class="op">.</span><span class="nam">inv</span><span class="op">(</span><span class="nam">fwd</span><span class="op">.</span><span class="nam">tocsc</span><span class="op">(</span><span class="op">)</span><span class="op">.</span><span class="nam">get</span><span class="op">(</span><span class="op">)</span><span class="op">)</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t270" href="#t270">270</a></span><span class="t">            <span class="key">except</span> <span class="nam">NotImplementedError</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t271" href="#t271">271</a></span><span class="t">                <span class="key">raise</span> <span class="nam">NotImplementedError</span> <span class="key">from</span> <span class="nam">E</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t272" href="#t272">272</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t273" href="#t273">273</a></span><span class="t">        <span class="nam">self</span><span class="op">.</span><span class="nam">cache</span><span class="op">[</span><span class="nam">name</span><span class="op">]</span> <span class="op">=</span> <span class="nam">mat</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t274" href="#t274">274</a></span><span class="t">        <span class="key">return</span> <span class="nam">mat</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t275" href="#t275">275</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t276" href="#t276">276</a></span><span class="t">    <span class="key">def</span> <span class="nam">get_basis_change_matrix</span><span class="op">(</span><span class="nam">self</span><span class="op">,</span> <span class="nam">conv</span><span class="op">=</span><span class="str">'T2T'</span><span class="op">,</span> <span class="op">**</span><span class="nam">kwargs</span><span class="op">)</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t277" href="#t277">277</a></span><span class="t">        <span class="str">"""</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t278" href="#t278">278</a></span><span class="t"><span class="str">        As the differentiation matrix in Chebychev-T base is dense but is sparse when simultaneously changing base to</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t279" href="#t279">279</a></span><span class="t"><span class="str">        Chebychev-U, you may need a basis change matrix to transfer the other matrices as well. This function returns a</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t280" href="#t280">280</a></span><span class="t"><span class="str">        conversion matrix from `ChebychevHelper.get_conv`. Not that `**kwargs` are used to absorb arguments for other</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t281" href="#t281">281</a></span><span class="t"><span class="str">        bases, see documentation of `SpectralHelper1D.get_basis_change_matrix`.</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t282" href="#t282">282</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t283" href="#t283">283</a></span><span class="t"><span class="str">        Args:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t284" href="#t284">284</a></span><span class="t"><span class="str">            conv (str): Conversion code, i.e. T2U</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t285" href="#t285">285</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t286" href="#t286">286</a></span><span class="t"><span class="str">        Returns:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t287" href="#t287">287</a></span><span class="t"><span class="str">            Sparse conversion matrix</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t288" href="#t288">288</a></span><span class="t"><span class="str">        """</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t289" href="#t289">289</a></span><span class="t">        <span class="key">return</span> <span class="nam">self</span><span class="op">.</span><span class="nam">get_conv</span><span class="op">(</span><span class="nam">conv</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t290" href="#t290">290</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t291" href="#t291">291</a></span><span class="t">    <span class="key">def</span> <span class="nam">get_integration_matrix</span><span class="op">(</span><span class="nam">self</span><span class="op">,</span> <span class="nam">lbnd</span><span class="op">=</span><span class="num">0</span><span class="op">)</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t292" href="#t292">292</a></span><span class="t">        <span class="str">"""</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t293" href="#t293">293</a></span><span class="t"><span class="str">        Get matrix for integration</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t294" href="#t294">294</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t295" href="#t295">295</a></span><span class="t"><span class="str">        Args:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t296" href="#t296">296</a></span><span class="t"><span class="str">            lbnd (float): Lower bound for integration, only 0 is currently implemented</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t297" href="#t297">297</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t298" href="#t298">298</a></span><span class="t"><span class="str">        Returns:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t299" href="#t299">299</a></span><span class="t"><span class="str">           Sparse integration matrix</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t300" href="#t300">300</a></span><span class="t"><span class="str">        """</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t301" href="#t301">301</a></span><span class="t">        <span class="nam">S</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">sparse_lib</span><span class="op">.</span><span class="nam">diags</span><span class="op">(</span><span class="num">1</span> <span class="op">/</span> <span class="op">(</span><span class="nam">self</span><span class="op">.</span><span class="nam">xp</span><span class="op">.</span><span class="nam">arange</span><span class="op">(</span><span class="nam">self</span><span class="op">.</span><span class="nam">N</span> <span class="op">-</span> <span class="num">1</span><span class="op">)</span> <span class="op">+</span> <span class="num">1</span><span class="op">)</span><span class="op">,</span> <span class="nam">offsets</span><span class="op">=</span><span class="op">-</span><span class="num">1</span><span class="op">)</span> <span class="op">@</span> <span class="nam">self</span><span class="op">.</span><span class="nam">get_conv</span><span class="op">(</span><span class="str">'T2U'</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t302" href="#t302">302</a></span><span class="t">        <span class="nam">n</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">xp</span><span class="op">.</span><span class="nam">arange</span><span class="op">(</span><span class="nam">self</span><span class="op">.</span><span class="nam">N</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t303" href="#t303">303</a></span><span class="t">        <span class="key">if</span> <span class="nam">lbnd</span> <span class="op">==</span> <span class="num">0</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t304" href="#t304">304</a></span><span class="t">            <span class="nam">S</span> <span class="op">=</span> <span class="nam">S</span><span class="op">.</span><span class="nam">tocsc</span><span class="op">(</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t305" href="#t305">305</a></span><span class="t">            <span class="nam">S</span><span class="op">[</span><span class="num">0</span><span class="op">,</span> <span class="num">1</span><span class="op">:</span><span class="op">:</span><span class="num">2</span><span class="op">]</span> <span class="op">=</span> <span class="op">(</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t306" href="#t306">306</a></span><span class="t">                <span class="op">(</span><span class="nam">n</span> <span class="op">/</span> <span class="op">(</span><span class="num">2</span> <span class="op">*</span> <span class="op">(</span><span class="nam">self</span><span class="op">.</span><span class="nam">xp</span><span class="op">.</span><span class="nam">arange</span><span class="op">(</span><span class="nam">self</span><span class="op">.</span><span class="nam">N</span><span class="op">)</span> <span class="op">+</span> <span class="num">1</span><span class="op">)</span><span class="op">)</span><span class="op">)</span><span class="op">[</span><span class="num">1</span><span class="op">:</span><span class="op">:</span><span class="num">2</span><span class="op">]</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t307" href="#t307">307</a></span><span class="t">                <span class="op">*</span> <span class="op">(</span><span class="op">-</span><span class="num">1</span><span class="op">)</span> <span class="op">**</span> <span class="op">(</span><span class="nam">self</span><span class="op">.</span><span class="nam">xp</span><span class="op">.</span><span class="nam">arange</span><span class="op">(</span><span class="nam">self</span><span class="op">.</span><span class="nam">N</span> <span class="op">//</span> <span class="num">2</span><span class="op">)</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t308" href="#t308">308</a></span><span class="t">                <span class="op">/</span> <span class="op">(</span><span class="nam">np</span><span class="op">.</span><span class="nam">append</span><span class="op">(</span><span class="op">[</span><span class="num">1</span><span class="op">]</span><span class="op">,</span> <span class="nam">self</span><span class="op">.</span><span class="nam">xp</span><span class="op">.</span><span class="nam">arange</span><span class="op">(</span><span class="nam">self</span><span class="op">.</span><span class="nam">N</span> <span class="op">//</span> <span class="num">2</span> <span class="op">-</span> <span class="num">1</span><span class="op">)</span> <span class="op">+</span> <span class="num">1</span><span class="op">)</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t309" href="#t309">309</a></span><span class="t">            <span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t310" href="#t310">310</a></span><span class="t">        <span class="key">else</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t311" href="#t311">311</a></span><span class="t">            <span class="key">raise</span> <span class="nam">NotImplementedError</span><span class="op">(</span><span class="fst">f'</span><span class="fst">This function allows to integrate only from x=0, you attempted from x=</span><span class="op">{</span><span class="nam">lbnd</span><span class="op">}</span><span class="fst">.</span><span class="fst">'</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t312" href="#t312">312</a></span><span class="t">        <span class="key">return</span> <span class="nam">S</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t313" href="#t313">313</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t314" href="#t314">314</a></span><span class="t">    <span class="key">def</span> <span class="nam">get_differentiation_matrix</span><span class="op">(</span><span class="nam">self</span><span class="op">,</span> <span class="nam">p</span><span class="op">=</span><span class="num">1</span><span class="op">)</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t315" href="#t315">315</a></span><span class="t">        <span class="str">'''</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t316" href="#t316">316</a></span><span class="t"><span class="str">        Keep in mind that the T2T differentiation matrix is dense.</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t317" href="#t317">317</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t318" href="#t318">318</a></span><span class="t"><span class="str">        Args:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t319" href="#t319">319</a></span><span class="t"><span class="str">            p (int): Derivative you want to compute</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t320" href="#t320">320</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t321" href="#t321">321</a></span><span class="t"><span class="str">        Returns:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t322" href="#t322">322</a></span><span class="t"><span class="str">            numpy.ndarray: Differentiation matrix</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t323" href="#t323">323</a></span><span class="t"><span class="str">        '''</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t324" href="#t324">324</a></span><span class="t">        <span class="nam">D</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">xp</span><span class="op">.</span><span class="nam">zeros</span><span class="op">(</span><span class="op">(</span><span class="nam">self</span><span class="op">.</span><span class="nam">N</span><span class="op">,</span> <span class="nam">self</span><span class="op">.</span><span class="nam">N</span><span class="op">)</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t325" href="#t325">325</a></span><span class="t">        <span class="key">for</span> <span class="nam">j</span> <span class="key">in</span> <span class="nam">range</span><span class="op">(</span><span class="nam">self</span><span class="op">.</span><span class="nam">N</span><span class="op">)</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t326" href="#t326">326</a></span><span class="t">            <span class="key">for</span> <span class="nam">k</span> <span class="key">in</span> <span class="nam">range</span><span class="op">(</span><span class="nam">j</span><span class="op">)</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t327" href="#t327">327</a></span><span class="t">                <span class="nam">D</span><span class="op">[</span><span class="nam">k</span><span class="op">,</span> <span class="nam">j</span><span class="op">]</span> <span class="op">=</span> <span class="num">2</span> <span class="op">*</span> <span class="nam">j</span> <span class="op">*</span> <span class="op">(</span><span class="op">(</span><span class="nam">j</span> <span class="op">-</span> <span class="nam">k</span><span class="op">)</span> <span class="op">%</span> <span class="num">2</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t328" href="#t328">328</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t329" href="#t329">329</a></span><span class="t">        <span class="nam">D</span><span class="op">[</span><span class="num">0</span><span class="op">,</span> <span class="op">:</span><span class="op">]</span> <span class="op">/=</span> <span class="num">2</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t330" href="#t330">330</a></span><span class="t">        <span class="key">return</span> <span class="nam">self</span><span class="op">.</span><span class="nam">sparse_lib</span><span class="op">.</span><span class="nam">csc_matrix</span><span class="op">(</span><span class="nam">self</span><span class="op">.</span><span class="nam">xp</span><span class="op">.</span><span class="nam">linalg</span><span class="op">.</span><span class="nam">matrix_power</span><span class="op">(</span><span class="nam">D</span><span class="op">,</span> <span class="nam">p</span><span class="op">)</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t331" href="#t331">331</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t332" href="#t332">332</a></span><span class="t">    <span class="key">def</span> <span class="nam">get_norm</span><span class="op">(</span><span class="nam">self</span><span class="op">,</span> <span class="nam">N</span><span class="op">=</span><span class="key">None</span><span class="op">)</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t333" href="#t333">333</a></span><span class="t">        <span class="str">'''</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t334" href="#t334">334</a></span><span class="t"><span class="str">        Get normalization for converting Chebychev coefficients and DCT</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t335" href="#t335">335</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t336" href="#t336">336</a></span><span class="t"><span class="str">        Args:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t337" href="#t337">337</a></span><span class="t"><span class="str">            N (int, optional): Resolution</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t338" href="#t338">338</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t339" href="#t339">339</a></span><span class="t"><span class="str">        Returns:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t340" href="#t340">340</a></span><span class="t"><span class="str">            self.xp.array: Normalization</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t341" href="#t341">341</a></span><span class="t"><span class="str">        '''</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t342" href="#t342">342</a></span><span class="t">        <span class="nam">N</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">N</span> <span class="key">if</span> <span class="nam">N</span> <span class="key">is</span> <span class="key">None</span> <span class="key">else</span> <span class="nam">N</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t343" href="#t343">343</a></span><span class="t">        <span class="nam">norm</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">xp</span><span class="op">.</span><span class="nam">ones</span><span class="op">(</span><span class="nam">N</span><span class="op">)</span> <span class="op">/</span> <span class="nam">N</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t344" href="#t344">344</a></span><span class="t">        <span class="nam">norm</span><span class="op">[</span><span class="num">0</span><span class="op">]</span> <span class="op">/=</span> <span class="num">2</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t345" href="#t345">345</a></span><span class="t">        <span class="key">return</span> <span class="nam">norm</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t346" href="#t346">346</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t347" href="#t347">347</a></span><span class="t">    <span class="key">def</span> <span class="nam">get_fft_shuffle</span><span class="op">(</span><span class="nam">self</span><span class="op">,</span> <span class="nam">forward</span><span class="op">,</span> <span class="nam">N</span><span class="op">)</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t348" href="#t348">348</a></span><span class="t">        <span class="str">"""</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t349" href="#t349">349</a></span><span class="t"><span class="str">        In order to more easily parallelize using distributed FFT libraries, we express the DCT via an FFT following</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t350" href="#t350">350</a></span><span class="t"><span class="str">        doi.org/10.1109/TASSP.1980.1163351. The idea is based on reshuffling the data to be periodic and rotating it</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t351" href="#t351">351</a></span><span class="t"><span class="str">        in the complex plane. This function returns a mask to do the shuffling.</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t352" href="#t352">352</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t353" href="#t353">353</a></span><span class="t"><span class="str">        Args:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t354" href="#t354">354</a></span><span class="t"><span class="str">            forward (bool): Whether you want the shuffle for forward transform or backward transform</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t355" href="#t355">355</a></span><span class="t"><span class="str">            N (int): size of the grid</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t356" href="#t356">356</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t357" href="#t357">357</a></span><span class="t"><span class="str">        Returns:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t358" href="#t358">358</a></span><span class="t"><span class="str">            self.xp.array: Use as mask</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t359" href="#t359">359</a></span><span class="t"><span class="str">        """</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t360" href="#t360">360</a></span><span class="t">        <span class="nam">xp</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">xp</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t361" href="#t361">361</a></span><span class="t">        <span class="key">if</span> <span class="nam">forward</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t362" href="#t362">362</a></span><span class="t">            <span class="key">return</span> <span class="nam">xp</span><span class="op">.</span><span class="nam">append</span><span class="op">(</span><span class="nam">xp</span><span class="op">.</span><span class="nam">arange</span><span class="op">(</span><span class="op">(</span><span class="nam">N</span> <span class="op">+</span> <span class="num">1</span><span class="op">)</span> <span class="op">//</span> <span class="num">2</span><span class="op">)</span> <span class="op">*</span> <span class="num">2</span><span class="op">,</span> <span class="op">-</span><span class="nam">xp</span><span class="op">.</span><span class="nam">arange</span><span class="op">(</span><span class="nam">N</span> <span class="op">//</span> <span class="num">2</span><span class="op">)</span> <span class="op">*</span> <span class="num">2</span> <span class="op">-</span> <span class="num">1</span> <span class="op">-</span> <span class="nam">N</span> <span class="op">%</span> <span class="num">2</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t363" href="#t363">363</a></span><span class="t">        <span class="key">else</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t364" href="#t364">364</a></span><span class="t">            <span class="nam">mask</span> <span class="op">=</span> <span class="nam">xp</span><span class="op">.</span><span class="nam">zeros</span><span class="op">(</span><span class="nam">N</span><span class="op">,</span> <span class="nam">dtype</span><span class="op">=</span><span class="nam">int</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t365" href="#t365">365</a></span><span class="t">            <span class="nam">mask</span><span class="op">[</span><span class="op">:</span> <span class="nam">N</span> <span class="op">-</span> <span class="nam">N</span> <span class="op">%</span> <span class="num">2</span> <span class="op">:</span> <span class="num">2</span><span class="op">]</span> <span class="op">=</span> <span class="nam">xp</span><span class="op">.</span><span class="nam">arange</span><span class="op">(</span><span class="nam">N</span> <span class="op">//</span> <span class="num">2</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t366" href="#t366">366</a></span><span class="t">            <span class="nam">mask</span><span class="op">[</span><span class="num">1</span><span class="op">:</span><span class="op">:</span><span class="num">2</span><span class="op">]</span> <span class="op">=</span> <span class="nam">N</span> <span class="op">-</span> <span class="nam">xp</span><span class="op">.</span><span class="nam">arange</span><span class="op">(</span><span class="nam">N</span> <span class="op">//</span> <span class="num">2</span><span class="op">)</span> <span class="op">-</span> <span class="num">1</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t367" href="#t367">367</a></span><span class="t">            <span class="nam">mask</span><span class="op">[</span><span class="op">-</span><span class="num">1</span><span class="op">]</span> <span class="op">=</span> <span class="nam">N</span> <span class="op">//</span> <span class="num">2</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t368" href="#t368">368</a></span><span class="t">            <span class="key">return</span> <span class="nam">mask</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t369" href="#t369">369</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t370" href="#t370">370</a></span><span class="t">    <span class="key">def</span> <span class="nam">get_fft_shift</span><span class="op">(</span><span class="nam">self</span><span class="op">,</span> <span class="nam">forward</span><span class="op">,</span> <span class="nam">N</span><span class="op">)</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t371" href="#t371">371</a></span><span class="t">        <span class="str">"""</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t372" href="#t372">372</a></span><span class="t"><span class="str">        As described in the docstring for `get_fft_shuffle`, we need to rotate in the complex plane in order to use FFT for DCT.</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t373" href="#t373">373</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t374" href="#t374">374</a></span><span class="t"><span class="str">        Args:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t375" href="#t375">375</a></span><span class="t"><span class="str">            forward (bool): Whether you want the rotation for forward transform or backward transform</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t376" href="#t376">376</a></span><span class="t"><span class="str">            N (int): size of the grid</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t377" href="#t377">377</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t378" href="#t378">378</a></span><span class="t"><span class="str">        Returns:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t379" href="#t379">379</a></span><span class="t"><span class="str">            self.xp.array: Rotation</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t380" href="#t380">380</a></span><span class="t"><span class="str">        """</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t381" href="#t381">381</a></span><span class="t">        <span class="nam">k</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">get_wavenumbers</span><span class="op">(</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t382" href="#t382">382</a></span><span class="t">        <span class="nam">norm</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">get_norm</span><span class="op">(</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t383" href="#t383">383</a></span><span class="t">        <span class="nam">xp</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">xp</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t384" href="#t384">384</a></span><span class="t">        <span class="key">if</span> <span class="nam">forward</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t385" href="#t385">385</a></span><span class="t">            <span class="key">return</span> <span class="num">2</span> <span class="op">*</span> <span class="nam">xp</span><span class="op">.</span><span class="nam">exp</span><span class="op">(</span><span class="op">-</span><span class="num">1j</span> <span class="op">*</span> <span class="nam">np</span><span class="op">.</span><span class="nam">pi</span> <span class="op">*</span> <span class="nam">k</span> <span class="op">/</span> <span class="op">(</span><span class="num">2</span> <span class="op">*</span> <span class="nam">N</span><span class="op">)</span> <span class="op">+</span> <span class="num">0j</span> <span class="op">*</span> <span class="nam">np</span><span class="op">.</span><span class="nam">pi</span> <span class="op">/</span> <span class="num">4</span><span class="op">)</span> <span class="op">*</span> <span class="nam">norm</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t386" href="#t386">386</a></span><span class="t">        <span class="key">else</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t387" href="#t387">387</a></span><span class="t">            <span class="nam">shift</span> <span class="op">=</span> <span class="nam">xp</span><span class="op">.</span><span class="nam">exp</span><span class="op">(</span><span class="num">1j</span> <span class="op">*</span> <span class="nam">np</span><span class="op">.</span><span class="nam">pi</span> <span class="op">*</span> <span class="nam">k</span> <span class="op">/</span> <span class="op">(</span><span class="num">2</span> <span class="op">*</span> <span class="nam">N</span><span class="op">)</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t388" href="#t388">388</a></span><span class="t">            <span class="nam">shift</span><span class="op">[</span><span class="num">0</span><span class="op">]</span> <span class="op">=</span> <span class="num">0.5</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t389" href="#t389">389</a></span><span class="t">            <span class="key">return</span> <span class="nam">shift</span> <span class="op">/</span> <span class="nam">norm</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t390" href="#t390">390</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t391" href="#t391">391</a></span><span class="t">    <span class="key">def</span> <span class="nam">get_fft_utils</span><span class="op">(</span><span class="nam">self</span><span class="op">)</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t392" href="#t392">392</a></span><span class="t">        <span class="str">"""</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t393" href="#t393">393</a></span><span class="t"><span class="str">        Get the required utilities for using FFT to do DCT as described in the docstring for `get_fft_shuffle` and keep</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t394" href="#t394">394</a></span><span class="t"><span class="str">        them cached.</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t395" href="#t395">395</a></span><span class="t"><span class="str">        """</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t396" href="#t396">396</a></span><span class="t">        <span class="nam">self</span><span class="op">.</span><span class="nam">fft_utils</span> <span class="op">=</span> <span class="op">{</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t397" href="#t397">397</a></span><span class="t">            <span class="str">'fwd'</span><span class="op">:</span> <span class="op">{</span><span class="op">}</span><span class="op">,</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t398" href="#t398">398</a></span><span class="t">            <span class="str">'bck'</span><span class="op">:</span> <span class="op">{</span><span class="op">}</span><span class="op">,</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t399" href="#t399">399</a></span><span class="t">        <span class="op">}</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t400" href="#t400">400</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t401" href="#t401">401</a></span><span class="t">        <span class="com"># forwards transform</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t402" href="#t402">402</a></span><span class="t">        <span class="nam">self</span><span class="op">.</span><span class="nam">fft_utils</span><span class="op">[</span><span class="str">'fwd'</span><span class="op">]</span><span class="op">[</span><span class="str">'shuffle'</span><span class="op">]</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">get_fft_shuffle</span><span class="op">(</span><span class="key">True</span><span class="op">,</span> <span class="nam">self</span><span class="op">.</span><span class="nam">N</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t403" href="#t403">403</a></span><span class="t">        <span class="nam">self</span><span class="op">.</span><span class="nam">fft_utils</span><span class="op">[</span><span class="str">'fwd'</span><span class="op">]</span><span class="op">[</span><span class="str">'shift'</span><span class="op">]</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">get_fft_shift</span><span class="op">(</span><span class="key">True</span><span class="op">,</span> <span class="nam">self</span><span class="op">.</span><span class="nam">N</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t404" href="#t404">404</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t405" href="#t405">405</a></span><span class="t">        <span class="com"># backwards transform</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t406" href="#t406">406</a></span><span class="t">        <span class="nam">self</span><span class="op">.</span><span class="nam">fft_utils</span><span class="op">[</span><span class="str">'bck'</span><span class="op">]</span><span class="op">[</span><span class="str">'shuffle'</span><span class="op">]</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">get_fft_shuffle</span><span class="op">(</span><span class="key">False</span><span class="op">,</span> <span class="nam">self</span><span class="op">.</span><span class="nam">N</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t407" href="#t407">407</a></span><span class="t">        <span class="nam">self</span><span class="op">.</span><span class="nam">fft_utils</span><span class="op">[</span><span class="str">'bck'</span><span class="op">]</span><span class="op">[</span><span class="str">'shift'</span><span class="op">]</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">get_fft_shift</span><span class="op">(</span><span class="key">False</span><span class="op">,</span> <span class="nam">self</span><span class="op">.</span><span class="nam">N</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t408" href="#t408">408</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t409" href="#t409">409</a></span><span class="t">        <span class="key">return</span> <span class="nam">self</span><span class="op">.</span><span class="nam">fft_utils</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t410" href="#t410">410</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t411" href="#t411">411</a></span><span class="t">    <span class="key">def</span> <span class="nam">transform</span><span class="op">(</span><span class="nam">self</span><span class="op">,</span> <span class="nam">u</span><span class="op">,</span> <span class="nam">axis</span><span class="op">=</span><span class="op">-</span><span class="num">1</span><span class="op">,</span> <span class="op">**</span><span class="nam">kwargs</span><span class="op">)</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t412" href="#t412">412</a></span><span class="t">        <span class="str">"""</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t413" href="#t413">413</a></span><span class="t"><span class="str">        1D DCT along axis. `kwargs` will be passed on to the FFT library.</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t414" href="#t414">414</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t415" href="#t415">415</a></span><span class="t"><span class="str">        Args:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t416" href="#t416">416</a></span><span class="t"><span class="str">            u: Data you want to transform</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t417" href="#t417">417</a></span><span class="t"><span class="str">            axis (int): Axis you want to transform along</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t418" href="#t418">418</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t419" href="#t419">419</a></span><span class="t"><span class="str">        Returns:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t420" href="#t420">420</a></span><span class="t"><span class="str">            Data in spectral space</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t421" href="#t421">421</a></span><span class="t"><span class="str">        """</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t422" href="#t422">422</a></span><span class="t">        <span class="key">if</span> <span class="nam">self</span><span class="op">.</span><span class="nam">transform_type</span><span class="op">.</span><span class="nam">lower</span><span class="op">(</span><span class="op">)</span> <span class="op">==</span> <span class="str">'dct'</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t423" href="#t423">423</a></span><span class="t">            <span class="key">return</span> <span class="nam">self</span><span class="op">.</span><span class="nam">fft_lib</span><span class="op">.</span><span class="nam">dct</span><span class="op">(</span><span class="nam">u</span><span class="op">,</span> <span class="nam">axis</span><span class="op">=</span><span class="nam">axis</span><span class="op">,</span> <span class="op">**</span><span class="nam">kwargs</span><span class="op">)</span> <span class="op">*</span> <span class="nam">self</span><span class="op">.</span><span class="nam">norm</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t424" href="#t424">424</a></span><span class="t">        <span class="key">elif</span> <span class="nam">self</span><span class="op">.</span><span class="nam">transform_type</span><span class="op">.</span><span class="nam">lower</span><span class="op">(</span><span class="op">)</span> <span class="op">==</span> <span class="str">'fft'</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t425" href="#t425">425</a></span><span class="t">            <span class="nam">result</span> <span class="op">=</span> <span class="nam">u</span><span class="op">.</span><span class="nam">copy</span><span class="op">(</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t426" href="#t426">426</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t427" href="#t427">427</a></span><span class="t">            <span class="nam">shuffle</span> <span class="op">=</span> <span class="op">[</span><span class="nam">slice</span><span class="op">(</span><span class="num">0</span><span class="op">,</span> <span class="nam">s</span><span class="op">,</span> <span class="num">1</span><span class="op">)</span> <span class="key">for</span> <span class="nam">s</span> <span class="key">in</span> <span class="nam">u</span><span class="op">.</span><span class="nam">shape</span><span class="op">]</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t428" href="#t428">428</a></span><span class="t">            <span class="nam">shuffle</span><span class="op">[</span><span class="nam">axis</span><span class="op">]</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">fft_utils</span><span class="op">[</span><span class="str">'fwd'</span><span class="op">]</span><span class="op">[</span><span class="str">'shuffle'</span><span class="op">]</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t429" href="#t429">429</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t430" href="#t430">430</a></span><span class="t">            <span class="nam">v</span> <span class="op">=</span> <span class="nam">u</span><span class="op">[</span><span class="op">(</span><span class="op">*</span><span class="nam">shuffle</span><span class="op">,</span><span class="op">)</span><span class="op">]</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t431" href="#t431">431</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t432" href="#t432">432</a></span><span class="t">            <span class="nam">V</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">fft_lib</span><span class="op">.</span><span class="nam">fft</span><span class="op">(</span><span class="nam">v</span><span class="op">,</span> <span class="nam">axis</span><span class="op">=</span><span class="nam">axis</span><span class="op">,</span> <span class="op">**</span><span class="nam">kwargs</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t433" href="#t433">433</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t434" href="#t434">434</a></span><span class="t">            <span class="nam">expansion</span> <span class="op">=</span> <span class="op">[</span><span class="nam">np</span><span class="op">.</span><span class="nam">newaxis</span> <span class="key">for</span> <span class="nam">_</span> <span class="key">in</span> <span class="nam">u</span><span class="op">.</span><span class="nam">shape</span><span class="op">]</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t435" href="#t435">435</a></span><span class="t">            <span class="nam">expansion</span><span class="op">[</span><span class="nam">axis</span><span class="op">]</span> <span class="op">=</span> <span class="nam">slice</span><span class="op">(</span><span class="num">0</span><span class="op">,</span> <span class="nam">u</span><span class="op">.</span><span class="nam">shape</span><span class="op">[</span><span class="nam">axis</span><span class="op">]</span><span class="op">,</span> <span class="num">1</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t436" href="#t436">436</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t437" href="#t437">437</a></span><span class="t">            <span class="nam">V</span> <span class="op">*=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">fft_utils</span><span class="op">[</span><span class="str">'fwd'</span><span class="op">]</span><span class="op">[</span><span class="str">'shift'</span><span class="op">]</span><span class="op">[</span><span class="op">(</span><span class="op">*</span><span class="nam">expansion</span><span class="op">,</span><span class="op">)</span><span class="op">]</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t438" href="#t438">438</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t439" href="#t439">439</a></span><span class="t">            <span class="nam">result</span><span class="op">.</span><span class="nam">real</span><span class="op">[</span><span class="op">...</span><span class="op">]</span> <span class="op">=</span> <span class="nam">V</span><span class="op">.</span><span class="nam">real</span><span class="op">[</span><span class="op">...</span><span class="op">]</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t440" href="#t440">440</a></span><span class="t">            <span class="key">return</span> <span class="nam">result</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t441" href="#t441">441</a></span><span class="t">        <span class="key">else</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t442" href="#t442">442</a></span><span class="t">            <span class="key">raise</span> <span class="nam">NotImplementedError</span><span class="op">(</span><span class="fst">f'</span><span class="fst">Please choose a transform type from fft and dct, not </span><span class="op">{</span><span class="nam">self</span><span class="op">.</span><span class="nam">transform_type</span><span class="op">=</span><span class="op">}</span><span class="fst">'</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t443" href="#t443">443</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t444" href="#t444">444</a></span><span class="t">    <span class="key">def</span> <span class="nam">itransform</span><span class="op">(</span><span class="nam">self</span><span class="op">,</span> <span class="nam">u</span><span class="op">,</span> <span class="nam">axis</span><span class="op">=</span><span class="op">-</span><span class="num">1</span><span class="op">)</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t445" href="#t445">445</a></span><span class="t">        <span class="str">"""</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t446" href="#t446">446</a></span><span class="t"><span class="str">        1D inverse DCT along axis.</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t447" href="#t447">447</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t448" href="#t448">448</a></span><span class="t"><span class="str">        Args:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t449" href="#t449">449</a></span><span class="t"><span class="str">            u: Data you want to transform</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t450" href="#t450">450</a></span><span class="t"><span class="str">            axis (int): Axis you want to transform along</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t451" href="#t451">451</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t452" href="#t452">452</a></span><span class="t"><span class="str">        Returns:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t453" href="#t453">453</a></span><span class="t"><span class="str">            Data in physical space</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t454" href="#t454">454</a></span><span class="t"><span class="str">        """</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t455" href="#t455">455</a></span><span class="t">        <span class="key">assert</span> <span class="nam">self</span><span class="op">.</span><span class="nam">norm</span><span class="op">.</span><span class="nam">shape</span><span class="op">[</span><span class="num">0</span><span class="op">]</span> <span class="op">==</span> <span class="nam">u</span><span class="op">.</span><span class="nam">shape</span><span class="op">[</span><span class="nam">axis</span><span class="op">]</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t456" href="#t456">456</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t457" href="#t457">457</a></span><span class="t">        <span class="key">if</span> <span class="nam">self</span><span class="op">.</span><span class="nam">transform_type</span> <span class="op">==</span> <span class="str">'dct'</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t458" href="#t458">458</a></span><span class="t">            <span class="key">return</span> <span class="nam">self</span><span class="op">.</span><span class="nam">fft_lib</span><span class="op">.</span><span class="nam">idct</span><span class="op">(</span><span class="nam">u</span> <span class="op">/</span> <span class="nam">self</span><span class="op">.</span><span class="nam">norm</span><span class="op">,</span> <span class="nam">axis</span><span class="op">=</span><span class="nam">axis</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t459" href="#t459">459</a></span><span class="t">        <span class="key">elif</span> <span class="nam">self</span><span class="op">.</span><span class="nam">transform_type</span> <span class="op">==</span> <span class="str">'fft'</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t460" href="#t460">460</a></span><span class="t">            <span class="nam">result</span> <span class="op">=</span> <span class="nam">u</span><span class="op">.</span><span class="nam">copy</span><span class="op">(</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t461" href="#t461">461</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t462" href="#t462">462</a></span><span class="t">            <span class="nam">expansion</span> <span class="op">=</span> <span class="op">[</span><span class="nam">np</span><span class="op">.</span><span class="nam">newaxis</span> <span class="key">for</span> <span class="nam">_</span> <span class="key">in</span> <span class="nam">u</span><span class="op">.</span><span class="nam">shape</span><span class="op">]</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t463" href="#t463">463</a></span><span class="t">            <span class="nam">expansion</span><span class="op">[</span><span class="nam">axis</span><span class="op">]</span> <span class="op">=</span> <span class="nam">slice</span><span class="op">(</span><span class="num">0</span><span class="op">,</span> <span class="nam">u</span><span class="op">.</span><span class="nam">shape</span><span class="op">[</span><span class="nam">axis</span><span class="op">]</span><span class="op">,</span> <span class="num">1</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t464" href="#t464">464</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t465" href="#t465">465</a></span><span class="t">            <span class="nam">v</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">fft_lib</span><span class="op">.</span><span class="nam">ifft</span><span class="op">(</span><span class="nam">u</span> <span class="op">*</span> <span class="nam">self</span><span class="op">.</span><span class="nam">fft_utils</span><span class="op">[</span><span class="str">'bck'</span><span class="op">]</span><span class="op">[</span><span class="str">'shift'</span><span class="op">]</span><span class="op">[</span><span class="op">(</span><span class="op">*</span><span class="nam">expansion</span><span class="op">,</span><span class="op">)</span><span class="op">]</span><span class="op">,</span> <span class="nam">axis</span><span class="op">=</span><span class="nam">axis</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t466" href="#t466">466</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t467" href="#t467">467</a></span><span class="t">            <span class="nam">shuffle</span> <span class="op">=</span> <span class="op">[</span><span class="nam">slice</span><span class="op">(</span><span class="num">0</span><span class="op">,</span> <span class="nam">s</span><span class="op">,</span> <span class="num">1</span><span class="op">)</span> <span class="key">for</span> <span class="nam">s</span> <span class="key">in</span> <span class="nam">u</span><span class="op">.</span><span class="nam">shape</span><span class="op">]</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t468" href="#t468">468</a></span><span class="t">            <span class="nam">shuffle</span><span class="op">[</span><span class="nam">axis</span><span class="op">]</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">fft_utils</span><span class="op">[</span><span class="str">'bck'</span><span class="op">]</span><span class="op">[</span><span class="str">'shuffle'</span><span class="op">]</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t469" href="#t469">469</a></span><span class="t">            <span class="nam">V</span> <span class="op">=</span> <span class="nam">v</span><span class="op">[</span><span class="op">(</span><span class="op">*</span><span class="nam">shuffle</span><span class="op">,</span><span class="op">)</span><span class="op">]</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t470" href="#t470">470</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t471" href="#t471">471</a></span><span class="t">            <span class="nam">result</span><span class="op">.</span><span class="nam">real</span><span class="op">[</span><span class="op">...</span><span class="op">]</span> <span class="op">=</span> <span class="nam">V</span><span class="op">.</span><span class="nam">real</span><span class="op">[</span><span class="op">...</span><span class="op">]</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t472" href="#t472">472</a></span><span class="t">            <span class="key">return</span> <span class="nam">result</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t473" href="#t473">473</a></span><span class="t">        <span class="key">else</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t474" href="#t474">474</a></span><span class="t">            <span class="key">raise</span> <span class="nam">NotImplementedError</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t475" href="#t475">475</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t476" href="#t476">476</a></span><span class="t">    <span class="key">def</span> <span class="nam">get_BC</span><span class="op">(</span><span class="nam">self</span><span class="op">,</span> <span class="nam">kind</span><span class="op">,</span> <span class="op">**</span><span class="nam">kwargs</span><span class="op">)</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t477" href="#t477">477</a></span><span class="t">        <span class="str">"""</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t478" href="#t478">478</a></span><span class="t"><span class="str">        Get boundary condition row for boundary bordering. `kwargs` will be passed on to implementations of the BC of</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t479" href="#t479">479</a></span><span class="t"><span class="str">        the kind you choose. Specifically, `x` for `'dirichlet'` boundary condition, which is the coordinate at which to</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t480" href="#t480">480</a></span><span class="t"><span class="str">        set the BC.</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t481" href="#t481">481</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t482" href="#t482">482</a></span><span class="t"><span class="str">        Args:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t483" href="#t483">483</a></span><span class="t"><span class="str">            kind ('integral' or 'dirichlet'): Kind of boundary condition you want</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t484" href="#t484">484</a></span><span class="t"><span class="str">        """</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t485" href="#t485">485</a></span><span class="t">        <span class="key">if</span> <span class="nam">kind</span><span class="op">.</span><span class="nam">lower</span><span class="op">(</span><span class="op">)</span> <span class="op">==</span> <span class="str">'integral'</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t486" href="#t486">486</a></span><span class="t">            <span class="key">return</span> <span class="nam">self</span><span class="op">.</span><span class="nam">get_integ_BC_row</span><span class="op">(</span><span class="op">**</span><span class="nam">kwargs</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t487" href="#t487">487</a></span><span class="t">        <span class="key">elif</span> <span class="nam">kind</span><span class="op">.</span><span class="nam">lower</span><span class="op">(</span><span class="op">)</span> <span class="op">==</span> <span class="str">'dirichlet'</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t488" href="#t488">488</a></span><span class="t">            <span class="key">return</span> <span class="nam">self</span><span class="op">.</span><span class="nam">get_Dirichlet_BC_row</span><span class="op">(</span><span class="op">**</span><span class="nam">kwargs</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t489" href="#t489">489</a></span><span class="t">        <span class="key">else</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t490" href="#t490">490</a></span><span class="t">            <span class="key">return</span> <span class="nam">super</span><span class="op">(</span><span class="op">)</span><span class="op">.</span><span class="nam">get_BC</span><span class="op">(</span><span class="nam">kind</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t491" href="#t491">491</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t492" href="#t492">492</a></span><span class="t">    <span class="key">def</span> <span class="nam">get_integ_BC_row</span><span class="op">(</span><span class="nam">self</span><span class="op">)</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t493" href="#t493">493</a></span><span class="t">        <span class="str">"""</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t494" href="#t494">494</a></span><span class="t"><span class="str">        Get a row for generating integral BCs with T polynomials.</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t495" href="#t495">495</a></span><span class="t"><span class="str">        It returns the values of the integrals of T polynomials over the entire interval.</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t496" href="#t496">496</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t497" href="#t497">497</a></span><span class="t"><span class="str">        Returns:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t498" href="#t498">498</a></span><span class="t"><span class="str">            self.xp.ndarray: Row to put into a matrix</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t499" href="#t499">499</a></span><span class="t"><span class="str">        """</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t500" href="#t500">500</a></span><span class="t">        <span class="nam">n</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">xp</span><span class="op">.</span><span class="nam">arange</span><span class="op">(</span><span class="nam">self</span><span class="op">.</span><span class="nam">N</span><span class="op">)</span> <span class="op">+</span> <span class="num">1</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t501" href="#t501">501</a></span><span class="t">        <span class="nam">me</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">xp</span><span class="op">.</span><span class="nam">zeros_like</span><span class="op">(</span><span class="nam">n</span><span class="op">)</span><span class="op">.</span><span class="nam">astype</span><span class="op">(</span><span class="nam">float</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t502" href="#t502">502</a></span><span class="t">        <span class="nam">me</span><span class="op">[</span><span class="num">2</span><span class="op">:</span><span class="op">]</span> <span class="op">=</span> <span class="op">(</span><span class="op">(</span><span class="op">-</span><span class="num">1</span><span class="op">)</span> <span class="op">**</span> <span class="nam">n</span><span class="op">[</span><span class="num">1</span><span class="op">:</span><span class="op">-</span><span class="num">1</span><span class="op">]</span> <span class="op">+</span> <span class="num">1</span><span class="op">)</span> <span class="op">/</span> <span class="op">(</span><span class="num">1</span> <span class="op">-</span> <span class="nam">n</span><span class="op">[</span><span class="num">1</span><span class="op">:</span><span class="op">-</span><span class="num">1</span><span class="op">]</span> <span class="op">**</span> <span class="num">2</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t503" href="#t503">503</a></span><span class="t">        <span class="nam">me</span><span class="op">[</span><span class="num">0</span><span class="op">]</span> <span class="op">=</span> <span class="num">2.0</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t504" href="#t504">504</a></span><span class="t">        <span class="key">return</span> <span class="nam">me</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t505" href="#t505">505</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t506" href="#t506">506</a></span><span class="t">    <span class="key">def</span> <span class="nam">get_Dirichlet_BC_row</span><span class="op">(</span><span class="nam">self</span><span class="op">,</span> <span class="nam">x</span><span class="op">)</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t507" href="#t507">507</a></span><span class="t">        <span class="str">"""</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t508" href="#t508">508</a></span><span class="t"><span class="str">        Get a row for generating Dirichlet BCs at x with T polynomials.</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t509" href="#t509">509</a></span><span class="t"><span class="str">        It returns the values of the T polynomials at x.</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t510" href="#t510">510</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t511" href="#t511">511</a></span><span class="t"><span class="str">        Args:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t512" href="#t512">512</a></span><span class="t"><span class="str">            x (float): Position of the boundary condition</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t513" href="#t513">513</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t514" href="#t514">514</a></span><span class="t"><span class="str">        Returns:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t515" href="#t515">515</a></span><span class="t"><span class="str">            self.xp.ndarray: Row to put into a matrix</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t516" href="#t516">516</a></span><span class="t"><span class="str">        """</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t517" href="#t517">517</a></span><span class="t">        <span class="key">if</span> <span class="nam">x</span> <span class="op">==</span> <span class="op">-</span><span class="num">1</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t518" href="#t518">518</a></span><span class="t">            <span class="key">return</span> <span class="op">(</span><span class="op">-</span><span class="num">1</span><span class="op">)</span> <span class="op">**</span> <span class="nam">self</span><span class="op">.</span><span class="nam">xp</span><span class="op">.</span><span class="nam">arange</span><span class="op">(</span><span class="nam">self</span><span class="op">.</span><span class="nam">N</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t519" href="#t519">519</a></span><span class="t">        <span class="key">elif</span> <span class="nam">x</span> <span class="op">==</span> <span class="num">1</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t520" href="#t520">520</a></span><span class="t">            <span class="key">return</span> <span class="nam">self</span><span class="op">.</span><span class="nam">xp</span><span class="op">.</span><span class="nam">ones</span><span class="op">(</span><span class="nam">self</span><span class="op">.</span><span class="nam">N</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t521" href="#t521">521</a></span><span class="t">        <span class="key">elif</span> <span class="nam">x</span> <span class="op">==</span> <span class="num">0</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t522" href="#t522">522</a></span><span class="t">            <span class="nam">n</span> <span class="op">=</span> <span class="op">(</span><span class="num">1</span> <span class="op">+</span> <span class="op">(</span><span class="op">-</span><span class="num">1</span><span class="op">)</span> <span class="op">**</span> <span class="nam">self</span><span class="op">.</span><span class="nam">xp</span><span class="op">.</span><span class="nam">arange</span><span class="op">(</span><span class="nam">self</span><span class="op">.</span><span class="nam">N</span><span class="op">)</span><span class="op">)</span> <span class="op">/</span> <span class="num">2</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t523" href="#t523">523</a></span><span class="t">            <span class="nam">n</span><span class="op">[</span><span class="num">2</span><span class="op">:</span><span class="op">:</span><span class="num">4</span><span class="op">]</span> <span class="op">*=</span> <span class="op">-</span><span class="num">1</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t524" href="#t524">524</a></span><span class="t">            <span class="key">return</span> <span class="nam">n</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t525" href="#t525">525</a></span><span class="t">        <span class="key">else</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t526" href="#t526">526</a></span><span class="t">            <span class="key">raise</span> <span class="nam">NotImplementedError</span><span class="op">(</span><span class="fst">f'</span><span class="fst">Don\'t know how to generate Dirichlet BC\'s at </span><span class="op">{</span><span class="nam">x</span><span class="op">=</span><span class="op">}</span><span class="fst">!</span><span class="fst">'</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t527" href="#t527">527</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t528" href="#t528">528</a></span><span class="t">    <span class="key">def</span> <span class="nam">get_Dirichlet_recombination_matrix</span><span class="op">(</span><span class="nam">self</span><span class="op">)</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t529" href="#t529">529</a></span><span class="t">        <span class="str">'''</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t530" href="#t530">530</a></span><span class="t"><span class="str">        Get matrix for Dirichlet recombination, which changes the basis to have sparse boundary conditions.</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t531" href="#t531">531</a></span><span class="t"><span class="str">        This makes for a good right preconditioner.</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t532" href="#t532">532</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t533" href="#t533">533</a></span><span class="t"><span class="str">        Returns:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t534" href="#t534">534</a></span><span class="t"><span class="str">            scipy.sparse: Sparse conversion matrix</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t535" href="#t535">535</a></span><span class="t"><span class="str">        '''</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t536" href="#t536">536</a></span><span class="t">        <span class="nam">N</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">N</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t537" href="#t537">537</a></span><span class="t">        <span class="nam">sp</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">sparse_lib</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t538" href="#t538">538</a></span><span class="t">        <span class="nam">xp</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">xp</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t539" href="#t539">539</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t540" href="#t540">540</a></span><span class="t">        <span class="key">return</span> <span class="nam">sp</span><span class="op">.</span><span class="nam">eye</span><span class="op">(</span><span class="nam">N</span><span class="op">)</span> <span class="op">-</span> <span class="nam">sp</span><span class="op">.</span><span class="nam">diags</span><span class="op">(</span><span class="nam">xp</span><span class="op">.</span><span class="nam">ones</span><span class="op">(</span><span class="nam">N</span> <span class="op">-</span> <span class="num">2</span><span class="op">)</span><span class="op">,</span> <span class="nam">offsets</span><span class="op">=</span><span class="op">+</span><span class="num">2</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t541" href="#t541">541</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t542" href="#t542">542</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t543" href="#t543">543</a></span><span class="t"><span class="key">class</span> <span class="nam">UltrasphericalHelper</span><span class="op">(</span><span class="nam">ChebychevHelper</span><span class="op">)</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t544" href="#t544">544</a></span><span class="t">    <span class="str">"""</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t545" href="#t545">545</a></span><span class="t"><span class="str">    This implementation follows https://doi.org/10.1137/120865458.</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t546" href="#t546">546</a></span><span class="t"><span class="str">    The ultraspherical method works in Chebychev polynomials as well, but also uses various Gegenbauer polynomials.</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t547" href="#t547">547</a></span><span class="t"><span class="str">    The idea is that for every derivative of Chebychev T polynomials, there is a basis of Gegenbauer polynomials where the differentiation matrix is a single off-diagonal.</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t548" href="#t548">548</a></span><span class="t"><span class="str">    There are also conversion operators from one derivative basis to the next that are sparse.</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t549" href="#t549">549</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t550" href="#t550">550</a></span><span class="t"><span class="str">    This basis is used like this: For every equation that you have, look for the highest derivative and bump all matrices to the correct basis. If your highest derivative is 2 and you have an identity, it needs to get bumped from 0 to 1 and from 1 to 2. If you have a first derivative as well, it needs to be bumped from 1 to 2.</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t551" href="#t551">551</a></span><span class="t"><span class="str">    You don't need the same resulting basis in all equations. You just need to take care that you translate the right hand side to the correct basis as well.</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t552" href="#t552">552</a></span><span class="t"><span class="str">    """</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t553" href="#t553">553</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t554" href="#t554">554</a></span><span class="t">    <span class="key">def</span> <span class="nam">get_differentiation_matrix</span><span class="op">(</span><span class="nam">self</span><span class="op">,</span> <span class="nam">p</span><span class="op">=</span><span class="num">1</span><span class="op">)</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t555" href="#t555">555</a></span><span class="t">        <span class="str">"""</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t556" href="#t556">556</a></span><span class="t"><span class="str">        Notice that while sparse, this matrix is not diagonal, which means the inversion cannot be parallelized easily.</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t557" href="#t557">557</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t558" href="#t558">558</a></span><span class="t"><span class="str">        Args:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t559" href="#t559">559</a></span><span class="t"><span class="str">            p (int): Order of the derivative</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t560" href="#t560">560</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t561" href="#t561">561</a></span><span class="t"><span class="str">        Returns:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t562" href="#t562">562</a></span><span class="t"><span class="str">            sparse differentiation matrix</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t563" href="#t563">563</a></span><span class="t"><span class="str">        """</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t564" href="#t564">564</a></span><span class="t">        <span class="nam">sp</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">sparse_lib</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t565" href="#t565">565</a></span><span class="t">        <span class="nam">xp</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">xp</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t566" href="#t566">566</a></span><span class="t">        <span class="nam">N</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">N</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t567" href="#t567">567</a></span><span class="t">        <span class="nam">l</span> <span class="op">=</span> <span class="nam">p</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t568" href="#t568">568</a></span><span class="t">        <span class="key">return</span> <span class="num">2</span> <span class="op">**</span> <span class="op">(</span><span class="nam">l</span> <span class="op">-</span> <span class="num">1</span><span class="op">)</span> <span class="op">*</span> <span class="nam">factorial</span><span class="op">(</span><span class="nam">l</span> <span class="op">-</span> <span class="num">1</span><span class="op">)</span> <span class="op">*</span> <span class="nam">sp</span><span class="op">.</span><span class="nam">diags</span><span class="op">(</span><span class="nam">xp</span><span class="op">.</span><span class="nam">arange</span><span class="op">(</span><span class="nam">N</span> <span class="op">-</span> <span class="nam">l</span><span class="op">)</span> <span class="op">+</span> <span class="nam">l</span><span class="op">,</span> <span class="nam">offsets</span><span class="op">=</span><span class="nam">l</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t569" href="#t569">569</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t570" href="#t570">570</a></span><span class="t">    <span class="key">def</span> <span class="nam">get_S</span><span class="op">(</span><span class="nam">self</span><span class="op">,</span> <span class="nam">lmbda</span><span class="op">)</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t571" href="#t571">571</a></span><span class="t">        <span class="str">"""</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t572" href="#t572">572</a></span><span class="t"><span class="str">        Get matrix for bumping the derivative base by one from lmbda to lmbda + 1. This is the same language as in</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t573" href="#t573">573</a></span><span class="t"><span class="str">        https://doi.org/10.1137/120865458.</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t574" href="#t574">574</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t575" href="#t575">575</a></span><span class="t"><span class="str">        Args:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t576" href="#t576">576</a></span><span class="t"><span class="str">            lmbda (int): Ingoing derivative base</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t577" href="#t577">577</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t578" href="#t578">578</a></span><span class="t"><span class="str">        Returns:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t579" href="#t579">579</a></span><span class="t"><span class="str">            sparse matrix: Conversion from derivative base lmbda to lmbda + 1</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t580" href="#t580">580</a></span><span class="t"><span class="str">        """</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t581" href="#t581">581</a></span><span class="t">        <span class="nam">N</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">N</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t582" href="#t582">582</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t583" href="#t583">583</a></span><span class="t">        <span class="key">if</span> <span class="nam">lmbda</span> <span class="op">==</span> <span class="num">0</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t584" href="#t584">584</a></span><span class="t">            <span class="nam">sp</span> <span class="op">=</span> <span class="nam">scipy</span><span class="op">.</span><span class="nam">sparse</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t585" href="#t585">585</a></span><span class="t">            <span class="nam">mat</span> <span class="op">=</span> <span class="op">(</span><span class="op">(</span><span class="nam">sp</span><span class="op">.</span><span class="nam">eye</span><span class="op">(</span><span class="nam">N</span><span class="op">)</span> <span class="op">-</span> <span class="nam">sp</span><span class="op">.</span><span class="nam">diags</span><span class="op">(</span><span class="nam">np</span><span class="op">.</span><span class="nam">ones</span><span class="op">(</span><span class="nam">N</span> <span class="op">-</span> <span class="num">2</span><span class="op">)</span><span class="op">,</span> <span class="nam">offsets</span><span class="op">=</span><span class="op">+</span><span class="num">2</span><span class="op">)</span><span class="op">)</span> <span class="op">/</span> <span class="num">2.0</span><span class="op">)</span><span class="op">.</span><span class="nam">tolil</span><span class="op">(</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t586" href="#t586">586</a></span><span class="t">            <span class="nam">mat</span><span class="op">[</span><span class="op">:</span><span class="op">,</span> <span class="num">0</span><span class="op">]</span> <span class="op">*=</span> <span class="num">2</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t587" href="#t587">587</a></span><span class="t">        <span class="key">else</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t588" href="#t588">588</a></span><span class="t">            <span class="nam">sp</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">sparse_lib</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t589" href="#t589">589</a></span><span class="t">            <span class="nam">xp</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">xp</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t590" href="#t590">590</a></span><span class="t">            <span class="nam">mat</span> <span class="op">=</span> <span class="nam">sp</span><span class="op">.</span><span class="nam">diags</span><span class="op">(</span><span class="nam">lmbda</span> <span class="op">/</span> <span class="op">(</span><span class="nam">lmbda</span> <span class="op">+</span> <span class="nam">xp</span><span class="op">.</span><span class="nam">arange</span><span class="op">(</span><span class="nam">N</span><span class="op">)</span><span class="op">)</span><span class="op">)</span> <span class="op">-</span> <span class="nam">sp</span><span class="op">.</span><span class="nam">diags</span><span class="op">(</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t591" href="#t591">591</a></span><span class="t">                <span class="nam">lmbda</span> <span class="op">/</span> <span class="op">(</span><span class="nam">lmbda</span> <span class="op">+</span> <span class="num">2</span> <span class="op">+</span> <span class="nam">xp</span><span class="op">.</span><span class="nam">arange</span><span class="op">(</span><span class="nam">N</span> <span class="op">-</span> <span class="num">2</span><span class="op">)</span><span class="op">)</span><span class="op">,</span> <span class="nam">offsets</span><span class="op">=</span><span class="op">+</span><span class="num">2</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t592" href="#t592">592</a></span><span class="t">            <span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t593" href="#t593">593</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t594" href="#t594">594</a></span><span class="t">        <span class="key">return</span> <span class="nam">self</span><span class="op">.</span><span class="nam">sparse_lib</span><span class="op">.</span><span class="nam">csc_matrix</span><span class="op">(</span><span class="nam">mat</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t595" href="#t595">595</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t596" href="#t596">596</a></span><span class="t">    <span class="key">def</span> <span class="nam">get_basis_change_matrix</span><span class="op">(</span><span class="nam">self</span><span class="op">,</span> <span class="nam">p_in</span><span class="op">=</span><span class="num">0</span><span class="op">,</span> <span class="nam">p_out</span><span class="op">=</span><span class="num">0</span><span class="op">,</span> <span class="op">**</span><span class="nam">kwargs</span><span class="op">)</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t597" href="#t597">597</a></span><span class="t">        <span class="str">"""</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t598" href="#t598">598</a></span><span class="t"><span class="str">        Get a conversion matrix from derivative base `p_in` to `p_out`.</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t599" href="#t599">599</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t600" href="#t600">600</a></span><span class="t"><span class="str">        Args:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t601" href="#t601">601</a></span><span class="t"><span class="str">            p_out (int): Resulting derivative base</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t602" href="#t602">602</a></span><span class="t"><span class="str">            p_in (int): Ingoing derivative base</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t603" href="#t603">603</a></span><span class="t"><span class="str">        """</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t604" href="#t604">604</a></span><span class="t">        <span class="nam">mat_fwd</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">sparse_lib</span><span class="op">.</span><span class="nam">eye</span><span class="op">(</span><span class="nam">self</span><span class="op">.</span><span class="nam">N</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t605" href="#t605">605</a></span><span class="t">        <span class="key">for</span> <span class="nam">i</span> <span class="key">in</span> <span class="nam">range</span><span class="op">(</span><span class="nam">min</span><span class="op">(</span><span class="op">[</span><span class="nam">p_in</span><span class="op">,</span> <span class="nam">p_out</span><span class="op">]</span><span class="op">)</span><span class="op">,</span> <span class="nam">max</span><span class="op">(</span><span class="op">[</span><span class="nam">p_in</span><span class="op">,</span> <span class="nam">p_out</span><span class="op">]</span><span class="op">)</span><span class="op">)</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t606" href="#t606">606</a></span><span class="t">            <span class="nam">mat_fwd</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">get_S</span><span class="op">(</span><span class="nam">i</span><span class="op">)</span> <span class="op">@</span> <span class="nam">mat_fwd</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t607" href="#t607">607</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t608" href="#t608">608</a></span><span class="t">        <span class="key">if</span> <span class="nam">p_out</span> <span class="op">></span> <span class="nam">p_in</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t609" href="#t609">609</a></span><span class="t">            <span class="key">return</span> <span class="nam">mat_fwd</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t610" href="#t610">610</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t611" href="#t611">611</a></span><span class="t">        <span class="key">else</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t612" href="#t612">612</a></span><span class="t">            <span class="com"># We have to invert the matrix on CPU because the GPU equivalent is not implemented in CuPy at the time of writing.</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t613" href="#t613">613</a></span><span class="t">            <span class="key">import</span> <span class="nam">scipy</span><span class="op">.</span><span class="nam">sparse</span> <span class="key">as</span> <span class="nam">sp</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t614" href="#t614">614</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t615" href="#t615">615</a></span><span class="t">            <span class="key">if</span> <span class="nam">self</span><span class="op">.</span><span class="nam">useGPU</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t616" href="#t616">616</a></span><span class="t">                <span class="nam">mat_fwd</span> <span class="op">=</span> <span class="nam">mat_fwd</span><span class="op">.</span><span class="nam">get</span><span class="op">(</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t617" href="#t617">617</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t618" href="#t618">618</a></span><span class="t">            <span class="nam">mat_bck</span> <span class="op">=</span> <span class="nam">sp</span><span class="op">.</span><span class="nam">linalg</span><span class="op">.</span><span class="nam">inv</span><span class="op">(</span><span class="nam">mat_fwd</span><span class="op">.</span><span class="nam">tocsc</span><span class="op">(</span><span class="op">)</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t619" href="#t619">619</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t620" href="#t620">620</a></span><span class="t">            <span class="key">return</span> <span class="nam">self</span><span class="op">.</span><span class="nam">sparse_lib</span><span class="op">.</span><span class="nam">csc_matrix</span><span class="op">(</span><span class="nam">mat_bck</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t621" href="#t621">621</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t622" href="#t622">622</a></span><span class="t">    <span class="key">def</span> <span class="nam">get_integration_matrix</span><span class="op">(</span><span class="nam">self</span><span class="op">)</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t623" href="#t623">623</a></span><span class="t">        <span class="str">"""</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t624" href="#t624">624</a></span><span class="t"><span class="str">        Get an integration matrix. Please use `UltrasphericalHelper.get_integration_constant` afterwards to compute the</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t625" href="#t625">625</a></span><span class="t"><span class="str">        integration constant such that integration starts from x=-1.</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t626" href="#t626">626</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t627" href="#t627">627</a></span><span class="t"><span class="str">        Example:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t628" href="#t628">628</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t629" href="#t629">629</a></span><span class="t"><span class="str">        .. code-block:: python</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t630" href="#t630">630</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t631" href="#t631">631</a></span><span class="t"><span class="str">            import numpy as np</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t632" href="#t632">632</a></span><span class="t"><span class="str">            from pySDC.helpers.spectral_helper import UltrasphericalHelper</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t633" href="#t633">633</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t634" href="#t634">634</a></span><span class="t"><span class="str">            N = 4</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t635" href="#t635">635</a></span><span class="t"><span class="str">            helper = UltrasphericalHelper(N)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t636" href="#t636">636</a></span><span class="t"><span class="str">            coeffs = np.random.random(N)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t637" href="#t637">637</a></span><span class="t"><span class="str">            coeffs[-1] = 0</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t638" href="#t638">638</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t639" href="#t639">639</a></span><span class="t"><span class="str">            poly = np.polynomial.Chebyshev(coeffs)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t640" href="#t640">640</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t641" href="#t641">641</a></span><span class="t"><span class="str">            S = helper.get_integration_matrix()</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t642" href="#t642">642</a></span><span class="t"><span class="str">            U_hat = S @ coeffs</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t643" href="#t643">643</a></span><span class="t"><span class="str">            U_hat[0] = helper.get_integration_constant(U_hat, axis=-1)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t644" href="#t644">644</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t645" href="#t645">645</a></span><span class="t"><span class="str">            assert np.allclose(poly.integ(lbnd=-1).coef[:-1], U_hat)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t646" href="#t646">646</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t647" href="#t647">647</a></span><span class="t"><span class="str">        Returns:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t648" href="#t648">648</a></span><span class="t"><span class="str">            sparse integration matrix</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t649" href="#t649">649</a></span><span class="t"><span class="str">        """</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t650" href="#t650">650</a></span><span class="t">        <span class="key">return</span> <span class="nam">self</span><span class="op">.</span><span class="nam">sparse_lib</span><span class="op">.</span><span class="nam">diags</span><span class="op">(</span><span class="num">1</span> <span class="op">/</span> <span class="op">(</span><span class="nam">self</span><span class="op">.</span><span class="nam">xp</span><span class="op">.</span><span class="nam">arange</span><span class="op">(</span><span class="nam">self</span><span class="op">.</span><span class="nam">N</span> <span class="op">-</span> <span class="num">1</span><span class="op">)</span> <span class="op">+</span> <span class="num">1</span><span class="op">)</span><span class="op">,</span> <span class="nam">offsets</span><span class="op">=</span><span class="op">-</span><span class="num">1</span><span class="op">)</span> <span class="op">@</span> <span class="nam">self</span><span class="op">.</span><span class="nam">get_basis_change_matrix</span><span class="op">(</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t651" href="#t651">651</a></span><span class="t">            <span class="nam">p_out</span><span class="op">=</span><span class="num">1</span><span class="op">,</span> <span class="nam">p_in</span><span class="op">=</span><span class="num">0</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t652" href="#t652">652</a></span><span class="t">        <span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t653" href="#t653">653</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t654" href="#t654">654</a></span><span class="t">    <span class="key">def</span> <span class="nam">get_integration_constant</span><span class="op">(</span><span class="nam">self</span><span class="op">,</span> <span class="nam">u_hat</span><span class="op">,</span> <span class="nam">axis</span><span class="op">)</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t655" href="#t655">655</a></span><span class="t">        <span class="str">"""</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t656" href="#t656">656</a></span><span class="t"><span class="str">        Get integration constant for lower bound of -1. See documentation of `UltrasphericalHelper.get_integration_matrix` for details.</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t657" href="#t657">657</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t658" href="#t658">658</a></span><span class="t"><span class="str">        Args:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t659" href="#t659">659</a></span><span class="t"><span class="str">            u_hat: Solution in spectral space</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t660" href="#t660">660</a></span><span class="t"><span class="str">            axis: Axis you want to integrate over</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t661" href="#t661">661</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t662" href="#t662">662</a></span><span class="t"><span class="str">        Returns:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t663" href="#t663">663</a></span><span class="t"><span class="str">            Integration constant, has one less dimension than `u_hat`</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t664" href="#t664">664</a></span><span class="t"><span class="str">        """</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t665" href="#t665">665</a></span><span class="t">        <span class="nam">slices</span> <span class="op">=</span> <span class="op">[</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t666" href="#t666">666</a></span><span class="t">            <span class="key">None</span><span class="op">,</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t667" href="#t667">667</a></span><span class="t">        <span class="op">]</span> <span class="op">*</span> <span class="nam">u_hat</span><span class="op">.</span><span class="nam">ndim</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t668" href="#t668">668</a></span><span class="t">        <span class="nam">slices</span><span class="op">[</span><span class="nam">axis</span><span class="op">]</span> <span class="op">=</span> <span class="nam">slice</span><span class="op">(</span><span class="num">1</span><span class="op">,</span> <span class="nam">u_hat</span><span class="op">.</span><span class="nam">shape</span><span class="op">[</span><span class="nam">axis</span><span class="op">]</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t669" href="#t669">669</a></span><span class="t">        <span class="key">return</span> <span class="nam">self</span><span class="op">.</span><span class="nam">xp</span><span class="op">.</span><span class="nam">sum</span><span class="op">(</span><span class="nam">u_hat</span><span class="op">[</span><span class="op">(</span><span class="op">*</span><span class="nam">slices</span><span class="op">,</span><span class="op">)</span><span class="op">]</span> <span class="op">*</span> <span class="op">(</span><span class="op">-</span><span class="num">1</span><span class="op">)</span> <span class="op">**</span> <span class="op">(</span><span class="nam">self</span><span class="op">.</span><span class="nam">xp</span><span class="op">.</span><span class="nam">arange</span><span class="op">(</span><span class="nam">u_hat</span><span class="op">.</span><span class="nam">shape</span><span class="op">[</span><span class="nam">axis</span><span class="op">]</span> <span class="op">-</span> <span class="num">1</span><span class="op">)</span><span class="op">)</span><span class="op">,</span> <span class="nam">axis</span><span class="op">=</span><span class="nam">axis</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t670" href="#t670">670</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t671" href="#t671">671</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t672" href="#t672">672</a></span><span class="t"><span class="key">class</span> <span class="nam">FFTHelper</span><span class="op">(</span><span class="nam">SpectralHelper1D</span><span class="op">)</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t673" href="#t673">673</a></span><span class="t">    <span class="key">def</span> <span class="nam">__init__</span><span class="op">(</span><span class="nam">self</span><span class="op">,</span> <span class="op">*</span><span class="nam">args</span><span class="op">,</span> <span class="nam">x0</span><span class="op">=</span><span class="num">0</span><span class="op">,</span> <span class="nam">x1</span><span class="op">=</span><span class="num">2</span> <span class="op">*</span> <span class="nam">np</span><span class="op">.</span><span class="nam">pi</span><span class="op">,</span> <span class="op">**</span><span class="nam">kwargs</span><span class="op">)</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t674" href="#t674">674</a></span><span class="t">        <span class="str">"""</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t675" href="#t675">675</a></span><span class="t"><span class="str">        Constructor.</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t676" href="#t676">676</a></span><span class="t"><span class="str">        Please refer to the parent class for additional arguments. Notably, you have to supply a resolution `N` and you</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t677" href="#t677">677</a></span><span class="t"><span class="str">        may choose to run on GPUs via the `useGPU` argument.</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t678" href="#t678">678</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t679" href="#t679">679</a></span><span class="t"><span class="str">        Args:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t680" href="#t680">680</a></span><span class="t"><span class="str">            transform_type ('fft' or 'dct'): Either use DCT functions directly implemented in the transform library or</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t681" href="#t681">681</a></span><span class="t"><span class="str">                                             use the FFT from the library to compute the DCT</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t682" href="#t682">682</a></span><span class="t"><span class="str">            x0 (float, optional): Coordinate of left boundary</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t683" href="#t683">683</a></span><span class="t"><span class="str">            x1 (float, optional): Coordinate of right boundary</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t684" href="#t684">684</a></span><span class="t"><span class="str">        """</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t685" href="#t685">685</a></span><span class="t">        <span class="nam">super</span><span class="op">(</span><span class="op">)</span><span class="op">.</span><span class="nam">__init__</span><span class="op">(</span><span class="op">*</span><span class="nam">args</span><span class="op">,</span> <span class="nam">x0</span><span class="op">=</span><span class="nam">x0</span><span class="op">,</span> <span class="nam">x1</span><span class="op">=</span><span class="nam">x1</span><span class="op">,</span> <span class="op">**</span><span class="nam">kwargs</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t686" href="#t686">686</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t687" href="#t687">687</a></span><span class="t">    <span class="key">def</span> <span class="nam">get_1dgrid</span><span class="op">(</span><span class="nam">self</span><span class="op">)</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t688" href="#t688">688</a></span><span class="t">        <span class="str">"""</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t689" href="#t689">689</a></span><span class="t"><span class="str">        We use equally spaced points including the left boundary and not including the right one, which is the left boundary.</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t690" href="#t690">690</a></span><span class="t"><span class="str">        """</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t691" href="#t691">691</a></span><span class="t">        <span class="nam">dx</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">L</span> <span class="op">/</span> <span class="nam">self</span><span class="op">.</span><span class="nam">N</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t692" href="#t692">692</a></span><span class="t">        <span class="key">return</span> <span class="nam">self</span><span class="op">.</span><span class="nam">xp</span><span class="op">.</span><span class="nam">arange</span><span class="op">(</span><span class="nam">self</span><span class="op">.</span><span class="nam">N</span><span class="op">)</span> <span class="op">*</span> <span class="nam">dx</span> <span class="op">+</span> <span class="nam">self</span><span class="op">.</span><span class="nam">x0</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t693" href="#t693">693</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t694" href="#t694">694</a></span><span class="t">    <span class="key">def</span> <span class="nam">get_wavenumbers</span><span class="op">(</span><span class="nam">self</span><span class="op">)</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t695" href="#t695">695</a></span><span class="t">        <span class="str">"""</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t696" href="#t696">696</a></span><span class="t"><span class="str">        Be careful that this ordering is very unintuitive.</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t697" href="#t697">697</a></span><span class="t"><span class="str">        """</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t698" href="#t698">698</a></span><span class="t">        <span class="key">return</span> <span class="nam">self</span><span class="op">.</span><span class="nam">xp</span><span class="op">.</span><span class="nam">fft</span><span class="op">.</span><span class="nam">fftfreq</span><span class="op">(</span><span class="nam">self</span><span class="op">.</span><span class="nam">N</span><span class="op">,</span> <span class="num">1.0</span> <span class="op">/</span> <span class="nam">self</span><span class="op">.</span><span class="nam">N</span><span class="op">)</span> <span class="op">*</span> <span class="num">2</span> <span class="op">*</span> <span class="nam">np</span><span class="op">.</span><span class="nam">pi</span> <span class="op">/</span> <span class="nam">self</span><span class="op">.</span><span class="nam">L</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t699" href="#t699">699</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t700" href="#t700">700</a></span><span class="t">    <span class="key">def</span> <span class="nam">get_differentiation_matrix</span><span class="op">(</span><span class="nam">self</span><span class="op">,</span> <span class="nam">p</span><span class="op">=</span><span class="num">1</span><span class="op">)</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t701" href="#t701">701</a></span><span class="t">        <span class="str">"""</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t702" href="#t702">702</a></span><span class="t"><span class="str">        This matrix is diagonal, allowing to invert concurrently.</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t703" href="#t703">703</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t704" href="#t704">704</a></span><span class="t"><span class="str">        Args:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t705" href="#t705">705</a></span><span class="t"><span class="str">            p (int): Order of the derivative</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t706" href="#t706">706</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t707" href="#t707">707</a></span><span class="t"><span class="str">        Returns:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t708" href="#t708">708</a></span><span class="t"><span class="str">            sparse differentiation matrix</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t709" href="#t709">709</a></span><span class="t"><span class="str">        """</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t710" href="#t710">710</a></span><span class="t">        <span class="nam">k</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">get_wavenumbers</span><span class="op">(</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t711" href="#t711">711</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t712" href="#t712">712</a></span><span class="t">        <span class="key">if</span> <span class="nam">self</span><span class="op">.</span><span class="nam">useGPU</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t713" href="#t713">713</a></span><span class="t">            <span class="com"># Have to raise the matrix to power p on CPU because the GPU equivalent is not implemented in CuPy at the time of writing.</span>&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t714" href="#t714">714</a></span><span class="t">            <span class="key">import</span> <span class="nam">scipy</span><span class="op">.</span><span class="nam">sparse</span> <span class="key">as</span> <span class="nam">sp</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t715" href="#t715">715</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t716" href="#t716">716</a></span><span class="t">            <span class="nam">D</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">sparse_lib</span><span class="op">.</span><span class="nam">diags</span><span class="op">(</span><span class="num">1j</span> <span class="op">*</span> <span class="nam">k</span><span class="op">)</span><span class="op">.</span><span class="nam">get</span><span class="op">(</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t717" href="#t717">717</a></span><span class="t">            <span class="key">return</span> <span class="nam">self</span><span class="op">.</span><span class="nam">sparse_lib</span><span class="op">.</span><span class="nam">csc_matrix</span><span class="op">(</span><span class="nam">sp</span><span class="op">.</span><span class="nam">linalg</span><span class="op">.</span><span class="nam">matrix_power</span><span class="op">(</span><span class="nam">D</span><span class="op">,</span> <span class="nam">p</span><span class="op">)</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t718" href="#t718">718</a></span><span class="t">        <span class="key">else</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t719" href="#t719">719</a></span><span class="t">            <span class="key">return</span> <span class="nam">self</span><span class="op">.</span><span class="nam">linalg</span><span class="op">.</span><span class="nam">matrix_power</span><span class="op">(</span><span class="nam">self</span><span class="op">.</span><span class="nam">sparse_lib</span><span class="op">.</span><span class="nam">diags</span><span class="op">(</span><span class="num">1j</span> <span class="op">*</span> <span class="nam">k</span><span class="op">)</span><span class="op">,</span> <span class="nam">p</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t720" href="#t720">720</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t721" href="#t721">721</a></span><span class="t">    <span class="key">def</span> <span class="nam">get_integration_matrix</span><span class="op">(</span><span class="nam">self</span><span class="op">,</span> <span class="nam">p</span><span class="op">=</span><span class="num">1</span><span class="op">)</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t722" href="#t722">722</a></span><span class="t">        <span class="str">"""</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t723" href="#t723">723</a></span><span class="t"><span class="str">        Get integration matrix to compute `p`-th integral over the entire domain.</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t724" href="#t724">724</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t725" href="#t725">725</a></span><span class="t"><span class="str">        Args:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t726" href="#t726">726</a></span><span class="t"><span class="str">            p (int): Order of integral you want to compute</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t727" href="#t727">727</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t728" href="#t728">728</a></span><span class="t"><span class="str">        Returns:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t729" href="#t729">729</a></span><span class="t"><span class="str">            sparse integration matrix</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t730" href="#t730">730</a></span><span class="t"><span class="str">        """</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t731" href="#t731">731</a></span><span class="t">        <span class="nam">k</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">xp</span><span class="op">.</span><span class="nam">array</span><span class="op">(</span><span class="nam">self</span><span class="op">.</span><span class="nam">get_wavenumbers</span><span class="op">(</span><span class="op">)</span><span class="op">,</span> <span class="nam">dtype</span><span class="op">=</span><span class="str">'complex128'</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t732" href="#t732">732</a></span><span class="t">        <span class="nam">k</span><span class="op">[</span><span class="num">0</span><span class="op">]</span> <span class="op">=</span> <span class="num">1j</span> <span class="op">*</span> <span class="nam">self</span><span class="op">.</span><span class="nam">L</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t733" href="#t733">733</a></span><span class="t">        <span class="key">return</span> <span class="nam">self</span><span class="op">.</span><span class="nam">linalg</span><span class="op">.</span><span class="nam">matrix_power</span><span class="op">(</span><span class="nam">self</span><span class="op">.</span><span class="nam">sparse_lib</span><span class="op">.</span><span class="nam">diags</span><span class="op">(</span><span class="num">1</span> <span class="op">/</span> <span class="op">(</span><span class="num">1j</span> <span class="op">*</span> <span class="nam">k</span><span class="op">)</span><span class="op">)</span><span class="op">,</span> <span class="nam">p</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t734" href="#t734">734</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t735" href="#t735">735</a></span><span class="t">    <span class="key">def</span> <span class="nam">transform</span><span class="op">(</span><span class="nam">self</span><span class="op">,</span> <span class="nam">u</span><span class="op">,</span> <span class="nam">axis</span><span class="op">=</span><span class="op">-</span><span class="num">1</span><span class="op">,</span> <span class="op">**</span><span class="nam">kwargs</span><span class="op">)</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t736" href="#t736">736</a></span><span class="t">        <span class="str">"""</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t737" href="#t737">737</a></span><span class="t"><span class="str">        1D FFT along axis. `kwargs` are passed on to the FFT library.</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t738" href="#t738">738</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t739" href="#t739">739</a></span><span class="t"><span class="str">        Args:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t740" href="#t740">740</a></span><span class="t"><span class="str">            u: Data you want to transform</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t741" href="#t741">741</a></span><span class="t"><span class="str">            axis (int): Axis you want to transform along</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t742" href="#t742">742</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t743" href="#t743">743</a></span><span class="t"><span class="str">        Returns:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t744" href="#t744">744</a></span><span class="t"><span class="str">            transformed data</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t745" href="#t745">745</a></span><span class="t"><span class="str">        """</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t746" href="#t746">746</a></span><span class="t">        <span class="key">return</span> <span class="nam">self</span><span class="op">.</span><span class="nam">fft_lib</span><span class="op">.</span><span class="nam">fft</span><span class="op">(</span><span class="nam">u</span><span class="op">,</span> <span class="nam">axis</span><span class="op">=</span><span class="nam">axis</span><span class="op">,</span> <span class="op">**</span><span class="nam">kwargs</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t747" href="#t747">747</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t748" href="#t748">748</a></span><span class="t">    <span class="key">def</span> <span class="nam">itransform</span><span class="op">(</span><span class="nam">self</span><span class="op">,</span> <span class="nam">u</span><span class="op">,</span> <span class="nam">axis</span><span class="op">=</span><span class="op">-</span><span class="num">1</span><span class="op">)</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t749" href="#t749">749</a></span><span class="t">        <span class="str">"""</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t750" href="#t750">750</a></span><span class="t"><span class="str">        Inverse 1D FFT.</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t751" href="#t751">751</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t752" href="#t752">752</a></span><span class="t"><span class="str">        Args:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t753" href="#t753">753</a></span><span class="t"><span class="str">            u: Data you want to transform</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t754" href="#t754">754</a></span><span class="t"><span class="str">            axis (int): Axis you want to transform along</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t755" href="#t755">755</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t756" href="#t756">756</a></span><span class="t"><span class="str">        Returns:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t757" href="#t757">757</a></span><span class="t"><span class="str">            transformed data</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t758" href="#t758">758</a></span><span class="t"><span class="str">        """</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t759" href="#t759">759</a></span><span class="t">        <span class="key">return</span> <span class="nam">self</span><span class="op">.</span><span class="nam">fft_lib</span><span class="op">.</span><span class="nam">ifft</span><span class="op">(</span><span class="nam">u</span><span class="op">,</span> <span class="nam">axis</span><span class="op">=</span><span class="nam">axis</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t760" href="#t760">760</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t761" href="#t761">761</a></span><span class="t">    <span class="key">def</span> <span class="nam">get_BC</span><span class="op">(</span><span class="nam">self</span><span class="op">,</span> <span class="nam">kind</span><span class="op">)</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t762" href="#t762">762</a></span><span class="t">        <span class="str">"""</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t763" href="#t763">763</a></span><span class="t"><span class="str">        Get a sort of boundary condition. You can use `kind=integral`, to fix the integral, or you can use `kind=Nyquist`.</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t764" href="#t764">764</a></span><span class="t"><span class="str">        The latter is not really a boundary condition, but is used to set the Nyquist mode to some value, preferably zero.</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t765" href="#t765">765</a></span><span class="t"><span class="str">        You should set the Nyquist mode zero when the solution in physical space is real and the resolution is even.</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t766" href="#t766">766</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t767" href="#t767">767</a></span><span class="t"><span class="str">        Args:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t768" href="#t768">768</a></span><span class="t"><span class="str">            kind ('integral' or 'nyquist'): Kind of BC</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t769" href="#t769">769</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t770" href="#t770">770</a></span><span class="t"><span class="str">        Returns:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t771" href="#t771">771</a></span><span class="t"><span class="str">            self.xp.ndarray: Boundary condition row</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t772" href="#t772">772</a></span><span class="t"><span class="str">        """</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t773" href="#t773">773</a></span><span class="t">        <span class="key">if</span> <span class="nam">kind</span><span class="op">.</span><span class="nam">lower</span><span class="op">(</span><span class="op">)</span> <span class="op">==</span> <span class="str">'integral'</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t774" href="#t774">774</a></span><span class="t">            <span class="key">return</span> <span class="nam">self</span><span class="op">.</span><span class="nam">get_integ_BC_row</span><span class="op">(</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t775" href="#t775">775</a></span><span class="t">        <span class="key">elif</span> <span class="nam">kind</span><span class="op">.</span><span class="nam">lower</span><span class="op">(</span><span class="op">)</span> <span class="op">==</span> <span class="str">'nyquist'</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t776" href="#t776">776</a></span><span class="t">            <span class="key">assert</span> <span class="op">(</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t777" href="#t777">777</a></span><span class="t">                <span class="nam">self</span><span class="op">.</span><span class="nam">N</span> <span class="op">%</span> <span class="num">2</span> <span class="op">==</span> <span class="num">0</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t778" href="#t778">778</a></span><span class="t">            <span class="op">)</span><span class="op">,</span> <span class="fst">f'</span><span class="fst">Do not eliminate the Nyquist mode with odd resolution as it is fully resolved. You chose </span><span class="op">{</span><span class="nam">self</span><span class="op">.</span><span class="nam">N</span><span class="op">}</span><span class="fst"> in this axis</span><span class="fst">'</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t779" href="#t779">779</a></span><span class="t">            <span class="nam">BC</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">xp</span><span class="op">.</span><span class="nam">zeros</span><span class="op">(</span><span class="nam">self</span><span class="op">.</span><span class="nam">N</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t780" href="#t780">780</a></span><span class="t">            <span class="nam">BC</span><span class="op">[</span><span class="nam">self</span><span class="op">.</span><span class="nam">get_Nyquist_mode_index</span><span class="op">(</span><span class="op">)</span><span class="op">]</span> <span class="op">=</span> <span class="num">1</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t781" href="#t781">781</a></span><span class="t">            <span class="key">return</span> <span class="nam">BC</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t782" href="#t782">782</a></span><span class="t">        <span class="key">else</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t783" href="#t783">783</a></span><span class="t">            <span class="key">return</span> <span class="nam">super</span><span class="op">(</span><span class="op">)</span><span class="op">.</span><span class="nam">get_BC</span><span class="op">(</span><span class="nam">kind</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t784" href="#t784">784</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t785" href="#t785">785</a></span><span class="t">    <span class="key">def</span> <span class="nam">get_Nyquist_mode_index</span><span class="op">(</span><span class="nam">self</span><span class="op">)</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t786" href="#t786">786</a></span><span class="t">        <span class="str">"""</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t787" href="#t787">787</a></span><span class="t"><span class="str">        Compute the index of the Nyquist mode, i.e. the mode with the lowest wavenumber, which doesn't have a positive</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t788" href="#t788">788</a></span><span class="t"><span class="str">        counterpart for even resolution. This means real waves of this wave number cannot be properly resolved and you</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t789" href="#t789">789</a></span><span class="t"><span class="str">        are best advised to set this mode zero if representing real functions on even-resolution grids is what you're</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t790" href="#t790">790</a></span><span class="t"><span class="str">        after.</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t791" href="#t791">791</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t792" href="#t792">792</a></span><span class="t"><span class="str">        Returns:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t793" href="#t793">793</a></span><span class="t"><span class="str">            int: Index of the Nyquist mode</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t794" href="#t794">794</a></span><span class="t"><span class="str">        """</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t795" href="#t795">795</a></span><span class="t">        <span class="nam">k</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">get_wavenumbers</span><span class="op">(</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t796" href="#t796">796</a></span><span class="t">        <span class="nam">Nyquist_mode</span> <span class="op">=</span> <span class="nam">min</span><span class="op">(</span><span class="nam">k</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t797" href="#t797">797</a></span><span class="t">        <span class="key">return</span> <span class="nam">self</span><span class="op">.</span><span class="nam">xp</span><span class="op">.</span><span class="nam">where</span><span class="op">(</span><span class="nam">k</span> <span class="op">==</span> <span class="nam">Nyquist_mode</span><span class="op">)</span><span class="op">[</span><span class="num">0</span><span class="op">]</span><span class="op">[</span><span class="num">0</span><span class="op">]</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t798" href="#t798">798</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t799" href="#t799">799</a></span><span class="t">    <span class="key">def</span> <span class="nam">get_integ_BC_row</span><span class="op">(</span><span class="nam">self</span><span class="op">)</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t800" href="#t800">800</a></span><span class="t">        <span class="str">"""</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t801" href="#t801">801</a></span><span class="t"><span class="str">        Only the 0-mode has non-zero integral with FFT basis in periodic BCs</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t802" href="#t802">802</a></span><span class="t"><span class="str">        """</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t803" href="#t803">803</a></span><span class="t">        <span class="nam">me</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">xp</span><span class="op">.</span><span class="nam">zeros</span><span class="op">(</span><span class="nam">self</span><span class="op">.</span><span class="nam">N</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t804" href="#t804">804</a></span><span class="t">        <span class="nam">me</span><span class="op">[</span><span class="num">0</span><span class="op">]</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">L</span> <span class="op">/</span> <span class="nam">self</span><span class="op">.</span><span class="nam">N</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t805" href="#t805">805</a></span><span class="t">        <span class="key">return</span> <span class="nam">me</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t806" href="#t806">806</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t807" href="#t807">807</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t808" href="#t808">808</a></span><span class="t"><span class="key">class</span> <span class="nam">SpectralHelper</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t809" href="#t809">809</a></span><span class="t">    <span class="str">"""</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t810" href="#t810">810</a></span><span class="t"><span class="str">    This class has three functions:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t811" href="#t811">811</a></span><span class="t"><span class="str">      - Easily assemble matrices containing multiple equations</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t812" href="#t812">812</a></span><span class="t"><span class="str">      - Direct product of 1D bases to solve problems in more dimensions</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t813" href="#t813">813</a></span><span class="t"><span class="str">      - Distribute the FFTs to facilitate concurrency.</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t814" href="#t814">814</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t815" href="#t815">815</a></span><span class="t"><span class="str">    Attributes:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t816" href="#t816">816</a></span><span class="t"><span class="str">        comm (mpi4py.Intracomm): MPI communicator</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t817" href="#t817">817</a></span><span class="t"><span class="str">        debug (bool): Perform additional checks at extra computational cost</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t818" href="#t818">818</a></span><span class="t"><span class="str">        useGPU (bool): Whether to use GPUs</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t819" href="#t819">819</a></span><span class="t"><span class="str">        axes (list): List of 1D bases</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t820" href="#t820">820</a></span><span class="t"><span class="str">        components (list): List of strings of the names of components in the equations</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t821" href="#t821">821</a></span><span class="t"><span class="str">        full_BCs (list): List of Dictionaries containing all information about the boundary conditions</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t822" href="#t822">822</a></span><span class="t"><span class="str">        BC_mat (list): List of lists of sparse matrices to put BCs into and eventually assemble the BC matrix from</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t823" href="#t823">823</a></span><span class="t"><span class="str">        BCs (sparse matrix): Matrix containing only the BCs</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t824" href="#t824">824</a></span><span class="t"><span class="str">        fft_cache (dict): Cache FFTs of various shapes here to facilitate padding and so on</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t825" href="#t825">825</a></span><span class="t"><span class="str">        BC_rhs_mask (self.xp.ndarray): Mask values that contain boundary conditions in the right hand side</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t826" href="#t826">826</a></span><span class="t"><span class="str">        BC_zero_index (self.xp.ndarray): Indeces of rows in the matrix that are replaced by BCs</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t827" href="#t827">827</a></span><span class="t"><span class="str">        BC_line_zero_matrix (sparse matrix): Matrix that zeros rows where we can then add the BCs in using `BCs`</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t828" href="#t828">828</a></span><span class="t"><span class="str">        rhs_BCs_hat (self.xp.ndarray): Boundary conditions in spectral space</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t829" href="#t829">829</a></span><span class="t"><span class="str">        global_shape (tuple): Global shape of the solution as in `mpi4py-fft`</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t830" href="#t830">830</a></span><span class="t"><span class="str">        local_slice (slice): Local slice of the solution as in `mpi4py-fft`</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t831" href="#t831">831</a></span><span class="t"><span class="str">        fft_obj: When using distributed FFTs, this will be a parallel transform object from `mpi4py-fft`</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t832" href="#t832">832</a></span><span class="t"><span class="str">        init (tuple): This is the same `init` that is used throughout the problem classes</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t833" href="#t833">833</a></span><span class="t"><span class="str">        init_forward (tuple): This is the equivalent of `init` in spectral space</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t834" href="#t834">834</a></span><span class="t"><span class="str">    """</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t835" href="#t835">835</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t836" href="#t836">836</a></span><span class="t">    <span class="nam">xp</span> <span class="op">=</span> <span class="nam">np</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t837" href="#t837">837</a></span><span class="t">    <span class="nam">fft_lib</span> <span class="op">=</span> <span class="nam">scipy</span><span class="op">.</span><span class="nam">fft</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t838" href="#t838">838</a></span><span class="t">    <span class="nam">sparse_lib</span> <span class="op">=</span> <span class="nam">scipy</span><span class="op">.</span><span class="nam">sparse</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t839" href="#t839">839</a></span><span class="t">    <span class="nam">linalg</span> <span class="op">=</span> <span class="nam">scipy</span><span class="op">.</span><span class="nam">sparse</span><span class="op">.</span><span class="nam">linalg</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t840" href="#t840">840</a></span><span class="t">    <span class="nam">dtype</span> <span class="op">=</span> <span class="nam">mesh</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t841" href="#t841">841</a></span><span class="t">    <span class="nam">fft_backend</span> <span class="op">=</span> <span class="str">'fftw'</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t842" href="#t842">842</a></span><span class="t">    <span class="nam">fft_comm_backend</span> <span class="op">=</span> <span class="str">'MPI'</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t843" href="#t843">843</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t844" href="#t844">844</a></span><span class="t">    <span class="op">@</span><span class="nam">classmethod</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t845" href="#t845">845</a></span><span class="t">    <span class="key">def</span> <span class="nam">setup_GPU</span><span class="op">(</span><span class="nam">cls</span><span class="op">)</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t846" href="#t846">846</a></span><span class="t">        <span class="str">"""switch to GPU modules"""</span>&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t847" href="#t847">847</a></span><span class="t">        <span class="key">import</span> <span class="nam">cupy</span> <span class="key">as</span> <span class="nam">cp</span>&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t848" href="#t848">848</a></span><span class="t">        <span class="key">import</span> <span class="nam">cupyx</span><span class="op">.</span><span class="nam">scipy</span><span class="op">.</span><span class="nam">sparse</span> <span class="key">as</span> <span class="nam">sparse_lib</span>&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t849" href="#t849">849</a></span><span class="t">        <span class="key">import</span> <span class="nam">cupyx</span><span class="op">.</span><span class="nam">scipy</span><span class="op">.</span><span class="nam">sparse</span><span class="op">.</span><span class="nam">linalg</span> <span class="key">as</span> <span class="nam">linalg</span>&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t850" href="#t850">850</a></span><span class="t">        <span class="key">from</span> <span class="nam">pySDC</span><span class="op">.</span><span class="nam">implementations</span><span class="op">.</span><span class="nam">datatype_classes</span><span class="op">.</span><span class="nam">cupy_mesh</span> <span class="key">import</span> <span class="nam">cupy_mesh</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t851" href="#t851">851</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t852" href="#t852">852</a></span><span class="t">        <span class="nam">cls</span><span class="op">.</span><span class="nam">xp</span> <span class="op">=</span> <span class="nam">cp</span>&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t853" href="#t853">853</a></span><span class="t">        <span class="nam">cls</span><span class="op">.</span><span class="nam">sparse_lib</span> <span class="op">=</span> <span class="nam">sparse_lib</span>&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t854" href="#t854">854</a></span><span class="t">        <span class="nam">cls</span><span class="op">.</span><span class="nam">linalg</span> <span class="op">=</span> <span class="nam">linalg</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t855" href="#t855">855</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t856" href="#t856">856</a></span><span class="t">        <span class="nam">cls</span><span class="op">.</span><span class="nam">fft_backend</span> <span class="op">=</span> <span class="str">'cupy'</span>&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t857" href="#t857">857</a></span><span class="t">        <span class="nam">cls</span><span class="op">.</span><span class="nam">fft_comm_backend</span> <span class="op">=</span> <span class="str">'NCCL'</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t858" href="#t858">858</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t859" href="#t859">859</a></span><span class="t">        <span class="nam">cls</span><span class="op">.</span><span class="nam">dtype</span> <span class="op">=</span> <span class="nam">cupy_mesh</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t860" href="#t860">860</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t861" href="#t861">861</a></span><span class="t">    <span class="key">def</span> <span class="nam">__init__</span><span class="op">(</span><span class="nam">self</span><span class="op">,</span> <span class="nam">comm</span><span class="op">=</span><span class="key">None</span><span class="op">,</span> <span class="nam">useGPU</span><span class="op">=</span><span class="key">False</span><span class="op">,</span> <span class="nam">debug</span><span class="op">=</span><span class="key">False</span><span class="op">)</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t862" href="#t862">862</a></span><span class="t">        <span class="str">"""</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t863" href="#t863">863</a></span><span class="t"><span class="str">        Constructor</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t864" href="#t864">864</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t865" href="#t865">865</a></span><span class="t"><span class="str">        Args:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t866" href="#t866">866</a></span><span class="t"><span class="str">            comm (mpi4py.Intracomm): MPI communicator</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t867" href="#t867">867</a></span><span class="t"><span class="str">            useGPU (bool): Whether to use GPUs</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t868" href="#t868">868</a></span><span class="t"><span class="str">            debug (bool): Perform additional checks at extra computational cost</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t869" href="#t869">869</a></span><span class="t"><span class="str">        """</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t870" href="#t870">870</a></span><span class="t">        <span class="nam">self</span><span class="op">.</span><span class="nam">comm</span> <span class="op">=</span> <span class="nam">comm</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t871" href="#t871">871</a></span><span class="t">        <span class="nam">self</span><span class="op">.</span><span class="nam">debug</span> <span class="op">=</span> <span class="nam">debug</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t872" href="#t872">872</a></span><span class="t">        <span class="nam">self</span><span class="op">.</span><span class="nam">useGPU</span> <span class="op">=</span> <span class="nam">useGPU</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t873" href="#t873">873</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t874" href="#t874">874</a></span><span class="t">        <span class="key">if</span> <span class="nam">useGPU</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t875" href="#t875">875</a></span><span class="t">            <span class="nam">self</span><span class="op">.</span><span class="nam">setup_GPU</span><span class="op">(</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t876" href="#t876">876</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t877" href="#t877">877</a></span><span class="t">        <span class="nam">self</span><span class="op">.</span><span class="nam">axes</span> <span class="op">=</span> <span class="op">[</span><span class="op">]</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t878" href="#t878">878</a></span><span class="t">        <span class="nam">self</span><span class="op">.</span><span class="nam">components</span> <span class="op">=</span> <span class="op">[</span><span class="op">]</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t879" href="#t879">879</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t880" href="#t880">880</a></span><span class="t">        <span class="nam">self</span><span class="op">.</span><span class="nam">full_BCs</span> <span class="op">=</span> <span class="op">[</span><span class="op">]</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t881" href="#t881">881</a></span><span class="t">        <span class="nam">self</span><span class="op">.</span><span class="nam">BC_mat</span> <span class="op">=</span> <span class="key">None</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t882" href="#t882">882</a></span><span class="t">        <span class="nam">self</span><span class="op">.</span><span class="nam">BCs</span> <span class="op">=</span> <span class="key">None</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t883" href="#t883">883</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t884" href="#t884">884</a></span><span class="t">        <span class="nam">self</span><span class="op">.</span><span class="nam">fft_cache</span> <span class="op">=</span> <span class="op">{</span><span class="op">}</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t885" href="#t885">885</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t886" href="#t886">886</a></span><span class="t">    <span class="op">@</span><span class="nam">property</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t887" href="#t887">887</a></span><span class="t">    <span class="key">def</span> <span class="nam">u_init</span><span class="op">(</span><span class="nam">self</span><span class="op">)</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t888" href="#t888">888</a></span><span class="t">        <span class="str">"""</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t889" href="#t889">889</a></span><span class="t"><span class="str">        Get empty data container in physical space</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t890" href="#t890">890</a></span><span class="t"><span class="str">        """</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t891" href="#t891">891</a></span><span class="t">        <span class="key">return</span> <span class="nam">self</span><span class="op">.</span><span class="nam">dtype</span><span class="op">(</span><span class="nam">self</span><span class="op">.</span><span class="nam">init</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t892" href="#t892">892</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t893" href="#t893">893</a></span><span class="t">    <span class="op">@</span><span class="nam">property</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t894" href="#t894">894</a></span><span class="t">    <span class="key">def</span> <span class="nam">u_init_forward</span><span class="op">(</span><span class="nam">self</span><span class="op">)</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t895" href="#t895">895</a></span><span class="t">        <span class="str">"""</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t896" href="#t896">896</a></span><span class="t"><span class="str">        Get empty data container in spectral space</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t897" href="#t897">897</a></span><span class="t"><span class="str">        """</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t898" href="#t898">898</a></span><span class="t">        <span class="key">return</span> <span class="nam">self</span><span class="op">.</span><span class="nam">dtype</span><span class="op">(</span><span class="nam">self</span><span class="op">.</span><span class="nam">init_forward</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t899" href="#t899">899</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t900" href="#t900">900</a></span><span class="t">    <span class="op">@</span><span class="nam">property</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t901" href="#t901">901</a></span><span class="t">    <span class="key">def</span> <span class="nam">shape</span><span class="op">(</span><span class="nam">self</span><span class="op">)</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t902" href="#t902">902</a></span><span class="t">        <span class="str">"""</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t903" href="#t903">903</a></span><span class="t"><span class="str">        Get shape of individual solution component</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t904" href="#t904">904</a></span><span class="t"><span class="str">        """</span>&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t905" href="#t905">905</a></span><span class="t">        <span class="key">return</span> <span class="nam">self</span><span class="op">.</span><span class="nam">init</span><span class="op">[</span><span class="num">0</span><span class="op">]</span><span class="op">[</span><span class="num">1</span><span class="op">:</span><span class="op">]</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t906" href="#t906">906</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t907" href="#t907">907</a></span><span class="t">    <span class="op">@</span><span class="nam">property</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t908" href="#t908">908</a></span><span class="t">    <span class="key">def</span> <span class="nam">ndim</span><span class="op">(</span><span class="nam">self</span><span class="op">)</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t909" href="#t909">909</a></span><span class="t">        <span class="key">return</span> <span class="nam">len</span><span class="op">(</span><span class="nam">self</span><span class="op">.</span><span class="nam">axes</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t910" href="#t910">910</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t911" href="#t911">911</a></span><span class="t">    <span class="op">@</span><span class="nam">property</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t912" href="#t912">912</a></span><span class="t">    <span class="key">def</span> <span class="nam">ncomponents</span><span class="op">(</span><span class="nam">self</span><span class="op">)</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t913" href="#t913">913</a></span><span class="t">        <span class="key">return</span> <span class="nam">len</span><span class="op">(</span><span class="nam">self</span><span class="op">.</span><span class="nam">components</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t914" href="#t914">914</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t915" href="#t915">915</a></span><span class="t">    <span class="op">@</span><span class="nam">property</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t916" href="#t916">916</a></span><span class="t">    <span class="key">def</span> <span class="nam">V</span><span class="op">(</span><span class="nam">self</span><span class="op">)</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t917" href="#t917">917</a></span><span class="t">        <span class="str">"""</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t918" href="#t918">918</a></span><span class="t"><span class="str">        Get domain volume</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t919" href="#t919">919</a></span><span class="t"><span class="str">        """</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t920" href="#t920">920</a></span><span class="t">        <span class="key">return</span> <span class="nam">np</span><span class="op">.</span><span class="nam">prod</span><span class="op">(</span><span class="op">[</span><span class="nam">me</span><span class="op">.</span><span class="nam">L</span> <span class="key">for</span> <span class="nam">me</span> <span class="key">in</span> <span class="nam">self</span><span class="op">.</span><span class="nam">axes</span><span class="op">]</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t921" href="#t921">921</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t922" href="#t922">922</a></span><span class="t">    <span class="key">def</span> <span class="nam">add_axis</span><span class="op">(</span><span class="nam">self</span><span class="op">,</span> <span class="nam">base</span><span class="op">,</span> <span class="op">*</span><span class="nam">args</span><span class="op">,</span> <span class="op">**</span><span class="nam">kwargs</span><span class="op">)</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t923" href="#t923">923</a></span><span class="t">        <span class="str">"""</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t924" href="#t924">924</a></span><span class="t"><span class="str">        Add an axis to the domain by deciding on suitable 1D base.</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t925" href="#t925">925</a></span><span class="t"><span class="str">        Arguments to the bases are forwarded using `*args` and `**kwargs`. Please refer to the documentation of the 1D</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t926" href="#t926">926</a></span><span class="t"><span class="str">        bases for possible arguments.</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t927" href="#t927">927</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t928" href="#t928">928</a></span><span class="t"><span class="str">        Args:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t929" href="#t929">929</a></span><span class="t"><span class="str">            base (str): 1D spectral method</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t930" href="#t930">930</a></span><span class="t"><span class="str">        """</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t931" href="#t931">931</a></span><span class="t">        <span class="nam">kwargs</span><span class="op">[</span><span class="str">'useGPU'</span><span class="op">]</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">useGPU</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t932" href="#t932">932</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t933" href="#t933">933</a></span><span class="t">        <span class="key">if</span> <span class="nam">base</span><span class="op">.</span><span class="nam">lower</span><span class="op">(</span><span class="op">)</span> <span class="key">in</span> <span class="op">[</span><span class="str">'chebychov'</span><span class="op">,</span> <span class="str">'chebychev'</span><span class="op">,</span> <span class="str">'cheby'</span><span class="op">,</span> <span class="str">'chebychovhelper'</span><span class="op">]</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t934" href="#t934">934</a></span><span class="t">            <span class="nam">kwargs</span><span class="op">[</span><span class="str">'transform_type'</span><span class="op">]</span> <span class="op">=</span> <span class="nam">kwargs</span><span class="op">.</span><span class="nam">get</span><span class="op">(</span><span class="str">'transform_type'</span><span class="op">,</span> <span class="str">'fft'</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t935" href="#t935">935</a></span><span class="t">            <span class="nam">self</span><span class="op">.</span><span class="nam">axes</span><span class="op">.</span><span class="nam">append</span><span class="op">(</span><span class="nam">ChebychevHelper</span><span class="op">(</span><span class="op">*</span><span class="nam">args</span><span class="op">,</span> <span class="op">**</span><span class="nam">kwargs</span><span class="op">)</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t936" href="#t936">936</a></span><span class="t">        <span class="key">elif</span> <span class="nam">base</span><span class="op">.</span><span class="nam">lower</span><span class="op">(</span><span class="op">)</span> <span class="key">in</span> <span class="op">[</span><span class="str">'fft'</span><span class="op">,</span> <span class="str">'fourier'</span><span class="op">,</span> <span class="str">'ffthelper'</span><span class="op">]</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t937" href="#t937">937</a></span><span class="t">            <span class="nam">self</span><span class="op">.</span><span class="nam">axes</span><span class="op">.</span><span class="nam">append</span><span class="op">(</span><span class="nam">FFTHelper</span><span class="op">(</span><span class="op">*</span><span class="nam">args</span><span class="op">,</span> <span class="op">**</span><span class="nam">kwargs</span><span class="op">)</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t938" href="#t938">938</a></span><span class="t">        <span class="key">elif</span> <span class="nam">base</span><span class="op">.</span><span class="nam">lower</span><span class="op">(</span><span class="op">)</span> <span class="key">in</span> <span class="op">[</span><span class="str">'ultraspherical'</span><span class="op">,</span> <span class="str">'gegenbauer'</span><span class="op">]</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t939" href="#t939">939</a></span><span class="t">            <span class="nam">self</span><span class="op">.</span><span class="nam">axes</span><span class="op">.</span><span class="nam">append</span><span class="op">(</span><span class="nam">UltrasphericalHelper</span><span class="op">(</span><span class="op">*</span><span class="nam">args</span><span class="op">,</span> <span class="op">**</span><span class="nam">kwargs</span><span class="op">)</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t940" href="#t940">940</a></span><span class="t">        <span class="key">else</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t941" href="#t941">941</a></span><span class="t">            <span class="key">raise</span> <span class="nam">NotImplementedError</span><span class="op">(</span><span class="fst">f'</span><span class="op">{</span><span class="nam">base</span><span class="op">=</span><span class="op">!</span><span class="nam">r</span><span class="op">}</span><span class="fst"> is not implemented!</span><span class="fst">'</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t942" href="#t942">942</a></span><span class="t">        <span class="nam">self</span><span class="op">.</span><span class="nam">axes</span><span class="op">[</span><span class="op">-</span><span class="num">1</span><span class="op">]</span><span class="op">.</span><span class="nam">xp</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">xp</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t943" href="#t943">943</a></span><span class="t">        <span class="nam">self</span><span class="op">.</span><span class="nam">axes</span><span class="op">[</span><span class="op">-</span><span class="num">1</span><span class="op">]</span><span class="op">.</span><span class="nam">sparse_lib</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">sparse_lib</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t944" href="#t944">944</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t945" href="#t945">945</a></span><span class="t">    <span class="key">def</span> <span class="nam">add_component</span><span class="op">(</span><span class="nam">self</span><span class="op">,</span> <span class="nam">name</span><span class="op">)</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t946" href="#t946">946</a></span><span class="t">        <span class="str">"""</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t947" href="#t947">947</a></span><span class="t"><span class="str">        Add solution component(s).</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t948" href="#t948">948</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t949" href="#t949">949</a></span><span class="t"><span class="str">        Args:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t950" href="#t950">950</a></span><span class="t"><span class="str">            name (str or list of strings): Name(s) of component(s)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t951" href="#t951">951</a></span><span class="t"><span class="str">        """</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t952" href="#t952">952</a></span><span class="t">        <span class="key">if</span> <span class="nam">type</span><span class="op">(</span><span class="nam">name</span><span class="op">)</span> <span class="key">in</span> <span class="op">[</span><span class="nam">list</span><span class="op">,</span> <span class="nam">tuple</span><span class="op">]</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t953" href="#t953">953</a></span><span class="t">            <span class="key">for</span> <span class="nam">me</span> <span class="key">in</span> <span class="nam">name</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t954" href="#t954">954</a></span><span class="t">                <span class="nam">self</span><span class="op">.</span><span class="nam">add_component</span><span class="op">(</span><span class="nam">me</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t955" href="#t955">955</a></span><span class="t">        <span class="key">elif</span> <span class="nam">type</span><span class="op">(</span><span class="nam">name</span><span class="op">)</span> <span class="key">in</span> <span class="op">[</span><span class="nam">str</span><span class="op">]</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t956" href="#t956">956</a></span><span class="t">            <span class="key">if</span> <span class="nam">name</span> <span class="key">in</span> <span class="nam">self</span><span class="op">.</span><span class="nam">components</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t957" href="#t957">957</a></span><span class="t">                <span class="key">raise</span> <span class="nam">Exception</span><span class="op">(</span><span class="fst">f'</span><span class="op">{</span><span class="nam">name</span><span class="op">=</span><span class="op">!</span><span class="nam">r</span><span class="op">}</span><span class="fst"> is already added to this problem!</span><span class="fst">'</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t958" href="#t958">958</a></span><span class="t">            <span class="nam">self</span><span class="op">.</span><span class="nam">components</span><span class="op">.</span><span class="nam">append</span><span class="op">(</span><span class="nam">name</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t959" href="#t959">959</a></span><span class="t">        <span class="key">else</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t960" href="#t960">960</a></span><span class="t">            <span class="key">raise</span> <span class="nam">NotImplementedError</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t961" href="#t961">961</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t962" href="#t962">962</a></span><span class="t">    <span class="key">def</span> <span class="nam">index</span><span class="op">(</span><span class="nam">self</span><span class="op">,</span> <span class="nam">name</span><span class="op">)</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t963" href="#t963">963</a></span><span class="t">        <span class="str">"""</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t964" href="#t964">964</a></span><span class="t"><span class="str">        Get the index of component `name`.</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t965" href="#t965">965</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t966" href="#t966">966</a></span><span class="t"><span class="str">        Args:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t967" href="#t967">967</a></span><span class="t"><span class="str">            name (str or list of strings): Name(s) of component(s)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t968" href="#t968">968</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t969" href="#t969">969</a></span><span class="t"><span class="str">        Returns:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t970" href="#t970">970</a></span><span class="t"><span class="str">            int: Index of the component</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t971" href="#t971">971</a></span><span class="t"><span class="str">        """</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t972" href="#t972">972</a></span><span class="t">        <span class="key">if</span> <span class="nam">type</span><span class="op">(</span><span class="nam">name</span><span class="op">)</span> <span class="key">in</span> <span class="op">[</span><span class="nam">str</span><span class="op">,</span> <span class="nam">int</span><span class="op">]</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t973" href="#t973">973</a></span><span class="t">            <span class="key">return</span> <span class="nam">self</span><span class="op">.</span><span class="nam">components</span><span class="op">.</span><span class="nam">index</span><span class="op">(</span><span class="nam">name</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t974" href="#t974">974</a></span><span class="t">        <span class="key">elif</span> <span class="nam">type</span><span class="op">(</span><span class="nam">name</span><span class="op">)</span> <span class="key">in</span> <span class="op">[</span><span class="nam">list</span><span class="op">,</span> <span class="nam">tuple</span><span class="op">]</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t975" href="#t975">975</a></span><span class="t">            <span class="key">return</span> <span class="op">(</span><span class="nam">self</span><span class="op">.</span><span class="nam">index</span><span class="op">(</span><span class="nam">me</span><span class="op">)</span> <span class="key">for</span> <span class="nam">me</span> <span class="key">in</span> <span class="nam">name</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t976" href="#t976">976</a></span><span class="t">        <span class="key">else</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t977" href="#t977">977</a></span><span class="t">            <span class="key">raise</span> <span class="nam">NotImplementedError</span><span class="op">(</span><span class="fst">f'</span><span class="fst">Don\'t know how to compute index for </span><span class="op">{</span><span class="nam">type</span><span class="op">(</span><span class="nam">name</span><span class="op">)</span><span class="op">=</span><span class="op">}</span><span class="fst">'</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t978" href="#t978">978</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t979" href="#t979">979</a></span><span class="t">    <span class="key">def</span> <span class="nam">get_empty_operator_matrix</span><span class="op">(</span><span class="nam">self</span><span class="op">)</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t980" href="#t980">980</a></span><span class="t">        <span class="str">"""</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t981" href="#t981">981</a></span><span class="t"><span class="str">        Return a matrix of operators to be filled with the connections between the solution components.</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t982" href="#t982">982</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t983" href="#t983">983</a></span><span class="t"><span class="str">        Returns:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t984" href="#t984">984</a></span><span class="t"><span class="str">            list containing sparse zeros</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t985" href="#t985">985</a></span><span class="t"><span class="str">        """</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t986" href="#t986">986</a></span><span class="t">        <span class="nam">S</span> <span class="op">=</span> <span class="nam">len</span><span class="op">(</span><span class="nam">self</span><span class="op">.</span><span class="nam">components</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t987" href="#t987">987</a></span><span class="t">        <span class="nam">O</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">get_Id</span><span class="op">(</span><span class="op">)</span> <span class="op">*</span> <span class="num">0</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t988" href="#t988">988</a></span><span class="t">        <span class="key">return</span> <span class="op">[</span><span class="op">[</span><span class="nam">O</span> <span class="key">for</span> <span class="nam">_</span> <span class="key">in</span> <span class="nam">range</span><span class="op">(</span><span class="nam">S</span><span class="op">)</span><span class="op">]</span> <span class="key">for</span> <span class="nam">_</span> <span class="key">in</span> <span class="nam">range</span><span class="op">(</span><span class="nam">S</span><span class="op">)</span><span class="op">]</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t989" href="#t989">989</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t990" href="#t990">990</a></span><span class="t">    <span class="key">def</span> <span class="nam">get_BC</span><span class="op">(</span><span class="nam">self</span><span class="op">,</span> <span class="nam">axis</span><span class="op">,</span> <span class="nam">kind</span><span class="op">,</span> <span class="nam">line</span><span class="op">=</span><span class="op">-</span><span class="num">1</span><span class="op">,</span> <span class="nam">scalar</span><span class="op">=</span><span class="key">False</span><span class="op">,</span> <span class="op">**</span><span class="nam">kwargs</span><span class="op">)</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t991" href="#t991">991</a></span><span class="t">        <span class="str">"""</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t992" href="#t992">992</a></span><span class="t"><span class="str">        Use this method for boundary bordering. It gets the respective matrix row and embeds it into a matrix.</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t993" href="#t993">993</a></span><span class="t"><span class="str">        Pay attention that if you have multiple BCs in a single equation, you need to put them in different lines.</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t994" href="#t994">994</a></span><span class="t"><span class="str">        Typically, the last line that does not contain a BC is the best choice.</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t995" href="#t995">995</a></span><span class="t"><span class="str">        Forward arguments for the boundary conditions using `kwargs`. Refer to documentation of 1D bases for details.</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t996" href="#t996">996</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t997" href="#t997">997</a></span><span class="t"><span class="str">        Args:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t998" href="#t998">998</a></span><span class="t"><span class="str">            axis (int): Axis you want to add the BC to</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t999" href="#t999">999</a></span><span class="t"><span class="str">            kind (str): kind of BC, e.g. Dirichlet</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1000" href="#t1000">1000</a></span><span class="t"><span class="str">            line (int): Line you want the BC to go in</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1001" href="#t1001">1001</a></span><span class="t"><span class="str">            scalar (bool): Put the BC in all space positions in the other direction</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1002" href="#t1002">1002</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1003" href="#t1003">1003</a></span><span class="t"><span class="str">        Returns:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1004" href="#t1004">1004</a></span><span class="t"><span class="str">            sparse matrix containing the BC</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1005" href="#t1005">1005</a></span><span class="t"><span class="str">        """</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1006" href="#t1006">1006</a></span><span class="t">        <span class="nam">sp</span> <span class="op">=</span> <span class="nam">scipy</span><span class="op">.</span><span class="nam">sparse</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1007" href="#t1007">1007</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1008" href="#t1008">1008</a></span><span class="t">        <span class="nam">base</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">axes</span><span class="op">[</span><span class="nam">axis</span><span class="op">]</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1009" href="#t1009">1009</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1010" href="#t1010">1010</a></span><span class="t">        <span class="nam">BC</span> <span class="op">=</span> <span class="nam">sp</span><span class="op">.</span><span class="nam">eye</span><span class="op">(</span><span class="nam">base</span><span class="op">.</span><span class="nam">N</span><span class="op">)</span><span class="op">.</span><span class="nam">tolil</span><span class="op">(</span><span class="op">)</span> <span class="op">*</span> <span class="num">0</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1011" href="#t1011">1011</a></span><span class="t">        <span class="key">if</span> <span class="nam">self</span><span class="op">.</span><span class="nam">useGPU</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t1012" href="#t1012">1012</a></span><span class="t">            <span class="nam">BC</span><span class="op">[</span><span class="nam">line</span><span class="op">,</span> <span class="op">:</span><span class="op">]</span> <span class="op">=</span> <span class="nam">base</span><span class="op">.</span><span class="nam">get_BC</span><span class="op">(</span><span class="nam">kind</span><span class="op">=</span><span class="nam">kind</span><span class="op">,</span> <span class="op">**</span><span class="nam">kwargs</span><span class="op">)</span><span class="op">.</span><span class="nam">get</span><span class="op">(</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1013" href="#t1013">1013</a></span><span class="t">        <span class="key">else</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1014" href="#t1014">1014</a></span><span class="t">            <span class="nam">BC</span><span class="op">[</span><span class="nam">line</span><span class="op">,</span> <span class="op">:</span><span class="op">]</span> <span class="op">=</span> <span class="nam">base</span><span class="op">.</span><span class="nam">get_BC</span><span class="op">(</span><span class="nam">kind</span><span class="op">=</span><span class="nam">kind</span><span class="op">,</span> <span class="op">**</span><span class="nam">kwargs</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1015" href="#t1015">1015</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1016" href="#t1016">1016</a></span><span class="t">        <span class="nam">ndim</span> <span class="op">=</span> <span class="nam">len</span><span class="op">(</span><span class="nam">self</span><span class="op">.</span><span class="nam">axes</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1017" href="#t1017">1017</a></span><span class="t">        <span class="key">if</span> <span class="nam">ndim</span> <span class="op">==</span> <span class="num">1</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1018" href="#t1018">1018</a></span><span class="t">            <span class="key">return</span> <span class="nam">self</span><span class="op">.</span><span class="nam">sparse_lib</span><span class="op">.</span><span class="nam">csc_matrix</span><span class="op">(</span><span class="nam">BC</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1019" href="#t1019">1019</a></span><span class="t">        <span class="key">elif</span> <span class="nam">ndim</span> <span class="op">==</span> <span class="num">2</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1020" href="#t1020">1020</a></span><span class="t">            <span class="nam">axis2</span> <span class="op">=</span> <span class="op">(</span><span class="nam">axis</span> <span class="op">+</span> <span class="num">1</span><span class="op">)</span> <span class="op">%</span> <span class="nam">ndim</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1021" href="#t1021">1021</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1022" href="#t1022">1022</a></span><span class="t">            <span class="key">if</span> <span class="nam">scalar</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1023" href="#t1023">1023</a></span><span class="t">                <span class="nam">_Id</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">sparse_lib</span><span class="op">.</span><span class="nam">diags</span><span class="op">(</span><span class="nam">self</span><span class="op">.</span><span class="nam">xp</span><span class="op">.</span><span class="nam">append</span><span class="op">(</span><span class="op">[</span><span class="num">1</span><span class="op">]</span><span class="op">,</span> <span class="nam">self</span><span class="op">.</span><span class="nam">xp</span><span class="op">.</span><span class="nam">zeros</span><span class="op">(</span><span class="nam">self</span><span class="op">.</span><span class="nam">axes</span><span class="op">[</span><span class="nam">axis2</span><span class="op">]</span><span class="op">.</span><span class="nam">N</span> <span class="op">-</span> <span class="num">1</span><span class="op">)</span><span class="op">)</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1024" href="#t1024">1024</a></span><span class="t">            <span class="key">else</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1025" href="#t1025">1025</a></span><span class="t">                <span class="nam">_Id</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">axes</span><span class="op">[</span><span class="nam">axis2</span><span class="op">]</span><span class="op">.</span><span class="nam">get_Id</span><span class="op">(</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1026" href="#t1026">1026</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1027" href="#t1027">1027</a></span><span class="t">            <span class="nam">Id</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">get_local_slice_of_1D_matrix</span><span class="op">(</span><span class="nam">self</span><span class="op">.</span><span class="nam">axes</span><span class="op">[</span><span class="nam">axis2</span><span class="op">]</span><span class="op">.</span><span class="nam">get_Id</span><span class="op">(</span><span class="op">)</span> <span class="op">@</span> <span class="nam">_Id</span><span class="op">,</span> <span class="nam">axis</span><span class="op">=</span><span class="nam">axis2</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1028" href="#t1028">1028</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1029" href="#t1029">1029</a></span><span class="t">            <span class="key">if</span> <span class="nam">self</span><span class="op">.</span><span class="nam">useGPU</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t1030" href="#t1030">1030</a></span><span class="t">                <span class="nam">Id</span> <span class="op">=</span> <span class="nam">Id</span><span class="op">.</span><span class="nam">get</span><span class="op">(</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1031" href="#t1031">1031</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1032" href="#t1032">1032</a></span><span class="t">            <span class="nam">mats</span> <span class="op">=</span> <span class="op">[</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1033" href="#t1033">1033</a></span><span class="t">                <span class="key">None</span><span class="op">,</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1034" href="#t1034">1034</a></span><span class="t">            <span class="op">]</span> <span class="op">*</span> <span class="nam">ndim</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1035" href="#t1035">1035</a></span><span class="t">            <span class="nam">mats</span><span class="op">[</span><span class="nam">axis</span><span class="op">]</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">get_local_slice_of_1D_matrix</span><span class="op">(</span><span class="nam">BC</span><span class="op">,</span> <span class="nam">axis</span><span class="op">=</span><span class="nam">axis</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1036" href="#t1036">1036</a></span><span class="t">            <span class="nam">mats</span><span class="op">[</span><span class="nam">axis2</span><span class="op">]</span> <span class="op">=</span> <span class="nam">Id</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1037" href="#t1037">1037</a></span><span class="t">            <span class="key">return</span> <span class="nam">self</span><span class="op">.</span><span class="nam">sparse_lib</span><span class="op">.</span><span class="nam">csc_matrix</span><span class="op">(</span><span class="nam">sp</span><span class="op">.</span><span class="nam">kron</span><span class="op">(</span><span class="op">*</span><span class="nam">mats</span><span class="op">)</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1038" href="#t1038">1038</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1039" href="#t1039">1039</a></span><span class="t">    <span class="key">def</span> <span class="nam">remove_BC</span><span class="op">(</span><span class="nam">self</span><span class="op">,</span> <span class="nam">component</span><span class="op">,</span> <span class="nam">equation</span><span class="op">,</span> <span class="nam">axis</span><span class="op">,</span> <span class="nam">kind</span><span class="op">,</span> <span class="nam">line</span><span class="op">=</span><span class="op">-</span><span class="num">1</span><span class="op">,</span> <span class="nam">scalar</span><span class="op">=</span><span class="key">False</span><span class="op">,</span> <span class="op">**</span><span class="nam">kwargs</span><span class="op">)</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1040" href="#t1040">1040</a></span><span class="t">        <span class="str">"""</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1041" href="#t1041">1041</a></span><span class="t"><span class="str">        Remove a BC from the matrix. This is useful e.g. when you add a non-scalar BC and then need to selectively</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1042" href="#t1042">1042</a></span><span class="t"><span class="str">        remove single BCs again, as in incompressible Navier-Stokes, for instance.</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1043" href="#t1043">1043</a></span><span class="t"><span class="str">        Forward arguments for the boundary conditions using `kwargs`. Refer to documentation of 1D bases for details.</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1044" href="#t1044">1044</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1045" href="#t1045">1045</a></span><span class="t"><span class="str">        Args:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1046" href="#t1046">1046</a></span><span class="t"><span class="str">            component (str): Name of the component the BC should act on</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1047" href="#t1047">1047</a></span><span class="t"><span class="str">            equation (str): Name of the equation for the component you want to put the BC in</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1048" href="#t1048">1048</a></span><span class="t"><span class="str">            axis (int): Axis you want to add the BC to</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1049" href="#t1049">1049</a></span><span class="t"><span class="str">            kind (str): kind of BC, e.g. Dirichlet</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1050" href="#t1050">1050</a></span><span class="t"><span class="str">            v: Value of the BC</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1051" href="#t1051">1051</a></span><span class="t"><span class="str">            line (int): Line you want the BC to go in</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1052" href="#t1052">1052</a></span><span class="t"><span class="str">            scalar (bool): Put the BC in all space positions in the other direction</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1053" href="#t1053">1053</a></span><span class="t"><span class="str">        """</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1054" href="#t1054">1054</a></span><span class="t">        <span class="nam">_BC</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">get_BC</span><span class="op">(</span><span class="nam">axis</span><span class="op">=</span><span class="nam">axis</span><span class="op">,</span> <span class="nam">kind</span><span class="op">=</span><span class="nam">kind</span><span class="op">,</span> <span class="nam">line</span><span class="op">=</span><span class="nam">line</span><span class="op">,</span> <span class="nam">scalar</span><span class="op">=</span><span class="nam">scalar</span><span class="op">,</span> <span class="op">**</span><span class="nam">kwargs</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1055" href="#t1055">1055</a></span><span class="t">        <span class="nam">self</span><span class="op">.</span><span class="nam">BC_mat</span><span class="op">[</span><span class="nam">self</span><span class="op">.</span><span class="nam">index</span><span class="op">(</span><span class="nam">equation</span><span class="op">)</span><span class="op">]</span><span class="op">[</span><span class="nam">self</span><span class="op">.</span><span class="nam">index</span><span class="op">(</span><span class="nam">component</span><span class="op">)</span><span class="op">]</span> <span class="op">-=</span> <span class="nam">_BC</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1056" href="#t1056">1056</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1057" href="#t1057">1057</a></span><span class="t">        <span class="key">if</span> <span class="nam">scalar</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1058" href="#t1058">1058</a></span><span class="t">            <span class="nam">slices</span> <span class="op">=</span> <span class="op">[</span><span class="nam">self</span><span class="op">.</span><span class="nam">index</span><span class="op">(</span><span class="nam">equation</span><span class="op">)</span><span class="op">]</span> <span class="op">+</span> <span class="op">[</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1059" href="#t1059">1059</a></span><span class="t">                <span class="num">0</span><span class="op">,</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1060" href="#t1060">1060</a></span><span class="t">            <span class="op">]</span> <span class="op">*</span> <span class="nam">self</span><span class="op">.</span><span class="nam">ndim</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1061" href="#t1061">1061</a></span><span class="t">            <span class="nam">slices</span><span class="op">[</span><span class="nam">axis</span> <span class="op">+</span> <span class="num">1</span><span class="op">]</span> <span class="op">=</span> <span class="nam">line</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1062" href="#t1062">1062</a></span><span class="t">        <span class="key">else</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t1063" href="#t1063">1063</a></span><span class="t">            <span class="nam">slices</span> <span class="op">=</span> <span class="op">(</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1064" href="#t1064">1064</a></span><span class="t">                <span class="op">[</span><span class="nam">self</span><span class="op">.</span><span class="nam">index</span><span class="op">(</span><span class="nam">equation</span><span class="op">)</span><span class="op">]</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1065" href="#t1065">1065</a></span><span class="t">                <span class="op">+</span> <span class="op">[</span><span class="nam">slice</span><span class="op">(</span><span class="num">0</span><span class="op">,</span> <span class="nam">self</span><span class="op">.</span><span class="nam">init</span><span class="op">[</span><span class="num">0</span><span class="op">]</span><span class="op">[</span><span class="nam">i</span> <span class="op">+</span> <span class="num">1</span><span class="op">]</span><span class="op">)</span> <span class="key">for</span> <span class="nam">i</span> <span class="key">in</span> <span class="nam">range</span><span class="op">(</span><span class="nam">axis</span><span class="op">)</span><span class="op">]</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1066" href="#t1066">1066</a></span><span class="t">                <span class="op">+</span> <span class="op">[</span><span class="nam">line</span><span class="op">]</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1067" href="#t1067">1067</a></span><span class="t">                <span class="op">+</span> <span class="op">[</span><span class="nam">slice</span><span class="op">(</span><span class="num">0</span><span class="op">,</span> <span class="nam">self</span><span class="op">.</span><span class="nam">init</span><span class="op">[</span><span class="num">0</span><span class="op">]</span><span class="op">[</span><span class="nam">i</span> <span class="op">+</span> <span class="num">1</span><span class="op">]</span><span class="op">)</span> <span class="key">for</span> <span class="nam">i</span> <span class="key">in</span> <span class="nam">range</span><span class="op">(</span><span class="nam">axis</span> <span class="op">+</span> <span class="num">1</span><span class="op">,</span> <span class="nam">len</span><span class="op">(</span><span class="nam">self</span><span class="op">.</span><span class="nam">axes</span><span class="op">)</span><span class="op">)</span><span class="op">]</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1068" href="#t1068">1068</a></span><span class="t">            <span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1069" href="#t1069">1069</a></span><span class="t">        <span class="nam">N</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">axes</span><span class="op">[</span><span class="nam">axis</span><span class="op">]</span><span class="op">.</span><span class="nam">N</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1070" href="#t1070">1070</a></span><span class="t">        <span class="key">if</span> <span class="op">(</span><span class="nam">N</span> <span class="op">+</span> <span class="nam">line</span><span class="op">)</span> <span class="op">%</span> <span class="nam">N</span> <span class="key">in</span> <span class="nam">self</span><span class="op">.</span><span class="nam">xp</span><span class="op">.</span><span class="nam">arange</span><span class="op">(</span><span class="nam">N</span><span class="op">)</span><span class="op">[</span><span class="nam">self</span><span class="op">.</span><span class="nam">local_slice</span><span class="op">[</span><span class="nam">axis</span><span class="op">]</span><span class="op">]</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1071" href="#t1071">1071</a></span><span class="t">            <span class="nam">self</span><span class="op">.</span><span class="nam">BC_rhs_mask</span><span class="op">[</span><span class="op">(</span><span class="op">*</span><span class="nam">slices</span><span class="op">,</span><span class="op">)</span><span class="op">]</span> <span class="op">=</span> <span class="key">False</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1072" href="#t1072">1072</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1073" href="#t1073">1073</a></span><span class="t">    <span class="key">def</span> <span class="nam">add_BC</span><span class="op">(</span><span class="nam">self</span><span class="op">,</span> <span class="nam">component</span><span class="op">,</span> <span class="nam">equation</span><span class="op">,</span> <span class="nam">axis</span><span class="op">,</span> <span class="nam">kind</span><span class="op">,</span> <span class="nam">v</span><span class="op">,</span> <span class="nam">line</span><span class="op">=</span><span class="op">-</span><span class="num">1</span><span class="op">,</span> <span class="nam">scalar</span><span class="op">=</span><span class="key">False</span><span class="op">,</span> <span class="op">**</span><span class="nam">kwargs</span><span class="op">)</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1074" href="#t1074">1074</a></span><span class="t">        <span class="str">"""</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1075" href="#t1075">1075</a></span><span class="t"><span class="str">        Add a BC to the matrix. Note that you need to convert the list of lists of BCs that this method generates to a</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1076" href="#t1076">1076</a></span><span class="t"><span class="str">        single sparse matrix by calling `setup_BCs` after adding/removing all BCs.</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1077" href="#t1077">1077</a></span><span class="t"><span class="str">        Forward arguments for the boundary conditions using `kwargs`. Refer to documentation of 1D bases for details.</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1078" href="#t1078">1078</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1079" href="#t1079">1079</a></span><span class="t"><span class="str">        Args:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1080" href="#t1080">1080</a></span><span class="t"><span class="str">            component (str): Name of the component the BC should act on</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1081" href="#t1081">1081</a></span><span class="t"><span class="str">            equation (str): Name of the equation for the component you want to put the BC in</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1082" href="#t1082">1082</a></span><span class="t"><span class="str">            axis (int): Axis you want to add the BC to</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1083" href="#t1083">1083</a></span><span class="t"><span class="str">            kind (str): kind of BC, e.g. Dirichlet</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1084" href="#t1084">1084</a></span><span class="t"><span class="str">            v: Value of the BC</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1085" href="#t1085">1085</a></span><span class="t"><span class="str">            line (int): Line you want the BC to go in</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1086" href="#t1086">1086</a></span><span class="t"><span class="str">            scalar (bool): Put the BC in all space positions in the other direction</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1087" href="#t1087">1087</a></span><span class="t"><span class="str">        """</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1088" href="#t1088">1088</a></span><span class="t">        <span class="nam">_BC</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">get_BC</span><span class="op">(</span><span class="nam">axis</span><span class="op">=</span><span class="nam">axis</span><span class="op">,</span> <span class="nam">kind</span><span class="op">=</span><span class="nam">kind</span><span class="op">,</span> <span class="nam">line</span><span class="op">=</span><span class="nam">line</span><span class="op">,</span> <span class="nam">scalar</span><span class="op">=</span><span class="nam">scalar</span><span class="op">,</span> <span class="op">**</span><span class="nam">kwargs</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1089" href="#t1089">1089</a></span><span class="t">        <span class="nam">self</span><span class="op">.</span><span class="nam">BC_mat</span><span class="op">[</span><span class="nam">self</span><span class="op">.</span><span class="nam">index</span><span class="op">(</span><span class="nam">equation</span><span class="op">)</span><span class="op">]</span><span class="op">[</span><span class="nam">self</span><span class="op">.</span><span class="nam">index</span><span class="op">(</span><span class="nam">component</span><span class="op">)</span><span class="op">]</span> <span class="op">+=</span> <span class="nam">_BC</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1090" href="#t1090">1090</a></span><span class="t">        <span class="nam">self</span><span class="op">.</span><span class="nam">full_BCs</span> <span class="op">+=</span> <span class="op">[</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1091" href="#t1091">1091</a></span><span class="t">            <span class="op">{</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1092" href="#t1092">1092</a></span><span class="t">                <span class="str">'component'</span><span class="op">:</span> <span class="nam">component</span><span class="op">,</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1093" href="#t1093">1093</a></span><span class="t">                <span class="str">'equation'</span><span class="op">:</span> <span class="nam">equation</span><span class="op">,</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1094" href="#t1094">1094</a></span><span class="t">                <span class="str">'axis'</span><span class="op">:</span> <span class="nam">axis</span><span class="op">,</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1095" href="#t1095">1095</a></span><span class="t">                <span class="str">'kind'</span><span class="op">:</span> <span class="nam">kind</span><span class="op">,</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1096" href="#t1096">1096</a></span><span class="t">                <span class="str">'v'</span><span class="op">:</span> <span class="nam">v</span><span class="op">,</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1097" href="#t1097">1097</a></span><span class="t">                <span class="str">'line'</span><span class="op">:</span> <span class="nam">line</span><span class="op">,</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1098" href="#t1098">1098</a></span><span class="t">                <span class="str">'scalar'</span><span class="op">:</span> <span class="nam">scalar</span><span class="op">,</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1099" href="#t1099">1099</a></span><span class="t">                <span class="op">**</span><span class="nam">kwargs</span><span class="op">,</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1100" href="#t1100">1100</a></span><span class="t">            <span class="op">}</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1101" href="#t1101">1101</a></span><span class="t">        <span class="op">]</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1102" href="#t1102">1102</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1103" href="#t1103">1103</a></span><span class="t">        <span class="key">if</span> <span class="nam">scalar</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1104" href="#t1104">1104</a></span><span class="t">            <span class="nam">slices</span> <span class="op">=</span> <span class="op">[</span><span class="nam">self</span><span class="op">.</span><span class="nam">index</span><span class="op">(</span><span class="nam">equation</span><span class="op">)</span><span class="op">]</span> <span class="op">+</span> <span class="op">[</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1105" href="#t1105">1105</a></span><span class="t">                <span class="num">0</span><span class="op">,</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1106" href="#t1106">1106</a></span><span class="t">            <span class="op">]</span> <span class="op">*</span> <span class="nam">self</span><span class="op">.</span><span class="nam">ndim</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1107" href="#t1107">1107</a></span><span class="t">            <span class="nam">slices</span><span class="op">[</span><span class="nam">axis</span> <span class="op">+</span> <span class="num">1</span><span class="op">]</span> <span class="op">=</span> <span class="nam">line</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1108" href="#t1108">1108</a></span><span class="t">            <span class="key">if</span> <span class="nam">self</span><span class="op">.</span><span class="nam">comm</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1109" href="#t1109">1109</a></span><span class="t">                <span class="key">if</span> <span class="nam">self</span><span class="op">.</span><span class="nam">comm</span><span class="op">.</span><span class="nam">rank</span> <span class="op">==</span> <span class="num">0</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1110" href="#t1110">1110</a></span><span class="t">                    <span class="nam">self</span><span class="op">.</span><span class="nam">BC_rhs_mask</span><span class="op">[</span><span class="op">(</span><span class="op">*</span><span class="nam">slices</span><span class="op">,</span><span class="op">)</span><span class="op">]</span> <span class="op">=</span> <span class="key">True</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1111" href="#t1111">1111</a></span><span class="t">            <span class="key">else</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t1112" href="#t1112">1112</a></span><span class="t">                <span class="nam">self</span><span class="op">.</span><span class="nam">BC_rhs_mask</span><span class="op">[</span><span class="op">(</span><span class="op">*</span><span class="nam">slices</span><span class="op">,</span><span class="op">)</span><span class="op">]</span> <span class="op">=</span> <span class="key">True</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1113" href="#t1113">1113</a></span><span class="t">        <span class="key">else</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1114" href="#t1114">1114</a></span><span class="t">            <span class="nam">slices</span> <span class="op">=</span> <span class="op">(</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1115" href="#t1115">1115</a></span><span class="t">                <span class="op">[</span><span class="nam">self</span><span class="op">.</span><span class="nam">index</span><span class="op">(</span><span class="nam">equation</span><span class="op">)</span><span class="op">]</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1116" href="#t1116">1116</a></span><span class="t">                <span class="op">+</span> <span class="op">[</span><span class="nam">slice</span><span class="op">(</span><span class="num">0</span><span class="op">,</span> <span class="nam">self</span><span class="op">.</span><span class="nam">init</span><span class="op">[</span><span class="num">0</span><span class="op">]</span><span class="op">[</span><span class="nam">i</span> <span class="op">+</span> <span class="num">1</span><span class="op">]</span><span class="op">)</span> <span class="key">for</span> <span class="nam">i</span> <span class="key">in</span> <span class="nam">range</span><span class="op">(</span><span class="nam">axis</span><span class="op">)</span><span class="op">]</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1117" href="#t1117">1117</a></span><span class="t">                <span class="op">+</span> <span class="op">[</span><span class="nam">line</span><span class="op">]</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1118" href="#t1118">1118</a></span><span class="t">                <span class="op">+</span> <span class="op">[</span><span class="nam">slice</span><span class="op">(</span><span class="num">0</span><span class="op">,</span> <span class="nam">self</span><span class="op">.</span><span class="nam">init</span><span class="op">[</span><span class="num">0</span><span class="op">]</span><span class="op">[</span><span class="nam">i</span> <span class="op">+</span> <span class="num">1</span><span class="op">]</span><span class="op">)</span> <span class="key">for</span> <span class="nam">i</span> <span class="key">in</span> <span class="nam">range</span><span class="op">(</span><span class="nam">axis</span> <span class="op">+</span> <span class="num">1</span><span class="op">,</span> <span class="nam">len</span><span class="op">(</span><span class="nam">self</span><span class="op">.</span><span class="nam">axes</span><span class="op">)</span><span class="op">)</span><span class="op">]</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1119" href="#t1119">1119</a></span><span class="t">            <span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1120" href="#t1120">1120</a></span><span class="t">            <span class="nam">N</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">axes</span><span class="op">[</span><span class="nam">axis</span><span class="op">]</span><span class="op">.</span><span class="nam">N</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1121" href="#t1121">1121</a></span><span class="t">            <span class="key">if</span> <span class="op">(</span><span class="nam">N</span> <span class="op">+</span> <span class="nam">line</span><span class="op">)</span> <span class="op">%</span> <span class="nam">N</span> <span class="key">in</span> <span class="nam">self</span><span class="op">.</span><span class="nam">xp</span><span class="op">.</span><span class="nam">arange</span><span class="op">(</span><span class="nam">N</span><span class="op">)</span><span class="op">[</span><span class="nam">self</span><span class="op">.</span><span class="nam">local_slice</span><span class="op">[</span><span class="nam">axis</span><span class="op">]</span><span class="op">]</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1122" href="#t1122">1122</a></span><span class="t">                <span class="nam">slices</span><span class="op">[</span><span class="nam">axis</span> <span class="op">+</span> <span class="num">1</span><span class="op">]</span> <span class="op">-=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">local_slice</span><span class="op">[</span><span class="nam">axis</span><span class="op">]</span><span class="op">.</span><span class="nam">start</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1123" href="#t1123">1123</a></span><span class="t">                <span class="nam">self</span><span class="op">.</span><span class="nam">BC_rhs_mask</span><span class="op">[</span><span class="op">(</span><span class="op">*</span><span class="nam">slices</span><span class="op">,</span><span class="op">)</span><span class="op">]</span> <span class="op">=</span> <span class="key">True</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1124" href="#t1124">1124</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1125" href="#t1125">1125</a></span><span class="t">    <span class="key">def</span> <span class="nam">setup_BCs</span><span class="op">(</span><span class="nam">self</span><span class="op">)</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1126" href="#t1126">1126</a></span><span class="t">        <span class="str">"""</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1127" href="#t1127">1127</a></span><span class="t"><span class="str">        Convert the list of lists of BCs to the boundary condition operator.</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1128" href="#t1128">1128</a></span><span class="t"><span class="str">        Also, boundary bordering requires to zero out all other entries in the matrix in rows containing a boundary</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1129" href="#t1129">1129</a></span><span class="t"><span class="str">        condition. This method sets up a suitable sparse matrix to do this.</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1130" href="#t1130">1130</a></span><span class="t"><span class="str">        """</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1131" href="#t1131">1131</a></span><span class="t">        <span class="nam">sp</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">sparse_lib</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1132" href="#t1132">1132</a></span><span class="t">        <span class="nam">self</span><span class="op">.</span><span class="nam">BCs</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">convert_operator_matrix_to_operator</span><span class="op">(</span><span class="nam">self</span><span class="op">.</span><span class="nam">BC_mat</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1133" href="#t1133">1133</a></span><span class="t">        <span class="nam">self</span><span class="op">.</span><span class="nam">BC_zero_index</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">xp</span><span class="op">.</span><span class="nam">arange</span><span class="op">(</span><span class="nam">np</span><span class="op">.</span><span class="nam">prod</span><span class="op">(</span><span class="nam">self</span><span class="op">.</span><span class="nam">init</span><span class="op">[</span><span class="num">0</span><span class="op">]</span><span class="op">)</span><span class="op">)</span><span class="op">[</span><span class="nam">self</span><span class="op">.</span><span class="nam">BC_rhs_mask</span><span class="op">.</span><span class="nam">flatten</span><span class="op">(</span><span class="op">)</span><span class="op">]</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1134" href="#t1134">1134</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1135" href="#t1135">1135</a></span><span class="t">        <span class="nam">diags</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">xp</span><span class="op">.</span><span class="nam">ones</span><span class="op">(</span><span class="nam">self</span><span class="op">.</span><span class="nam">BCs</span><span class="op">.</span><span class="nam">shape</span><span class="op">[</span><span class="num">0</span><span class="op">]</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1136" href="#t1136">1136</a></span><span class="t">        <span class="nam">diags</span><span class="op">[</span><span class="nam">self</span><span class="op">.</span><span class="nam">BC_zero_index</span><span class="op">]</span> <span class="op">=</span> <span class="num">0</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1137" href="#t1137">1137</a></span><span class="t">        <span class="nam">self</span><span class="op">.</span><span class="nam">BC_line_zero_matrix</span> <span class="op">=</span> <span class="nam">sp</span><span class="op">.</span><span class="nam">diags</span><span class="op">(</span><span class="nam">diags</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1138" href="#t1138">1138</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1139" href="#t1139">1139</a></span><span class="t">        <span class="com"># prepare BCs in spectral space to easily add to the RHS</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1140" href="#t1140">1140</a></span><span class="t">        <span class="nam">rhs_BCs</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">put_BCs_in_rhs</span><span class="op">(</span><span class="nam">self</span><span class="op">.</span><span class="nam">u_init</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1141" href="#t1141">1141</a></span><span class="t">        <span class="nam">self</span><span class="op">.</span><span class="nam">rhs_BCs_hat</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">transform</span><span class="op">(</span><span class="nam">rhs_BCs</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1142" href="#t1142">1142</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1143" href="#t1143">1143</a></span><span class="t">    <span class="key">def</span> <span class="nam">check_BCs</span><span class="op">(</span><span class="nam">self</span><span class="op">,</span> <span class="nam">u</span><span class="op">)</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1144" href="#t1144">1144</a></span><span class="t">        <span class="str">"""</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1145" href="#t1145">1145</a></span><span class="t"><span class="str">        Check that the solution satisfies the boundary conditions</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1146" href="#t1146">1146</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1147" href="#t1147">1147</a></span><span class="t"><span class="str">        Args:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1148" href="#t1148">1148</a></span><span class="t"><span class="str">            u: The solution you want to check</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1149" href="#t1149">1149</a></span><span class="t"><span class="str">        """</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1150" href="#t1150">1150</a></span><span class="t">        <span class="key">assert</span> <span class="nam">self</span><span class="op">.</span><span class="nam">ndim</span> <span class="op">&lt;</span> <span class="num">3</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1151" href="#t1151">1151</a></span><span class="t">        <span class="key">for</span> <span class="nam">axis</span> <span class="key">in</span> <span class="nam">range</span><span class="op">(</span><span class="nam">self</span><span class="op">.</span><span class="nam">ndim</span><span class="op">)</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1152" href="#t1152">1152</a></span><span class="t">            <span class="nam">BCs</span> <span class="op">=</span> <span class="op">[</span><span class="nam">me</span> <span class="key">for</span> <span class="nam">me</span> <span class="key">in</span> <span class="nam">self</span><span class="op">.</span><span class="nam">full_BCs</span> <span class="key">if</span> <span class="nam">me</span><span class="op">[</span><span class="str">"axis"</span><span class="op">]</span> <span class="op">==</span> <span class="nam">axis</span> <span class="key">and</span> <span class="key">not</span> <span class="nam">me</span><span class="op">[</span><span class="str">"scalar"</span><span class="op">]</span><span class="op">]</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1153" href="#t1153">1153</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1154" href="#t1154">1154</a></span><span class="t">            <span class="key">if</span> <span class="nam">len</span><span class="op">(</span><span class="nam">BCs</span><span class="op">)</span> <span class="op">></span> <span class="num">0</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1155" href="#t1155">1155</a></span><span class="t">                <span class="nam">u_hat</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">transform</span><span class="op">(</span><span class="nam">u</span><span class="op">,</span> <span class="nam">axes</span><span class="op">=</span><span class="op">(</span><span class="nam">axis</span> <span class="op">-</span> <span class="nam">self</span><span class="op">.</span><span class="nam">ndim</span><span class="op">,</span><span class="op">)</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1156" href="#t1156">1156</a></span><span class="t">                <span class="key">for</span> <span class="nam">BC</span> <span class="key">in</span> <span class="nam">BCs</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1157" href="#t1157">1157</a></span><span class="t">                    <span class="nam">kwargs</span> <span class="op">=</span> <span class="op">{</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1158" href="#t1158">1158</a></span><span class="t">                        <span class="nam">key</span><span class="op">:</span> <span class="nam">value</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1159" href="#t1159">1159</a></span><span class="t">                        <span class="key">for</span> <span class="nam">key</span><span class="op">,</span> <span class="nam">value</span> <span class="key">in</span> <span class="nam">BC</span><span class="op">.</span><span class="nam">items</span><span class="op">(</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1160" href="#t1160">1160</a></span><span class="t">                        <span class="key">if</span> <span class="nam">key</span> <span class="key">not</span> <span class="key">in</span> <span class="op">[</span><span class="str">'component'</span><span class="op">,</span> <span class="str">'equation'</span><span class="op">,</span> <span class="str">'axis'</span><span class="op">,</span> <span class="str">'v'</span><span class="op">,</span> <span class="str">'line'</span><span class="op">,</span> <span class="str">'scalar'</span><span class="op">]</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1161" href="#t1161">1161</a></span><span class="t">                    <span class="op">}</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1162" href="#t1162">1162</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1163" href="#t1163">1163</a></span><span class="t">                    <span class="key">if</span> <span class="nam">axis</span> <span class="op">==</span> <span class="num">0</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1164" href="#t1164">1164</a></span><span class="t">                        <span class="nam">get</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">axes</span><span class="op">[</span><span class="nam">axis</span><span class="op">]</span><span class="op">.</span><span class="nam">get_BC</span><span class="op">(</span><span class="op">**</span><span class="nam">kwargs</span><span class="op">)</span> <span class="op">@</span> <span class="nam">u_hat</span><span class="op">[</span><span class="nam">self</span><span class="op">.</span><span class="nam">index</span><span class="op">(</span><span class="nam">BC</span><span class="op">[</span><span class="str">'component'</span><span class="op">]</span><span class="op">)</span><span class="op">]</span>&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t1165" href="#t1165">1165</a></span><span class="t">                    <span class="key">elif</span> <span class="nam">axis</span> <span class="op">==</span> <span class="num">1</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t1166" href="#t1166">1166</a></span><span class="t">                        <span class="nam">get</span> <span class="op">=</span> <span class="nam">u_hat</span><span class="op">[</span><span class="nam">self</span><span class="op">.</span><span class="nam">index</span><span class="op">(</span><span class="nam">BC</span><span class="op">[</span><span class="str">'component'</span><span class="op">]</span><span class="op">)</span><span class="op">]</span> <span class="op">@</span> <span class="nam">self</span><span class="op">.</span><span class="nam">axes</span><span class="op">[</span><span class="nam">axis</span><span class="op">]</span><span class="op">.</span><span class="nam">get_BC</span><span class="op">(</span><span class="op">**</span><span class="nam">kwargs</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1167" href="#t1167">1167</a></span><span class="t">                    <span class="nam">want</span> <span class="op">=</span> <span class="nam">BC</span><span class="op">[</span><span class="str">'v'</span><span class="op">]</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1168" href="#t1168">1168</a></span><span class="t">                    <span class="key">assert</span> <span class="nam">self</span><span class="op">.</span><span class="nam">xp</span><span class="op">.</span><span class="nam">allclose</span><span class="op">(</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1169" href="#t1169">1169</a></span><span class="t">                        <span class="nam">get</span><span class="op">,</span> <span class="nam">want</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1170" href="#t1170">1170</a></span><span class="t">                    <span class="op">)</span><span class="op">,</span> <span class="fst">f'</span><span class="fst">Unexpected BC in </span><span class="op">{</span><span class="nam">BC</span><span class="op">[</span><span class="str">"component"</span><span class="op">]</span><span class="op">}</span><span class="fst"> in equation </span><span class="op">{</span><span class="nam">BC</span><span class="op">[</span><span class="str">"equation"</span><span class="op">]</span><span class="op">}</span><span class="fst">, line </span><span class="op">{</span><span class="nam">BC</span><span class="op">[</span><span class="str">"line"</span><span class="op">]</span><span class="op">}</span><span class="fst">! Got </span><span class="op">{</span><span class="nam">get</span><span class="op">}</span><span class="fst">, wanted </span><span class="op">{</span><span class="nam">want</span><span class="op">}</span><span class="fst">'</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1171" href="#t1171">1171</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1172" href="#t1172">1172</a></span><span class="t">    <span class="key">def</span> <span class="nam">put_BCs_in_matrix</span><span class="op">(</span><span class="nam">self</span><span class="op">,</span> <span class="nam">A</span><span class="op">)</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1173" href="#t1173">1173</a></span><span class="t">        <span class="str">"""</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1174" href="#t1174">1174</a></span><span class="t"><span class="str">        Put the boundary conditions in a matrix by replacing rows with BCs.</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1175" href="#t1175">1175</a></span><span class="t"><span class="str">        """</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1176" href="#t1176">1176</a></span><span class="t">        <span class="key">return</span> <span class="nam">self</span><span class="op">.</span><span class="nam">BC_line_zero_matrix</span> <span class="op">@</span> <span class="nam">A</span> <span class="op">+</span> <span class="nam">self</span><span class="op">.</span><span class="nam">BCs</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1177" href="#t1177">1177</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1178" href="#t1178">1178</a></span><span class="t">    <span class="key">def</span> <span class="nam">put_BCs_in_rhs_hat</span><span class="op">(</span><span class="nam">self</span><span class="op">,</span> <span class="nam">rhs_hat</span><span class="op">)</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1179" href="#t1179">1179</a></span><span class="t">        <span class="str">"""</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1180" href="#t1180">1180</a></span><span class="t"><span class="str">        Put the BCs in the right hand side in spectral space for solving.</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1181" href="#t1181">1181</a></span><span class="t"><span class="str">        This function needs no transforms.</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1182" href="#t1182">1182</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1183" href="#t1183">1183</a></span><span class="t"><span class="str">        Args:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1184" href="#t1184">1184</a></span><span class="t"><span class="str">            rhs_hat: Right hand side in spectral space</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1185" href="#t1185">1185</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1186" href="#t1186">1186</a></span><span class="t"><span class="str">        Returns:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1187" href="#t1187">1187</a></span><span class="t"><span class="str">            rhs in spectral space with BCs</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1188" href="#t1188">1188</a></span><span class="t"><span class="str">        """</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1189" href="#t1189">1189</a></span><span class="t">        <span class="nam">ndim</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">ndim</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1190" href="#t1190">1190</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1191" href="#t1191">1191</a></span><span class="t">        <span class="key">for</span> <span class="nam">axis</span> <span class="key">in</span> <span class="nam">range</span><span class="op">(</span><span class="nam">ndim</span><span class="op">)</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1192" href="#t1192">1192</a></span><span class="t">            <span class="key">for</span> <span class="nam">bc</span> <span class="key">in</span> <span class="nam">self</span><span class="op">.</span><span class="nam">full_BCs</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1193" href="#t1193">1193</a></span><span class="t">                <span class="nam">slices</span> <span class="op">=</span> <span class="op">(</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1194" href="#t1194">1194</a></span><span class="t">                    <span class="op">[</span><span class="nam">slice</span><span class="op">(</span><span class="num">0</span><span class="op">,</span> <span class="nam">self</span><span class="op">.</span><span class="nam">init</span><span class="op">[</span><span class="num">0</span><span class="op">]</span><span class="op">[</span><span class="nam">i</span> <span class="op">+</span> <span class="num">1</span><span class="op">]</span><span class="op">)</span> <span class="key">for</span> <span class="nam">i</span> <span class="key">in</span> <span class="nam">range</span><span class="op">(</span><span class="nam">axis</span><span class="op">)</span><span class="op">]</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1195" href="#t1195">1195</a></span><span class="t">                    <span class="op">+</span> <span class="op">[</span><span class="nam">bc</span><span class="op">[</span><span class="str">'line'</span><span class="op">]</span><span class="op">]</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1196" href="#t1196">1196</a></span><span class="t">                    <span class="op">+</span> <span class="op">[</span><span class="nam">slice</span><span class="op">(</span><span class="num">0</span><span class="op">,</span> <span class="nam">self</span><span class="op">.</span><span class="nam">init</span><span class="op">[</span><span class="num">0</span><span class="op">]</span><span class="op">[</span><span class="nam">i</span> <span class="op">+</span> <span class="num">1</span><span class="op">]</span><span class="op">)</span> <span class="key">for</span> <span class="nam">i</span> <span class="key">in</span> <span class="nam">range</span><span class="op">(</span><span class="nam">axis</span> <span class="op">+</span> <span class="num">1</span><span class="op">,</span> <span class="nam">len</span><span class="op">(</span><span class="nam">self</span><span class="op">.</span><span class="nam">axes</span><span class="op">)</span><span class="op">)</span><span class="op">]</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1197" href="#t1197">1197</a></span><span class="t">                <span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1198" href="#t1198">1198</a></span><span class="t">                <span class="key">if</span> <span class="nam">axis</span> <span class="op">==</span> <span class="nam">bc</span><span class="op">[</span><span class="str">'axis'</span><span class="op">]</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1199" href="#t1199">1199</a></span><span class="t">                    <span class="nam">_slice</span> <span class="op">=</span> <span class="op">[</span><span class="nam">self</span><span class="op">.</span><span class="nam">index</span><span class="op">(</span><span class="nam">bc</span><span class="op">[</span><span class="str">'equation'</span><span class="op">]</span><span class="op">)</span><span class="op">]</span> <span class="op">+</span> <span class="nam">slices</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1200" href="#t1200">1200</a></span><span class="t">                    <span class="nam">N</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">axes</span><span class="op">[</span><span class="nam">axis</span><span class="op">]</span><span class="op">.</span><span class="nam">N</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1201" href="#t1201">1201</a></span><span class="t">                    <span class="key">if</span> <span class="op">(</span><span class="nam">N</span> <span class="op">+</span> <span class="nam">bc</span><span class="op">[</span><span class="str">'line'</span><span class="op">]</span><span class="op">)</span> <span class="op">%</span> <span class="nam">N</span> <span class="key">in</span> <span class="nam">self</span><span class="op">.</span><span class="nam">xp</span><span class="op">.</span><span class="nam">arange</span><span class="op">(</span><span class="nam">N</span><span class="op">)</span><span class="op">[</span><span class="nam">self</span><span class="op">.</span><span class="nam">local_slice</span><span class="op">[</span><span class="nam">axis</span><span class="op">]</span><span class="op">]</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1202" href="#t1202">1202</a></span><span class="t">                        <span class="nam">_slice</span><span class="op">[</span><span class="nam">axis</span> <span class="op">+</span> <span class="num">1</span><span class="op">]</span> <span class="op">-=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">local_slice</span><span class="op">[</span><span class="nam">axis</span><span class="op">]</span><span class="op">.</span><span class="nam">start</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1203" href="#t1203">1203</a></span><span class="t">                        <span class="nam">rhs_hat</span><span class="op">[</span><span class="op">(</span><span class="op">*</span><span class="nam">_slice</span><span class="op">,</span><span class="op">)</span><span class="op">]</span> <span class="op">=</span> <span class="num">0</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1204" href="#t1204">1204</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1205" href="#t1205">1205</a></span><span class="t">        <span class="key">return</span> <span class="nam">rhs_hat</span> <span class="op">+</span> <span class="nam">self</span><span class="op">.</span><span class="nam">rhs_BCs_hat</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1206" href="#t1206">1206</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1207" href="#t1207">1207</a></span><span class="t">    <span class="key">def</span> <span class="nam">put_BCs_in_rhs</span><span class="op">(</span><span class="nam">self</span><span class="op">,</span> <span class="nam">rhs</span><span class="op">)</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1208" href="#t1208">1208</a></span><span class="t">        <span class="str">"""</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1209" href="#t1209">1209</a></span><span class="t"><span class="str">        Put the BCs in the right hand side for solving.</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1210" href="#t1210">1210</a></span><span class="t"><span class="str">        This function will transform along each axis individually and add all BCs in that axis.</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1211" href="#t1211">1211</a></span><span class="t"><span class="str">        Consider `put_BCs_in_rhs_hat` to add BCs with no extra transforms needed.</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1212" href="#t1212">1212</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1213" href="#t1213">1213</a></span><span class="t"><span class="str">        Args:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1214" href="#t1214">1214</a></span><span class="t"><span class="str">            rhs: Right hand side in physical space</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1215" href="#t1215">1215</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1216" href="#t1216">1216</a></span><span class="t"><span class="str">        Returns:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1217" href="#t1217">1217</a></span><span class="t"><span class="str">            rhs in physical space with BCs</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1218" href="#t1218">1218</a></span><span class="t"><span class="str">        """</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1219" href="#t1219">1219</a></span><span class="t">        <span class="key">assert</span> <span class="nam">rhs</span><span class="op">.</span><span class="nam">ndim</span> <span class="op">></span> <span class="num">1</span><span class="op">,</span> <span class="str">'rhs must not be flattened here!'</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1220" href="#t1220">1220</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1221" href="#t1221">1221</a></span><span class="t">        <span class="nam">ndim</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">ndim</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1222" href="#t1222">1222</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1223" href="#t1223">1223</a></span><span class="t">        <span class="key">for</span> <span class="nam">axis</span> <span class="key">in</span> <span class="nam">range</span><span class="op">(</span><span class="nam">ndim</span><span class="op">)</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1224" href="#t1224">1224</a></span><span class="t">            <span class="nam">_rhs_hat</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">transform</span><span class="op">(</span><span class="nam">rhs</span><span class="op">,</span> <span class="nam">axes</span><span class="op">=</span><span class="op">(</span><span class="nam">axis</span> <span class="op">-</span> <span class="nam">ndim</span><span class="op">,</span><span class="op">)</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1225" href="#t1225">1225</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1226" href="#t1226">1226</a></span><span class="t">            <span class="key">for</span> <span class="nam">bc</span> <span class="key">in</span> <span class="nam">self</span><span class="op">.</span><span class="nam">full_BCs</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1227" href="#t1227">1227</a></span><span class="t">                <span class="nam">slices</span> <span class="op">=</span> <span class="op">(</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1228" href="#t1228">1228</a></span><span class="t">                    <span class="op">[</span><span class="nam">slice</span><span class="op">(</span><span class="num">0</span><span class="op">,</span> <span class="nam">self</span><span class="op">.</span><span class="nam">init</span><span class="op">[</span><span class="num">0</span><span class="op">]</span><span class="op">[</span><span class="nam">i</span> <span class="op">+</span> <span class="num">1</span><span class="op">]</span><span class="op">)</span> <span class="key">for</span> <span class="nam">i</span> <span class="key">in</span> <span class="nam">range</span><span class="op">(</span><span class="nam">axis</span><span class="op">)</span><span class="op">]</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1229" href="#t1229">1229</a></span><span class="t">                    <span class="op">+</span> <span class="op">[</span><span class="nam">bc</span><span class="op">[</span><span class="str">'line'</span><span class="op">]</span><span class="op">]</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1230" href="#t1230">1230</a></span><span class="t">                    <span class="op">+</span> <span class="op">[</span><span class="nam">slice</span><span class="op">(</span><span class="num">0</span><span class="op">,</span> <span class="nam">self</span><span class="op">.</span><span class="nam">init</span><span class="op">[</span><span class="num">0</span><span class="op">]</span><span class="op">[</span><span class="nam">i</span> <span class="op">+</span> <span class="num">1</span><span class="op">]</span><span class="op">)</span> <span class="key">for</span> <span class="nam">i</span> <span class="key">in</span> <span class="nam">range</span><span class="op">(</span><span class="nam">axis</span> <span class="op">+</span> <span class="num">1</span><span class="op">,</span> <span class="nam">len</span><span class="op">(</span><span class="nam">self</span><span class="op">.</span><span class="nam">axes</span><span class="op">)</span><span class="op">)</span><span class="op">]</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1231" href="#t1231">1231</a></span><span class="t">                <span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1232" href="#t1232">1232</a></span><span class="t">                <span class="key">if</span> <span class="nam">axis</span> <span class="op">==</span> <span class="nam">bc</span><span class="op">[</span><span class="str">'axis'</span><span class="op">]</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1233" href="#t1233">1233</a></span><span class="t">                    <span class="nam">_slice</span> <span class="op">=</span> <span class="op">[</span><span class="nam">self</span><span class="op">.</span><span class="nam">index</span><span class="op">(</span><span class="nam">bc</span><span class="op">[</span><span class="str">'equation'</span><span class="op">]</span><span class="op">)</span><span class="op">]</span> <span class="op">+</span> <span class="nam">slices</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1234" href="#t1234">1234</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1235" href="#t1235">1235</a></span><span class="t">                    <span class="nam">N</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">axes</span><span class="op">[</span><span class="nam">axis</span><span class="op">]</span><span class="op">.</span><span class="nam">N</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1236" href="#t1236">1236</a></span><span class="t">                    <span class="key">if</span> <span class="op">(</span><span class="nam">N</span> <span class="op">+</span> <span class="nam">bc</span><span class="op">[</span><span class="str">'line'</span><span class="op">]</span><span class="op">)</span> <span class="op">%</span> <span class="nam">N</span> <span class="key">in</span> <span class="nam">self</span><span class="op">.</span><span class="nam">xp</span><span class="op">.</span><span class="nam">arange</span><span class="op">(</span><span class="nam">N</span><span class="op">)</span><span class="op">[</span><span class="nam">self</span><span class="op">.</span><span class="nam">local_slice</span><span class="op">[</span><span class="nam">axis</span><span class="op">]</span><span class="op">]</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1237" href="#t1237">1237</a></span><span class="t">                        <span class="nam">_slice</span><span class="op">[</span><span class="nam">axis</span> <span class="op">+</span> <span class="num">1</span><span class="op">]</span> <span class="op">-=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">local_slice</span><span class="op">[</span><span class="nam">axis</span><span class="op">]</span><span class="op">.</span><span class="nam">start</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1238" href="#t1238">1238</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1239" href="#t1239">1239</a></span><span class="t">                        <span class="nam">_rhs_hat</span><span class="op">[</span><span class="op">(</span><span class="op">*</span><span class="nam">_slice</span><span class="op">,</span><span class="op">)</span><span class="op">]</span> <span class="op">=</span> <span class="nam">bc</span><span class="op">[</span><span class="str">'v'</span><span class="op">]</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1240" href="#t1240">1240</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1241" href="#t1241">1241</a></span><span class="t">            <span class="nam">rhs</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">itransform</span><span class="op">(</span><span class="nam">_rhs_hat</span><span class="op">,</span> <span class="nam">axes</span><span class="op">=</span><span class="op">(</span><span class="nam">axis</span> <span class="op">-</span> <span class="nam">ndim</span><span class="op">,</span><span class="op">)</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1242" href="#t1242">1242</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1243" href="#t1243">1243</a></span><span class="t">        <span class="key">return</span> <span class="nam">rhs</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1244" href="#t1244">1244</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1245" href="#t1245">1245</a></span><span class="t">    <span class="key">def</span> <span class="nam">add_equation_lhs</span><span class="op">(</span><span class="nam">self</span><span class="op">,</span> <span class="nam">A</span><span class="op">,</span> <span class="nam">equation</span><span class="op">,</span> <span class="nam">relations</span><span class="op">)</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1246" href="#t1246">1246</a></span><span class="t">        <span class="str">"""</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1247" href="#t1247">1247</a></span><span class="t"><span class="str">        Add the left hand part (that you want to solve implicitly) of an equation to a list of lists of sparse matrices</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1248" href="#t1248">1248</a></span><span class="t"><span class="str">        that you will convert to an operator later.</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1249" href="#t1249">1249</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1250" href="#t1250">1250</a></span><span class="t"><span class="str">        Example:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1251" href="#t1251">1251</a></span><span class="t"><span class="str">        Setup linear operator `L` for 1D heat equation using Chebychev method in first order form and T-to-U</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1252" href="#t1252">1252</a></span><span class="t"><span class="str">        preconditioning:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1253" href="#t1253">1253</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1254" href="#t1254">1254</a></span><span class="t"><span class="str">        .. code-block:: python</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1255" href="#t1255">1255</a></span><span class="t"><span class="str">            helper = SpectralHelper()</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1256" href="#t1256">1256</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1257" href="#t1257">1257</a></span><span class="t"><span class="str">            helper.add_axis(base='chebychev', N=8)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1258" href="#t1258">1258</a></span><span class="t"><span class="str">            helper.add_component(['u', 'ux'])</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1259" href="#t1259">1259</a></span><span class="t"><span class="str">            helper.setup_fft()</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1260" href="#t1260">1260</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1261" href="#t1261">1261</a></span><span class="t"><span class="str">            I = helper.get_Id()</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1262" href="#t1262">1262</a></span><span class="t"><span class="str">            Dx = helper.get_differentiation_matrix(axes=(0,))</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1263" href="#t1263">1263</a></span><span class="t"><span class="str">            T2U = helper.get_basis_change_matrix('T2U')</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1264" href="#t1264">1264</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1265" href="#t1265">1265</a></span><span class="t"><span class="str">            L_lhs = {</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1266" href="#t1266">1266</a></span><span class="t"><span class="str">                'ux': {'u': -T2U @ Dx, 'ux': T2U @ I},</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1267" href="#t1267">1267</a></span><span class="t"><span class="str">                'u': {'ux': -(T2U @ Dx)},</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1268" href="#t1268">1268</a></span><span class="t"><span class="str">            }</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1269" href="#t1269">1269</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1270" href="#t1270">1270</a></span><span class="t"><span class="str">            operator = helper.get_empty_operator_matrix()</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1271" href="#t1271">1271</a></span><span class="t"><span class="str">            for line, equation in L_lhs.items():</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1272" href="#t1272">1272</a></span><span class="t"><span class="str">                helper.add_equation_lhs(operator, line, equation)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1273" href="#t1273">1273</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1274" href="#t1274">1274</a></span><span class="t"><span class="str">            L = helper.convert_operator_matrix_to_operator(operator)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1275" href="#t1275">1275</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1276" href="#t1276">1276</a></span><span class="t"><span class="str">        Args:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1277" href="#t1277">1277</a></span><span class="t"><span class="str">            A (list of lists of sparse matrices): The operator to be</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1278" href="#t1278">1278</a></span><span class="t"><span class="str">            equation (str): The equation of the component you want this in</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1279" href="#t1279">1279</a></span><span class="t"><span class="str">            relations: (dict): Relations between quantities</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1280" href="#t1280">1280</a></span><span class="t"><span class="str">        """</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1281" href="#t1281">1281</a></span><span class="t">        <span class="key">for</span> <span class="nam">k</span><span class="op">,</span> <span class="nam">v</span> <span class="key">in</span> <span class="nam">relations</span><span class="op">.</span><span class="nam">items</span><span class="op">(</span><span class="op">)</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1282" href="#t1282">1282</a></span><span class="t">            <span class="nam">A</span><span class="op">[</span><span class="nam">self</span><span class="op">.</span><span class="nam">index</span><span class="op">(</span><span class="nam">equation</span><span class="op">)</span><span class="op">]</span><span class="op">[</span><span class="nam">self</span><span class="op">.</span><span class="nam">index</span><span class="op">(</span><span class="nam">k</span><span class="op">)</span><span class="op">]</span> <span class="op">=</span> <span class="nam">v</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1283" href="#t1283">1283</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1284" href="#t1284">1284</a></span><span class="t">    <span class="key">def</span> <span class="nam">convert_operator_matrix_to_operator</span><span class="op">(</span><span class="nam">self</span><span class="op">,</span> <span class="nam">M</span><span class="op">)</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1285" href="#t1285">1285</a></span><span class="t">        <span class="str">"""</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1286" href="#t1286">1286</a></span><span class="t"><span class="str">        Promote the list of lists of sparse matrices to a single sparse matrix that can be used as linear operator.</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1287" href="#t1287">1287</a></span><span class="t"><span class="str">        See documentation of `SpectralHelper.add_equation_lhs` for an example.</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1288" href="#t1288">1288</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1289" href="#t1289">1289</a></span><span class="t"><span class="str">        Args:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1290" href="#t1290">1290</a></span><span class="t"><span class="str">            M (list of lists of sparse matrices): The operator to be</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1291" href="#t1291">1291</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1292" href="#t1292">1292</a></span><span class="t"><span class="str">        Returns:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1293" href="#t1293">1293</a></span><span class="t"><span class="str">            sparse linear operator</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1294" href="#t1294">1294</a></span><span class="t"><span class="str">        """</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1295" href="#t1295">1295</a></span><span class="t">        <span class="key">if</span> <span class="nam">len</span><span class="op">(</span><span class="nam">self</span><span class="op">.</span><span class="nam">components</span><span class="op">)</span> <span class="op">==</span> <span class="num">1</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1296" href="#t1296">1296</a></span><span class="t">            <span class="key">return</span> <span class="nam">M</span><span class="op">[</span><span class="num">0</span><span class="op">]</span><span class="op">[</span><span class="num">0</span><span class="op">]</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1297" href="#t1297">1297</a></span><span class="t">        <span class="key">else</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1298" href="#t1298">1298</a></span><span class="t">            <span class="key">return</span> <span class="nam">self</span><span class="op">.</span><span class="nam">sparse_lib</span><span class="op">.</span><span class="nam">bmat</span><span class="op">(</span><span class="nam">M</span><span class="op">,</span> <span class="nam">format</span><span class="op">=</span><span class="str">'csc'</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1299" href="#t1299">1299</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1300" href="#t1300">1300</a></span><span class="t">    <span class="key">def</span> <span class="nam">get_wavenumbers</span><span class="op">(</span><span class="nam">self</span><span class="op">)</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1301" href="#t1301">1301</a></span><span class="t">        <span class="str">"""</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1302" href="#t1302">1302</a></span><span class="t"><span class="str">        Get grid in spectral space</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1303" href="#t1303">1303</a></span><span class="t"><span class="str">        """</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1304" href="#t1304">1304</a></span><span class="t">        <span class="nam">grids</span> <span class="op">=</span> <span class="op">[</span><span class="nam">self</span><span class="op">.</span><span class="nam">axes</span><span class="op">[</span><span class="nam">i</span><span class="op">]</span><span class="op">.</span><span class="nam">get_wavenumbers</span><span class="op">(</span><span class="op">)</span><span class="op">[</span><span class="nam">self</span><span class="op">.</span><span class="nam">local_slice</span><span class="op">[</span><span class="nam">i</span><span class="op">]</span><span class="op">]</span> <span class="key">for</span> <span class="nam">i</span> <span class="key">in</span> <span class="nam">range</span><span class="op">(</span><span class="nam">len</span><span class="op">(</span><span class="nam">self</span><span class="op">.</span><span class="nam">axes</span><span class="op">)</span><span class="op">)</span><span class="op">]</span><span class="op">[</span><span class="op">:</span><span class="op">:</span><span class="op">-</span><span class="num">1</span><span class="op">]</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1305" href="#t1305">1305</a></span><span class="t">        <span class="key">return</span> <span class="nam">self</span><span class="op">.</span><span class="nam">xp</span><span class="op">.</span><span class="nam">meshgrid</span><span class="op">(</span><span class="op">*</span><span class="nam">grids</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1306" href="#t1306">1306</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1307" href="#t1307">1307</a></span><span class="t">    <span class="key">def</span> <span class="nam">get_grid</span><span class="op">(</span><span class="nam">self</span><span class="op">)</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1308" href="#t1308">1308</a></span><span class="t">        <span class="str">"""</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1309" href="#t1309">1309</a></span><span class="t"><span class="str">        Get grid in physical space</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1310" href="#t1310">1310</a></span><span class="t"><span class="str">        """</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1311" href="#t1311">1311</a></span><span class="t">        <span class="nam">grids</span> <span class="op">=</span> <span class="op">[</span><span class="nam">self</span><span class="op">.</span><span class="nam">axes</span><span class="op">[</span><span class="nam">i</span><span class="op">]</span><span class="op">.</span><span class="nam">get_1dgrid</span><span class="op">(</span><span class="op">)</span><span class="op">[</span><span class="nam">self</span><span class="op">.</span><span class="nam">local_slice</span><span class="op">[</span><span class="nam">i</span><span class="op">]</span><span class="op">]</span> <span class="key">for</span> <span class="nam">i</span> <span class="key">in</span> <span class="nam">range</span><span class="op">(</span><span class="nam">len</span><span class="op">(</span><span class="nam">self</span><span class="op">.</span><span class="nam">axes</span><span class="op">)</span><span class="op">)</span><span class="op">]</span><span class="op">[</span><span class="op">:</span><span class="op">:</span><span class="op">-</span><span class="num">1</span><span class="op">]</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1312" href="#t1312">1312</a></span><span class="t">        <span class="key">return</span> <span class="nam">self</span><span class="op">.</span><span class="nam">xp</span><span class="op">.</span><span class="nam">meshgrid</span><span class="op">(</span><span class="op">*</span><span class="nam">grids</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1313" href="#t1313">1313</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1314" href="#t1314">1314</a></span><span class="t">    <span class="key">def</span> <span class="nam">get_fft</span><span class="op">(</span><span class="nam">self</span><span class="op">,</span> <span class="nam">axes</span><span class="op">=</span><span class="key">None</span><span class="op">,</span> <span class="nam">direction</span><span class="op">=</span><span class="str">'object'</span><span class="op">,</span> <span class="nam">padding</span><span class="op">=</span><span class="key">None</span><span class="op">,</span> <span class="nam">shape</span><span class="op">=</span><span class="key">None</span><span class="op">)</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1315" href="#t1315">1315</a></span><span class="t">        <span class="str">"""</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1316" href="#t1316">1316</a></span><span class="t"><span class="str">        When using MPI, we use `PFFT` objects generated by mpi4py-fft</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1317" href="#t1317">1317</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1318" href="#t1318">1318</a></span><span class="t"><span class="str">        Args:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1319" href="#t1319">1319</a></span><span class="t"><span class="str">            axes (tuple): Axes you want to transform over</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1320" href="#t1320">1320</a></span><span class="t"><span class="str">            direction (str): use "forward" or "backward" to get functions for performing the transforms or "object" to get the PFFT object</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1321" href="#t1321">1321</a></span><span class="t"><span class="str">            padding (tuple): Padding for dealiasing</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1322" href="#t1322">1322</a></span><span class="t"><span class="str">            shape (tuple): Shape of the transform</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1323" href="#t1323">1323</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1324" href="#t1324">1324</a></span><span class="t"><span class="str">        Returns:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1325" href="#t1325">1325</a></span><span class="t"><span class="str">            transform</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1326" href="#t1326">1326</a></span><span class="t"><span class="str">        """</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1327" href="#t1327">1327</a></span><span class="t">        <span class="nam">axes</span> <span class="op">=</span> <span class="nam">tuple</span><span class="op">(</span><span class="op">-</span><span class="nam">i</span> <span class="op">-</span> <span class="num">1</span> <span class="key">for</span> <span class="nam">i</span> <span class="key">in</span> <span class="nam">range</span><span class="op">(</span><span class="nam">self</span><span class="op">.</span><span class="nam">ndim</span><span class="op">)</span><span class="op">)</span> <span class="key">if</span> <span class="nam">axes</span> <span class="key">is</span> <span class="key">None</span> <span class="key">else</span> <span class="nam">axes</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1328" href="#t1328">1328</a></span><span class="t">        <span class="nam">shape</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">global_shape</span><span class="op">[</span><span class="num">1</span><span class="op">:</span><span class="op">]</span> <span class="key">if</span> <span class="nam">shape</span> <span class="key">is</span> <span class="key">None</span> <span class="key">else</span> <span class="nam">shape</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1329" href="#t1329">1329</a></span><span class="t">        <span class="nam">padding</span> <span class="op">=</span> <span class="op">(</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1330" href="#t1330">1330</a></span><span class="t">            <span class="op">[</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1331" href="#t1331">1331</a></span><span class="t">                <span class="num">1</span><span class="op">,</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1332" href="#t1332">1332</a></span><span class="t">            <span class="op">]</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1333" href="#t1333">1333</a></span><span class="t">            <span class="op">*</span> <span class="nam">self</span><span class="op">.</span><span class="nam">ndim</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1334" href="#t1334">1334</a></span><span class="t">            <span class="key">if</span> <span class="nam">padding</span> <span class="key">is</span> <span class="key">None</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1335" href="#t1335">1335</a></span><span class="t">            <span class="key">else</span> <span class="nam">padding</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1336" href="#t1336">1336</a></span><span class="t">        <span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1337" href="#t1337">1337</a></span><span class="t">        <span class="nam">key</span> <span class="op">=</span> <span class="op">(</span><span class="nam">axes</span><span class="op">,</span> <span class="nam">direction</span><span class="op">,</span> <span class="nam">tuple</span><span class="op">(</span><span class="nam">padding</span><span class="op">)</span><span class="op">,</span> <span class="nam">tuple</span><span class="op">(</span><span class="nam">shape</span><span class="op">)</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1338" href="#t1338">1338</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1339" href="#t1339">1339</a></span><span class="t">        <span class="key">if</span> <span class="nam">key</span> <span class="key">not</span> <span class="key">in</span> <span class="nam">self</span><span class="op">.</span><span class="nam">fft_cache</span><span class="op">.</span><span class="nam">keys</span><span class="op">(</span><span class="op">)</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1340" href="#t1340">1340</a></span><span class="t">            <span class="key">if</span> <span class="nam">self</span><span class="op">.</span><span class="nam">comm</span> <span class="key">is</span> <span class="key">None</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1341" href="#t1341">1341</a></span><span class="t">                <span class="key">assert</span> <span class="nam">np</span><span class="op">.</span><span class="nam">allclose</span><span class="op">(</span><span class="nam">padding</span><span class="op">,</span> <span class="num">1</span><span class="op">)</span><span class="op">,</span> <span class="str">'Zero padding is not implemented for non-MPI transforms'</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1342" href="#t1342">1342</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1343" href="#t1343">1343</a></span><span class="t">                <span class="key">if</span> <span class="nam">direction</span> <span class="op">==</span> <span class="str">'forward'</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1344" href="#t1344">1344</a></span><span class="t">                    <span class="nam">self</span><span class="op">.</span><span class="nam">fft_cache</span><span class="op">[</span><span class="nam">key</span><span class="op">]</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">xp</span><span class="op">.</span><span class="nam">fft</span><span class="op">.</span><span class="nam">fftn</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1345" href="#t1345">1345</a></span><span class="t">                <span class="key">elif</span> <span class="nam">direction</span> <span class="op">==</span> <span class="str">'backward'</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1346" href="#t1346">1346</a></span><span class="t">                    <span class="nam">self</span><span class="op">.</span><span class="nam">fft_cache</span><span class="op">[</span><span class="nam">key</span><span class="op">]</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">xp</span><span class="op">.</span><span class="nam">fft</span><span class="op">.</span><span class="nam">ifftn</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1347" href="#t1347">1347</a></span><span class="t">                <span class="key">elif</span> <span class="nam">direction</span> <span class="op">==</span> <span class="str">'object'</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1348" href="#t1348">1348</a></span><span class="t">                    <span class="nam">self</span><span class="op">.</span><span class="nam">fft_cache</span><span class="op">[</span><span class="nam">key</span><span class="op">]</span> <span class="op">=</span> <span class="key">None</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1349" href="#t1349">1349</a></span><span class="t">            <span class="key">else</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1350" href="#t1350">1350</a></span><span class="t">                <span class="key">from</span> <span class="nam">mpi4py_fft</span> <span class="key">import</span> <span class="nam">PFFT</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1351" href="#t1351">1351</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1352" href="#t1352">1352</a></span><span class="t">                <span class="nam">_fft</span> <span class="op">=</span> <span class="nam">PFFT</span><span class="op">(</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1353" href="#t1353">1353</a></span><span class="t">                    <span class="nam">comm</span><span class="op">=</span><span class="nam">self</span><span class="op">.</span><span class="nam">comm</span><span class="op">,</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1354" href="#t1354">1354</a></span><span class="t">                    <span class="nam">shape</span><span class="op">=</span><span class="nam">shape</span><span class="op">,</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1355" href="#t1355">1355</a></span><span class="t">                    <span class="nam">axes</span><span class="op">=</span><span class="nam">sorted</span><span class="op">(</span><span class="nam">axes</span><span class="op">)</span><span class="op">,</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1356" href="#t1356">1356</a></span><span class="t">                    <span class="nam">dtype</span><span class="op">=</span><span class="str">'D'</span><span class="op">,</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1357" href="#t1357">1357</a></span><span class="t">                    <span class="nam">collapse</span><span class="op">=</span><span class="key">False</span><span class="op">,</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1358" href="#t1358">1358</a></span><span class="t">                    <span class="nam">backend</span><span class="op">=</span><span class="nam">self</span><span class="op">.</span><span class="nam">fft_backend</span><span class="op">,</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1359" href="#t1359">1359</a></span><span class="t">                    <span class="nam">comm_backend</span><span class="op">=</span><span class="nam">self</span><span class="op">.</span><span class="nam">fft_comm_backend</span><span class="op">,</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1360" href="#t1360">1360</a></span><span class="t">                    <span class="nam">padding</span><span class="op">=</span><span class="nam">padding</span><span class="op">,</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1361" href="#t1361">1361</a></span><span class="t">                <span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1362" href="#t1362">1362</a></span><span class="t">                <span class="key">if</span> <span class="nam">direction</span> <span class="op">==</span> <span class="str">'forward'</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1363" href="#t1363">1363</a></span><span class="t">                    <span class="nam">self</span><span class="op">.</span><span class="nam">fft_cache</span><span class="op">[</span><span class="nam">key</span><span class="op">]</span> <span class="op">=</span> <span class="nam">_fft</span><span class="op">.</span><span class="nam">forward</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1364" href="#t1364">1364</a></span><span class="t">                <span class="key">elif</span> <span class="nam">direction</span> <span class="op">==</span> <span class="str">'backward'</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1365" href="#t1365">1365</a></span><span class="t">                    <span class="nam">self</span><span class="op">.</span><span class="nam">fft_cache</span><span class="op">[</span><span class="nam">key</span><span class="op">]</span> <span class="op">=</span> <span class="nam">_fft</span><span class="op">.</span><span class="nam">backward</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1366" href="#t1366">1366</a></span><span class="t">                <span class="key">elif</span> <span class="nam">direction</span> <span class="op">==</span> <span class="str">'object'</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1367" href="#t1367">1367</a></span><span class="t">                    <span class="nam">self</span><span class="op">.</span><span class="nam">fft_cache</span><span class="op">[</span><span class="nam">key</span><span class="op">]</span> <span class="op">=</span> <span class="nam">_fft</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1368" href="#t1368">1368</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1369" href="#t1369">1369</a></span><span class="t">        <span class="key">return</span> <span class="nam">self</span><span class="op">.</span><span class="nam">fft_cache</span><span class="op">[</span><span class="nam">key</span><span class="op">]</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1370" href="#t1370">1370</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1371" href="#t1371">1371</a></span><span class="t">    <span class="key">def</span> <span class="nam">setup_fft</span><span class="op">(</span><span class="nam">self</span><span class="op">,</span> <span class="nam">real_spectral_coefficients</span><span class="op">=</span><span class="key">False</span><span class="op">)</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1372" href="#t1372">1372</a></span><span class="t">        <span class="str">"""</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1373" href="#t1373">1373</a></span><span class="t"><span class="str">        This function must be called after all axes have been setup in order to prepare the local shapes of the data.</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1374" href="#t1374">1374</a></span><span class="t"><span class="str">        This must also be called before setting up any BCs.</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1375" href="#t1375">1375</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1376" href="#t1376">1376</a></span><span class="t"><span class="str">        Args:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1377" href="#t1377">1377</a></span><span class="t"><span class="str">            real_spectral_coefficients (bool): Allow only real coefficients in spectral space</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1378" href="#t1378">1378</a></span><span class="t"><span class="str">        """</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1379" href="#t1379">1379</a></span><span class="t">        <span class="key">if</span> <span class="nam">len</span><span class="op">(</span><span class="nam">self</span><span class="op">.</span><span class="nam">components</span><span class="op">)</span> <span class="op">==</span> <span class="num">0</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1380" href="#t1380">1380</a></span><span class="t">            <span class="nam">self</span><span class="op">.</span><span class="nam">add_component</span><span class="op">(</span><span class="str">'u'</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1381" href="#t1381">1381</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1382" href="#t1382">1382</a></span><span class="t">        <span class="nam">self</span><span class="op">.</span><span class="nam">global_shape</span> <span class="op">=</span> <span class="op">(</span><span class="nam">len</span><span class="op">(</span><span class="nam">self</span><span class="op">.</span><span class="nam">components</span><span class="op">)</span><span class="op">,</span><span class="op">)</span> <span class="op">+</span> <span class="nam">tuple</span><span class="op">(</span><span class="nam">me</span><span class="op">.</span><span class="nam">N</span> <span class="key">for</span> <span class="nam">me</span> <span class="key">in</span> <span class="nam">self</span><span class="op">.</span><span class="nam">axes</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1383" href="#t1383">1383</a></span><span class="t">        <span class="nam">self</span><span class="op">.</span><span class="nam">local_slice</span> <span class="op">=</span> <span class="op">[</span><span class="nam">slice</span><span class="op">(</span><span class="num">0</span><span class="op">,</span> <span class="nam">me</span><span class="op">.</span><span class="nam">N</span><span class="op">)</span> <span class="key">for</span> <span class="nam">me</span> <span class="key">in</span> <span class="nam">self</span><span class="op">.</span><span class="nam">axes</span><span class="op">]</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1384" href="#t1384">1384</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1385" href="#t1385">1385</a></span><span class="t">        <span class="nam">axes</span> <span class="op">=</span> <span class="nam">tuple</span><span class="op">(</span><span class="nam">i</span> <span class="key">for</span> <span class="nam">i</span> <span class="key">in</span> <span class="nam">range</span><span class="op">(</span><span class="nam">len</span><span class="op">(</span><span class="nam">self</span><span class="op">.</span><span class="nam">axes</span><span class="op">)</span><span class="op">)</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1386" href="#t1386">1386</a></span><span class="t">        <span class="nam">self</span><span class="op">.</span><span class="nam">fft_obj</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">get_fft</span><span class="op">(</span><span class="nam">axes</span><span class="op">=</span><span class="nam">axes</span><span class="op">,</span> <span class="nam">direction</span><span class="op">=</span><span class="str">'object'</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1387" href="#t1387">1387</a></span><span class="t">        <span class="key">if</span> <span class="nam">self</span><span class="op">.</span><span class="nam">fft_obj</span> <span class="key">is</span> <span class="key">not</span> <span class="key">None</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1388" href="#t1388">1388</a></span><span class="t">            <span class="nam">self</span><span class="op">.</span><span class="nam">local_slice</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">fft_obj</span><span class="op">.</span><span class="nam">local_slice</span><span class="op">(</span><span class="key">False</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1389" href="#t1389">1389</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1390" href="#t1390">1390</a></span><span class="t">        <span class="nam">self</span><span class="op">.</span><span class="nam">init</span> <span class="op">=</span> <span class="op">(</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1391" href="#t1391">1391</a></span><span class="t">            <span class="nam">np</span><span class="op">.</span><span class="nam">empty</span><span class="op">(</span><span class="nam">shape</span><span class="op">=</span><span class="nam">self</span><span class="op">.</span><span class="nam">global_shape</span><span class="op">)</span><span class="op">[</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1392" href="#t1392">1392</a></span><span class="t">                <span class="op">(</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1393" href="#t1393">1393</a></span><span class="t">                    <span class="op">...</span><span class="op">,</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1394" href="#t1394">1394</a></span><span class="t">                    <span class="op">*</span><span class="nam">self</span><span class="op">.</span><span class="nam">local_slice</span><span class="op">,</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1395" href="#t1395">1395</a></span><span class="t">                <span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1396" href="#t1396">1396</a></span><span class="t">            <span class="op">]</span><span class="op">.</span><span class="nam">shape</span><span class="op">,</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1397" href="#t1397">1397</a></span><span class="t">            <span class="nam">self</span><span class="op">.</span><span class="nam">comm</span><span class="op">,</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1398" href="#t1398">1398</a></span><span class="t">            <span class="nam">np</span><span class="op">.</span><span class="nam">dtype</span><span class="op">(</span><span class="str">'float'</span><span class="op">)</span><span class="op">,</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1399" href="#t1399">1399</a></span><span class="t">        <span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1400" href="#t1400">1400</a></span><span class="t">        <span class="nam">self</span><span class="op">.</span><span class="nam">init_forward</span> <span class="op">=</span> <span class="op">(</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1401" href="#t1401">1401</a></span><span class="t">            <span class="nam">np</span><span class="op">.</span><span class="nam">empty</span><span class="op">(</span><span class="nam">shape</span><span class="op">=</span><span class="nam">self</span><span class="op">.</span><span class="nam">global_shape</span><span class="op">)</span><span class="op">[</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1402" href="#t1402">1402</a></span><span class="t">                <span class="op">(</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1403" href="#t1403">1403</a></span><span class="t">                    <span class="op">...</span><span class="op">,</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1404" href="#t1404">1404</a></span><span class="t">                    <span class="op">*</span><span class="nam">self</span><span class="op">.</span><span class="nam">local_slice</span><span class="op">,</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1405" href="#t1405">1405</a></span><span class="t">                <span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1406" href="#t1406">1406</a></span><span class="t">            <span class="op">]</span><span class="op">.</span><span class="nam">shape</span><span class="op">,</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1407" href="#t1407">1407</a></span><span class="t">            <span class="nam">self</span><span class="op">.</span><span class="nam">comm</span><span class="op">,</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1408" href="#t1408">1408</a></span><span class="t">            <span class="nam">np</span><span class="op">.</span><span class="nam">dtype</span><span class="op">(</span><span class="str">'float'</span><span class="op">)</span> <span class="key">if</span> <span class="nam">real_spectral_coefficients</span> <span class="key">else</span> <span class="nam">np</span><span class="op">.</span><span class="nam">dtype</span><span class="op">(</span><span class="str">'complex128'</span><span class="op">)</span><span class="op">,</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1409" href="#t1409">1409</a></span><span class="t">        <span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1410" href="#t1410">1410</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1411" href="#t1411">1411</a></span><span class="t">        <span class="nam">self</span><span class="op">.</span><span class="nam">BC_mat</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">get_empty_operator_matrix</span><span class="op">(</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1412" href="#t1412">1412</a></span><span class="t">        <span class="nam">self</span><span class="op">.</span><span class="nam">BC_rhs_mask</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">xp</span><span class="op">.</span><span class="nam">zeros</span><span class="op">(</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1413" href="#t1413">1413</a></span><span class="t">            <span class="nam">shape</span><span class="op">=</span><span class="nam">self</span><span class="op">.</span><span class="nam">init</span><span class="op">[</span><span class="num">0</span><span class="op">]</span><span class="op">,</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1414" href="#t1414">1414</a></span><span class="t">            <span class="nam">dtype</span><span class="op">=</span><span class="nam">bool</span><span class="op">,</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1415" href="#t1415">1415</a></span><span class="t">        <span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1416" href="#t1416">1416</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1417" href="#t1417">1417</a></span><span class="t">    <span class="key">def</span> <span class="nam">_transform_fft</span><span class="op">(</span><span class="nam">self</span><span class="op">,</span> <span class="nam">u</span><span class="op">,</span> <span class="nam">axes</span><span class="op">,</span> <span class="op">**</span><span class="nam">kwargs</span><span class="op">)</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1418" href="#t1418">1418</a></span><span class="t">        <span class="str">"""</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1419" href="#t1419">1419</a></span><span class="t"><span class="str">        FFT along `axes`</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1420" href="#t1420">1420</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1421" href="#t1421">1421</a></span><span class="t"><span class="str">        Args:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1422" href="#t1422">1422</a></span><span class="t"><span class="str">            u: The solution</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1423" href="#t1423">1423</a></span><span class="t"><span class="str">            axes (tuple): Axes you want to transform over</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1424" href="#t1424">1424</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1425" href="#t1425">1425</a></span><span class="t"><span class="str">        Returns:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1426" href="#t1426">1426</a></span><span class="t"><span class="str">            transformed solution</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1427" href="#t1427">1427</a></span><span class="t"><span class="str">        """</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1428" href="#t1428">1428</a></span><span class="t">        <span class="com"># TODO: clean up and try putting more of this in the 1D bases</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1429" href="#t1429">1429</a></span><span class="t">        <span class="nam">fft</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">get_fft</span><span class="op">(</span><span class="nam">axes</span><span class="op">,</span> <span class="str">'forward'</span><span class="op">,</span> <span class="op">**</span><span class="nam">kwargs</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1430" href="#t1430">1430</a></span><span class="t">        <span class="key">return</span> <span class="nam">fft</span><span class="op">(</span><span class="nam">u</span><span class="op">,</span> <span class="nam">axes</span><span class="op">=</span><span class="nam">axes</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1431" href="#t1431">1431</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1432" href="#t1432">1432</a></span><span class="t">    <span class="key">def</span> <span class="nam">_transform_dct</span><span class="op">(</span><span class="nam">self</span><span class="op">,</span> <span class="nam">u</span><span class="op">,</span> <span class="nam">axes</span><span class="op">,</span> <span class="nam">padding</span><span class="op">=</span><span class="key">None</span><span class="op">,</span> <span class="op">**</span><span class="nam">kwargs</span><span class="op">)</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1433" href="#t1433">1433</a></span><span class="t">        <span class="str">'''</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1434" href="#t1434">1434</a></span><span class="t"><span class="str">        DCT along `axes`.</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1435" href="#t1435">1435</a></span><span class="t"><span class="str">        This will only return real values!</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1436" href="#t1436">1436</a></span><span class="t"><span class="str">        When padding the solution, we cannot just use the mpi4py-fft implementation, because of the unusual ordering of</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1437" href="#t1437">1437</a></span><span class="t"><span class="str">        wavenumbers in FFTs.</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1438" href="#t1438">1438</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1439" href="#t1439">1439</a></span><span class="t"><span class="str">        Args:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1440" href="#t1440">1440</a></span><span class="t"><span class="str">            u: The solution</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1441" href="#t1441">1441</a></span><span class="t"><span class="str">            axes (tuple): Axes you want to transform over</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1442" href="#t1442">1442</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1443" href="#t1443">1443</a></span><span class="t"><span class="str">        Returns:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1444" href="#t1444">1444</a></span><span class="t"><span class="str">            transformed solution</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1445" href="#t1445">1445</a></span><span class="t"><span class="str">        '''</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1446" href="#t1446">1446</a></span><span class="t">        <span class="com"># TODO: clean up and try putting more of this in the 1D bases</span>&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t1447" href="#t1447">1447</a></span><span class="t">        <span class="key">if</span> <span class="nam">self</span><span class="op">.</span><span class="nam">debug</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t1448" href="#t1448">1448</a></span><span class="t">            <span class="key">assert</span> <span class="nam">self</span><span class="op">.</span><span class="nam">xp</span><span class="op">.</span><span class="nam">allclose</span><span class="op">(</span><span class="nam">u</span><span class="op">.</span><span class="nam">imag</span><span class="op">,</span> <span class="num">0</span><span class="op">)</span><span class="op">,</span> <span class="str">'This function can only handle real input.'</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1449" href="#t1449">1449</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1450" href="#t1450">1450</a></span><span class="t">        <span class="key">if</span> <span class="nam">len</span><span class="op">(</span><span class="nam">axes</span><span class="op">)</span> <span class="op">></span> <span class="num">1</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1451" href="#t1451">1451</a></span><span class="t">            <span class="nam">v</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">_transform_dct</span><span class="op">(</span><span class="nam">self</span><span class="op">.</span><span class="nam">_transform_dct</span><span class="op">(</span><span class="nam">u</span><span class="op">,</span> <span class="nam">axes</span><span class="op">[</span><span class="num">1</span><span class="op">:</span><span class="op">]</span><span class="op">,</span> <span class="op">**</span><span class="nam">kwargs</span><span class="op">)</span><span class="op">,</span> <span class="op">(</span><span class="nam">axes</span><span class="op">[</span><span class="num">0</span><span class="op">]</span><span class="op">,</span><span class="op">)</span><span class="op">,</span> <span class="op">**</span><span class="nam">kwargs</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1452" href="#t1452">1452</a></span><span class="t">        <span class="key">else</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1453" href="#t1453">1453</a></span><span class="t">            <span class="nam">v</span> <span class="op">=</span> <span class="nam">u</span><span class="op">.</span><span class="nam">copy</span><span class="op">(</span><span class="op">)</span><span class="op">.</span><span class="nam">astype</span><span class="op">(</span><span class="nam">complex</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1454" href="#t1454">1454</a></span><span class="t">            <span class="nam">axis</span> <span class="op">=</span> <span class="nam">axes</span><span class="op">[</span><span class="num">0</span><span class="op">]</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1455" href="#t1455">1455</a></span><span class="t">            <span class="nam">base</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">axes</span><span class="op">[</span><span class="nam">axis</span><span class="op">]</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1456" href="#t1456">1456</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1457" href="#t1457">1457</a></span><span class="t">            <span class="nam">shuffle</span> <span class="op">=</span> <span class="op">[</span><span class="nam">slice</span><span class="op">(</span><span class="num">0</span><span class="op">,</span> <span class="nam">s</span><span class="op">,</span> <span class="num">1</span><span class="op">)</span> <span class="key">for</span> <span class="nam">s</span> <span class="key">in</span> <span class="nam">u</span><span class="op">.</span><span class="nam">shape</span><span class="op">]</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1458" href="#t1458">1458</a></span><span class="t">            <span class="nam">shuffle</span><span class="op">[</span><span class="nam">axis</span><span class="op">]</span> <span class="op">=</span> <span class="nam">base</span><span class="op">.</span><span class="nam">get_fft_shuffle</span><span class="op">(</span><span class="key">True</span><span class="op">,</span> <span class="nam">N</span><span class="op">=</span><span class="nam">v</span><span class="op">.</span><span class="nam">shape</span><span class="op">[</span><span class="nam">axis</span><span class="op">]</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1459" href="#t1459">1459</a></span><span class="t">            <span class="nam">v</span> <span class="op">=</span> <span class="nam">v</span><span class="op">[</span><span class="op">(</span><span class="op">*</span><span class="nam">shuffle</span><span class="op">,</span><span class="op">)</span><span class="op">]</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1460" href="#t1460">1460</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1461" href="#t1461">1461</a></span><span class="t">            <span class="key">if</span> <span class="nam">padding</span> <span class="key">is</span> <span class="key">not</span> <span class="key">None</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1462" href="#t1462">1462</a></span><span class="t">                <span class="nam">shape</span> <span class="op">=</span> <span class="nam">list</span><span class="op">(</span><span class="nam">v</span><span class="op">.</span><span class="nam">shape</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1463" href="#t1463">1463</a></span><span class="t">                <span class="key">if</span> <span class="nam">self</span><span class="op">.</span><span class="nam">comm</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1464" href="#t1464">1464</a></span><span class="t">                    <span class="nam">shape</span><span class="op">[</span><span class="num">0</span><span class="op">]</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">comm</span><span class="op">.</span><span class="nam">allreduce</span><span class="op">(</span><span class="nam">v</span><span class="op">.</span><span class="nam">shape</span><span class="op">[</span><span class="num">0</span><span class="op">]</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1465" href="#t1465">1465</a></span><span class="t">                <span class="nam">fft</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">get_fft</span><span class="op">(</span><span class="nam">axes</span><span class="op">,</span> <span class="str">'forward'</span><span class="op">,</span> <span class="nam">shape</span><span class="op">=</span><span class="nam">shape</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1466" href="#t1466">1466</a></span><span class="t">            <span class="key">else</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1467" href="#t1467">1467</a></span><span class="t">                <span class="nam">fft</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">get_fft</span><span class="op">(</span><span class="nam">axes</span><span class="op">,</span> <span class="str">'forward'</span><span class="op">,</span> <span class="op">**</span><span class="nam">kwargs</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1468" href="#t1468">1468</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1469" href="#t1469">1469</a></span><span class="t">            <span class="nam">v</span> <span class="op">=</span> <span class="nam">fft</span><span class="op">(</span><span class="nam">v</span><span class="op">,</span> <span class="nam">axes</span><span class="op">=</span><span class="nam">axes</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1470" href="#t1470">1470</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1471" href="#t1471">1471</a></span><span class="t">            <span class="nam">expansion</span> <span class="op">=</span> <span class="op">[</span><span class="nam">np</span><span class="op">.</span><span class="nam">newaxis</span> <span class="key">for</span> <span class="nam">_</span> <span class="key">in</span> <span class="nam">u</span><span class="op">.</span><span class="nam">shape</span><span class="op">]</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1472" href="#t1472">1472</a></span><span class="t">            <span class="nam">expansion</span><span class="op">[</span><span class="nam">axis</span><span class="op">]</span> <span class="op">=</span> <span class="nam">slice</span><span class="op">(</span><span class="num">0</span><span class="op">,</span> <span class="nam">v</span><span class="op">.</span><span class="nam">shape</span><span class="op">[</span><span class="nam">axis</span><span class="op">]</span><span class="op">,</span> <span class="num">1</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1473" href="#t1473">1473</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1474" href="#t1474">1474</a></span><span class="t">            <span class="key">if</span> <span class="nam">padding</span> <span class="key">is</span> <span class="key">not</span> <span class="key">None</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1475" href="#t1475">1475</a></span><span class="t">                <span class="nam">shift</span> <span class="op">=</span> <span class="nam">base</span><span class="op">.</span><span class="nam">get_fft_shift</span><span class="op">(</span><span class="key">True</span><span class="op">,</span> <span class="nam">v</span><span class="op">.</span><span class="nam">shape</span><span class="op">[</span><span class="nam">axis</span><span class="op">]</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1476" href="#t1476">1476</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1477" href="#t1477">1477</a></span><span class="t">                <span class="key">if</span> <span class="nam">padding</span><span class="op">[</span><span class="nam">axis</span><span class="op">]</span> <span class="op">!=</span> <span class="num">1</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1478" href="#t1478">1478</a></span><span class="t">                    <span class="nam">N</span> <span class="op">=</span> <span class="nam">int</span><span class="op">(</span><span class="nam">np</span><span class="op">.</span><span class="nam">ceil</span><span class="op">(</span><span class="nam">v</span><span class="op">.</span><span class="nam">shape</span><span class="op">[</span><span class="nam">axis</span><span class="op">]</span> <span class="op">/</span> <span class="nam">padding</span><span class="op">[</span><span class="nam">axis</span><span class="op">]</span><span class="op">)</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1479" href="#t1479">1479</a></span><span class="t">                    <span class="nam">_expansion</span> <span class="op">=</span> <span class="op">[</span><span class="nam">slice</span><span class="op">(</span><span class="num">0</span><span class="op">,</span> <span class="nam">n</span><span class="op">)</span> <span class="key">for</span> <span class="nam">n</span> <span class="key">in</span> <span class="nam">v</span><span class="op">.</span><span class="nam">shape</span><span class="op">]</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1480" href="#t1480">1480</a></span><span class="t">                    <span class="nam">_expansion</span><span class="op">[</span><span class="nam">axis</span><span class="op">]</span> <span class="op">=</span> <span class="nam">slice</span><span class="op">(</span><span class="num">0</span><span class="op">,</span> <span class="nam">N</span><span class="op">,</span> <span class="num">1</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1481" href="#t1481">1481</a></span><span class="t">                    <span class="nam">v</span> <span class="op">=</span> <span class="nam">v</span><span class="op">[</span><span class="op">(</span><span class="op">*</span><span class="nam">_expansion</span><span class="op">,</span><span class="op">)</span><span class="op">]</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1482" href="#t1482">1482</a></span><span class="t">            <span class="key">else</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1483" href="#t1483">1483</a></span><span class="t">                <span class="nam">shift</span> <span class="op">=</span> <span class="nam">base</span><span class="op">.</span><span class="nam">fft_utils</span><span class="op">[</span><span class="str">'fwd'</span><span class="op">]</span><span class="op">[</span><span class="str">'shift'</span><span class="op">]</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1484" href="#t1484">1484</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1485" href="#t1485">1485</a></span><span class="t">            <span class="nam">v</span> <span class="op">*=</span> <span class="nam">shift</span><span class="op">[</span><span class="op">(</span><span class="op">*</span><span class="nam">expansion</span><span class="op">,</span><span class="op">)</span><span class="op">]</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1486" href="#t1486">1486</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1487" href="#t1487">1487</a></span><span class="t">        <span class="key">return</span> <span class="nam">v</span><span class="op">.</span><span class="nam">real</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1488" href="#t1488">1488</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1489" href="#t1489">1489</a></span><span class="t">    <span class="key">def</span> <span class="nam">transform_single_component</span><span class="op">(</span><span class="nam">self</span><span class="op">,</span> <span class="nam">u</span><span class="op">,</span> <span class="nam">axes</span><span class="op">=</span><span class="key">None</span><span class="op">,</span> <span class="nam">padding</span><span class="op">=</span><span class="key">None</span><span class="op">)</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1490" href="#t1490">1490</a></span><span class="t">        <span class="str">"""</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1491" href="#t1491">1491</a></span><span class="t"><span class="str">        Transform a single component of the solution</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1492" href="#t1492">1492</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1493" href="#t1493">1493</a></span><span class="t"><span class="str">        Args:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1494" href="#t1494">1494</a></span><span class="t"><span class="str">            u data to transform:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1495" href="#t1495">1495</a></span><span class="t"><span class="str">            axes (tuple): Axes over which to transform</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1496" href="#t1496">1496</a></span><span class="t"><span class="str">            padding (list): Padding factor for transform. E.g. a padding factor of 2 will discard the upper half of modes after transforming</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1497" href="#t1497">1497</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1498" href="#t1498">1498</a></span><span class="t"><span class="str">        Returns:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1499" href="#t1499">1499</a></span><span class="t"><span class="str">            Transformed data</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1500" href="#t1500">1500</a></span><span class="t"><span class="str">        """</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1501" href="#t1501">1501</a></span><span class="t">        <span class="com"># TODO: clean up and try putting more of this in the 1D bases</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1502" href="#t1502">1502</a></span><span class="t">        <span class="nam">trfs</span> <span class="op">=</span> <span class="op">{</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1503" href="#t1503">1503</a></span><span class="t">            <span class="nam">ChebychevHelper</span><span class="op">:</span> <span class="nam">self</span><span class="op">.</span><span class="nam">_transform_dct</span><span class="op">,</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1504" href="#t1504">1504</a></span><span class="t">            <span class="nam">UltrasphericalHelper</span><span class="op">:</span> <span class="nam">self</span><span class="op">.</span><span class="nam">_transform_dct</span><span class="op">,</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1505" href="#t1505">1505</a></span><span class="t">            <span class="nam">FFTHelper</span><span class="op">:</span> <span class="nam">self</span><span class="op">.</span><span class="nam">_transform_fft</span><span class="op">,</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1506" href="#t1506">1506</a></span><span class="t">        <span class="op">}</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1507" href="#t1507">1507</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1508" href="#t1508">1508</a></span><span class="t">        <span class="nam">axes</span> <span class="op">=</span> <span class="nam">tuple</span><span class="op">(</span><span class="op">-</span><span class="nam">i</span> <span class="op">-</span> <span class="num">1</span> <span class="key">for</span> <span class="nam">i</span> <span class="key">in</span> <span class="nam">range</span><span class="op">(</span><span class="nam">self</span><span class="op">.</span><span class="nam">ndim</span><span class="op">)</span><span class="op">[</span><span class="op">:</span><span class="op">:</span><span class="op">-</span><span class="num">1</span><span class="op">]</span><span class="op">)</span> <span class="key">if</span> <span class="nam">axes</span> <span class="key">is</span> <span class="key">None</span> <span class="key">else</span> <span class="nam">axes</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1509" href="#t1509">1509</a></span><span class="t">        <span class="nam">padding</span> <span class="op">=</span> <span class="op">(</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1510" href="#t1510">1510</a></span><span class="t">            <span class="op">[</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1511" href="#t1511">1511</a></span><span class="t">                <span class="num">1</span><span class="op">,</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1512" href="#t1512">1512</a></span><span class="t">            <span class="op">]</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1513" href="#t1513">1513</a></span><span class="t">            <span class="op">*</span> <span class="nam">self</span><span class="op">.</span><span class="nam">ndim</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1514" href="#t1514">1514</a></span><span class="t">            <span class="key">if</span> <span class="nam">padding</span> <span class="key">is</span> <span class="key">None</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1515" href="#t1515">1515</a></span><span class="t">            <span class="key">else</span> <span class="nam">padding</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1516" href="#t1516">1516</a></span><span class="t">        <span class="op">)</span>  <span class="com"># You know, sometimes I feel very strongly about Black still. This atrocious formatting is readable by Sauron only.</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1517" href="#t1517">1517</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1518" href="#t1518">1518</a></span><span class="t">        <span class="nam">result</span> <span class="op">=</span> <span class="nam">u</span><span class="op">.</span><span class="nam">copy</span><span class="op">(</span><span class="op">)</span><span class="op">.</span><span class="nam">astype</span><span class="op">(</span><span class="nam">complex</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1519" href="#t1519">1519</a></span><span class="t">        <span class="nam">alignment</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">ndim</span> <span class="op">-</span> <span class="num">1</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1520" href="#t1520">1520</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1521" href="#t1521">1521</a></span><span class="t">        <span class="nam">axes_collapsed</span> <span class="op">=</span> <span class="op">[</span><span class="nam">tuple</span><span class="op">(</span><span class="nam">sorted</span><span class="op">(</span><span class="nam">me</span> <span class="key">for</span> <span class="nam">me</span> <span class="key">in</span> <span class="nam">axes</span> <span class="key">if</span> <span class="nam">type</span><span class="op">(</span><span class="nam">self</span><span class="op">.</span><span class="nam">axes</span><span class="op">[</span><span class="nam">me</span><span class="op">]</span><span class="op">)</span> <span class="op">==</span> <span class="nam">base</span><span class="op">)</span><span class="op">)</span> <span class="key">for</span> <span class="nam">base</span> <span class="key">in</span> <span class="nam">trfs</span><span class="op">.</span><span class="nam">keys</span><span class="op">(</span><span class="op">)</span><span class="op">]</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1522" href="#t1522">1522</a></span><span class="t">        <span class="nam">bases</span> <span class="op">=</span> <span class="op">[</span><span class="nam">list</span><span class="op">(</span><span class="nam">trfs</span><span class="op">.</span><span class="nam">keys</span><span class="op">(</span><span class="op">)</span><span class="op">)</span><span class="op">[</span><span class="nam">i</span><span class="op">]</span> <span class="key">for</span> <span class="nam">i</span> <span class="key">in</span> <span class="nam">range</span><span class="op">(</span><span class="nam">len</span><span class="op">(</span><span class="nam">axes_collapsed</span><span class="op">)</span><span class="op">)</span> <span class="key">if</span> <span class="nam">len</span><span class="op">(</span><span class="nam">axes_collapsed</span><span class="op">[</span><span class="nam">i</span><span class="op">]</span><span class="op">)</span> <span class="op">></span> <span class="num">0</span><span class="op">]</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1523" href="#t1523">1523</a></span><span class="t">        <span class="nam">axes_collapsed</span> <span class="op">=</span> <span class="op">[</span><span class="nam">me</span> <span class="key">for</span> <span class="nam">me</span> <span class="key">in</span> <span class="nam">axes_collapsed</span> <span class="key">if</span> <span class="nam">len</span><span class="op">(</span><span class="nam">me</span><span class="op">)</span> <span class="op">></span> <span class="num">0</span><span class="op">]</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1524" href="#t1524">1524</a></span><span class="t">        <span class="nam">shape</span> <span class="op">=</span> <span class="op">[</span><span class="nam">max</span><span class="op">(</span><span class="nam">u</span><span class="op">.</span><span class="nam">shape</span><span class="op">[</span><span class="nam">i</span><span class="op">]</span><span class="op">,</span> <span class="nam">self</span><span class="op">.</span><span class="nam">global_shape</span><span class="op">[</span><span class="num">1</span> <span class="op">+</span> <span class="nam">i</span><span class="op">]</span><span class="op">)</span> <span class="key">for</span> <span class="nam">i</span> <span class="key">in</span> <span class="nam">range</span><span class="op">(</span><span class="nam">self</span><span class="op">.</span><span class="nam">ndim</span><span class="op">)</span><span class="op">]</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1525" href="#t1525">1525</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1526" href="#t1526">1526</a></span><span class="t">        <span class="nam">fft</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">get_fft</span><span class="op">(</span><span class="nam">axes</span><span class="op">=</span><span class="nam">axes</span><span class="op">,</span> <span class="nam">padding</span><span class="op">=</span><span class="nam">padding</span><span class="op">,</span> <span class="nam">direction</span><span class="op">=</span><span class="str">'object'</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1527" href="#t1527">1527</a></span><span class="t">        <span class="key">if</span> <span class="nam">fft</span> <span class="key">is</span> <span class="key">not</span> <span class="key">None</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1528" href="#t1528">1528</a></span><span class="t">            <span class="nam">shape</span> <span class="op">=</span> <span class="nam">list</span><span class="op">(</span><span class="nam">fft</span><span class="op">.</span><span class="nam">global_shape</span><span class="op">(</span><span class="key">False</span><span class="op">)</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1529" href="#t1529">1529</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1530" href="#t1530">1530</a></span><span class="t">        <span class="key">for</span> <span class="nam">trf</span> <span class="key">in</span> <span class="nam">range</span><span class="op">(</span><span class="nam">len</span><span class="op">(</span><span class="nam">axes_collapsed</span><span class="op">)</span><span class="op">)</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1531" href="#t1531">1531</a></span><span class="t">            <span class="nam">_axes</span> <span class="op">=</span> <span class="nam">axes_collapsed</span><span class="op">[</span><span class="nam">trf</span><span class="op">]</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1532" href="#t1532">1532</a></span><span class="t">            <span class="nam">base</span> <span class="op">=</span> <span class="nam">bases</span><span class="op">[</span><span class="nam">trf</span><span class="op">]</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1533" href="#t1533">1533</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1534" href="#t1534">1534</a></span><span class="t">            <span class="key">if</span> <span class="nam">len</span><span class="op">(</span><span class="nam">_axes</span><span class="op">)</span> <span class="op">==</span> <span class="num">0</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t1535" href="#t1535">1535</a></span><span class="t">                <span class="key">continue</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1536" href="#t1536">1536</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1537" href="#t1537">1537</a></span><span class="t">            <span class="key">for</span> <span class="nam">_ax</span> <span class="key">in</span> <span class="nam">_axes</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1538" href="#t1538">1538</a></span><span class="t">                <span class="nam">shape</span><span class="op">[</span><span class="nam">_ax</span><span class="op">]</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">global_shape</span><span class="op">[</span><span class="num">1</span> <span class="op">+</span> <span class="nam">self</span><span class="op">.</span><span class="nam">ndim</span> <span class="op">+</span> <span class="nam">_ax</span><span class="op">]</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1539" href="#t1539">1539</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1540" href="#t1540">1540</a></span><span class="t">            <span class="nam">fft</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">get_fft</span><span class="op">(</span><span class="nam">_axes</span><span class="op">,</span> <span class="str">'object'</span><span class="op">,</span> <span class="nam">padding</span><span class="op">=</span><span class="nam">padding</span><span class="op">,</span> <span class="nam">shape</span><span class="op">=</span><span class="nam">shape</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1541" href="#t1541">1541</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1542" href="#t1542">1542</a></span><span class="t">            <span class="nam">_in</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">get_aligned</span><span class="op">(</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1543" href="#t1543">1543</a></span><span class="t">                <span class="nam">result</span><span class="op">,</span> <span class="nam">axis_in</span><span class="op">=</span><span class="nam">alignment</span><span class="op">,</span> <span class="nam">axis_out</span><span class="op">=</span><span class="nam">self</span><span class="op">.</span><span class="nam">ndim</span> <span class="op">+</span> <span class="nam">_axes</span><span class="op">[</span><span class="op">-</span><span class="num">1</span><span class="op">]</span><span class="op">,</span> <span class="nam">forward</span><span class="op">=</span><span class="key">False</span><span class="op">,</span> <span class="nam">fft</span><span class="op">=</span><span class="nam">fft</span><span class="op">,</span> <span class="nam">shape</span><span class="op">=</span><span class="nam">shape</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1544" href="#t1544">1544</a></span><span class="t">            <span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1545" href="#t1545">1545</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1546" href="#t1546">1546</a></span><span class="t">            <span class="nam">alignment</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">ndim</span> <span class="op">+</span> <span class="nam">_axes</span><span class="op">[</span><span class="op">-</span><span class="num">1</span><span class="op">]</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1547" href="#t1547">1547</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1548" href="#t1548">1548</a></span><span class="t">            <span class="nam">_out</span> <span class="op">=</span> <span class="nam">trfs</span><span class="op">[</span><span class="nam">base</span><span class="op">]</span><span class="op">(</span><span class="nam">_in</span><span class="op">,</span> <span class="nam">axes</span><span class="op">=</span><span class="nam">_axes</span><span class="op">,</span> <span class="nam">padding</span><span class="op">=</span><span class="nam">padding</span><span class="op">,</span> <span class="nam">shape</span><span class="op">=</span><span class="nam">shape</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1549" href="#t1549">1549</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1550" href="#t1550">1550</a></span><span class="t">            <span class="key">if</span> <span class="nam">self</span><span class="op">.</span><span class="nam">comm</span> <span class="key">is</span> <span class="key">not</span> <span class="key">None</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1551" href="#t1551">1551</a></span><span class="t">                <span class="nam">_out</span> <span class="op">*=</span> <span class="nam">np</span><span class="op">.</span><span class="nam">prod</span><span class="op">(</span><span class="op">[</span><span class="nam">self</span><span class="op">.</span><span class="nam">axes</span><span class="op">[</span><span class="nam">i</span><span class="op">]</span><span class="op">.</span><span class="nam">N</span> <span class="key">for</span> <span class="nam">i</span> <span class="key">in</span> <span class="nam">_axes</span><span class="op">]</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1552" href="#t1552">1552</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1553" href="#t1553">1553</a></span><span class="t">            <span class="nam">axes_next_base</span> <span class="op">=</span> <span class="nam">axes_collapsed</span><span class="op">[</span><span class="op">(</span><span class="nam">trf</span> <span class="op">+</span> <span class="num">1</span><span class="op">)</span> <span class="op">%</span> <span class="nam">len</span><span class="op">(</span><span class="nam">axes_collapsed</span><span class="op">)</span><span class="op">]</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1554" href="#t1554">1554</a></span><span class="t">            <span class="nam">alignment</span> <span class="op">=</span> <span class="nam">alignment</span> <span class="key">if</span> <span class="nam">len</span><span class="op">(</span><span class="nam">axes_next_base</span><span class="op">)</span> <span class="op">==</span> <span class="num">0</span> <span class="key">else</span> <span class="nam">self</span><span class="op">.</span><span class="nam">ndim</span> <span class="op">+</span> <span class="nam">axes_next_base</span><span class="op">[</span><span class="op">-</span><span class="num">1</span><span class="op">]</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1555" href="#t1555">1555</a></span><span class="t">            <span class="nam">result</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">get_aligned</span><span class="op">(</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1556" href="#t1556">1556</a></span><span class="t">                <span class="nam">_out</span><span class="op">,</span> <span class="nam">axis_in</span><span class="op">=</span><span class="nam">self</span><span class="op">.</span><span class="nam">ndim</span> <span class="op">+</span> <span class="nam">_axes</span><span class="op">[</span><span class="num">0</span><span class="op">]</span><span class="op">,</span> <span class="nam">axis_out</span><span class="op">=</span><span class="nam">alignment</span><span class="op">,</span> <span class="nam">fft</span><span class="op">=</span><span class="nam">fft</span><span class="op">,</span> <span class="nam">forward</span><span class="op">=</span><span class="key">True</span><span class="op">,</span> <span class="nam">shape</span><span class="op">=</span><span class="nam">shape</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1557" href="#t1557">1557</a></span><span class="t">            <span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1558" href="#t1558">1558</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1559" href="#t1559">1559</a></span><span class="t">        <span class="nam">fft</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">get_fft</span><span class="op">(</span><span class="nam">axes</span><span class="op">=</span><span class="nam">axes</span><span class="op">,</span> <span class="nam">padding</span><span class="op">=</span><span class="nam">padding</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1560" href="#t1560">1560</a></span><span class="t">        <span class="key">return</span> <span class="nam">self</span><span class="op">.</span><span class="nam">get_aligned</span><span class="op">(</span><span class="nam">result</span><span class="op">,</span> <span class="nam">axis_in</span><span class="op">=</span><span class="nam">alignment</span><span class="op">,</span> <span class="nam">axis_out</span><span class="op">=</span><span class="nam">self</span><span class="op">.</span><span class="nam">ndim</span> <span class="op">-</span> <span class="num">1</span><span class="op">,</span> <span class="nam">fft</span><span class="op">=</span><span class="nam">fft</span><span class="op">,</span> <span class="nam">forward</span><span class="op">=</span><span class="key">True</span><span class="op">,</span> <span class="nam">shape</span><span class="op">=</span><span class="nam">shape</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1561" href="#t1561">1561</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1562" href="#t1562">1562</a></span><span class="t">    <span class="key">def</span> <span class="nam">transform</span><span class="op">(</span><span class="nam">self</span><span class="op">,</span> <span class="nam">u</span><span class="op">,</span> <span class="nam">axes</span><span class="op">=</span><span class="key">None</span><span class="op">,</span> <span class="nam">padding</span><span class="op">=</span><span class="key">None</span><span class="op">)</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1563" href="#t1563">1563</a></span><span class="t">        <span class="str">"""</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1564" href="#t1564">1564</a></span><span class="t"><span class="str">        Transform all components from physical space to spectral space</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1565" href="#t1565">1565</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1566" href="#t1566">1566</a></span><span class="t"><span class="str">        Args:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1567" href="#t1567">1567</a></span><span class="t"><span class="str">            u data to transform:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1568" href="#t1568">1568</a></span><span class="t"><span class="str">            axes (tuple): Axes over which to transform</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1569" href="#t1569">1569</a></span><span class="t"><span class="str">            padding (list): Padding factor for transform. E.g. a padding factor of 2 will discard the upper half of modes after transforming</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1570" href="#t1570">1570</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1571" href="#t1571">1571</a></span><span class="t"><span class="str">        Returns:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1572" href="#t1572">1572</a></span><span class="t"><span class="str">            Transformed data</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1573" href="#t1573">1573</a></span><span class="t"><span class="str">        """</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1574" href="#t1574">1574</a></span><span class="t">        <span class="nam">axes</span> <span class="op">=</span> <span class="nam">tuple</span><span class="op">(</span><span class="op">-</span><span class="nam">i</span> <span class="op">-</span> <span class="num">1</span> <span class="key">for</span> <span class="nam">i</span> <span class="key">in</span> <span class="nam">range</span><span class="op">(</span><span class="nam">self</span><span class="op">.</span><span class="nam">ndim</span><span class="op">)</span><span class="op">[</span><span class="op">:</span><span class="op">:</span><span class="op">-</span><span class="num">1</span><span class="op">]</span><span class="op">)</span> <span class="key">if</span> <span class="nam">axes</span> <span class="key">is</span> <span class="key">None</span> <span class="key">else</span> <span class="nam">axes</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1575" href="#t1575">1575</a></span><span class="t">        <span class="nam">padding</span> <span class="op">=</span> <span class="op">(</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1576" href="#t1576">1576</a></span><span class="t">            <span class="op">[</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1577" href="#t1577">1577</a></span><span class="t">                <span class="num">1</span><span class="op">,</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1578" href="#t1578">1578</a></span><span class="t">            <span class="op">]</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1579" href="#t1579">1579</a></span><span class="t">            <span class="op">*</span> <span class="nam">self</span><span class="op">.</span><span class="nam">ndim</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1580" href="#t1580">1580</a></span><span class="t">            <span class="key">if</span> <span class="nam">padding</span> <span class="key">is</span> <span class="key">None</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1581" href="#t1581">1581</a></span><span class="t">            <span class="key">else</span> <span class="nam">padding</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1582" href="#t1582">1582</a></span><span class="t">        <span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1583" href="#t1583">1583</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1584" href="#t1584">1584</a></span><span class="t">        <span class="nam">result</span> <span class="op">=</span> <span class="op">[</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1585" href="#t1585">1585</a></span><span class="t">            <span class="key">None</span><span class="op">,</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1586" href="#t1586">1586</a></span><span class="t">        <span class="op">]</span> <span class="op">*</span> <span class="nam">self</span><span class="op">.</span><span class="nam">ncomponents</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1587" href="#t1587">1587</a></span><span class="t">        <span class="key">for</span> <span class="nam">comp</span> <span class="key">in</span> <span class="nam">self</span><span class="op">.</span><span class="nam">components</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1588" href="#t1588">1588</a></span><span class="t">            <span class="nam">i</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">index</span><span class="op">(</span><span class="nam">comp</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1589" href="#t1589">1589</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1590" href="#t1590">1590</a></span><span class="t">            <span class="nam">result</span><span class="op">[</span><span class="nam">i</span><span class="op">]</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">transform_single_component</span><span class="op">(</span><span class="nam">u</span><span class="op">[</span><span class="nam">i</span><span class="op">]</span><span class="op">,</span> <span class="nam">axes</span><span class="op">=</span><span class="nam">axes</span><span class="op">,</span> <span class="nam">padding</span><span class="op">=</span><span class="nam">padding</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1591" href="#t1591">1591</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1592" href="#t1592">1592</a></span><span class="t">        <span class="key">return</span> <span class="nam">self</span><span class="op">.</span><span class="nam">xp</span><span class="op">.</span><span class="nam">stack</span><span class="op">(</span><span class="nam">result</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1593" href="#t1593">1593</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1594" href="#t1594">1594</a></span><span class="t">    <span class="key">def</span> <span class="nam">_transform_ifft</span><span class="op">(</span><span class="nam">self</span><span class="op">,</span> <span class="nam">u</span><span class="op">,</span> <span class="nam">axes</span><span class="op">,</span> <span class="op">**</span><span class="nam">kwargs</span><span class="op">)</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1595" href="#t1595">1595</a></span><span class="t">        <span class="com"># TODO: clean up and try putting more of this in the 1D bases</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1596" href="#t1596">1596</a></span><span class="t">        <span class="nam">ifft</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">get_fft</span><span class="op">(</span><span class="nam">axes</span><span class="op">,</span> <span class="str">'backward'</span><span class="op">,</span> <span class="op">**</span><span class="nam">kwargs</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1597" href="#t1597">1597</a></span><span class="t">        <span class="key">return</span> <span class="nam">ifft</span><span class="op">(</span><span class="nam">u</span><span class="op">,</span> <span class="nam">axes</span><span class="op">=</span><span class="nam">axes</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1598" href="#t1598">1598</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1599" href="#t1599">1599</a></span><span class="t">    <span class="key">def</span> <span class="nam">_transform_idct</span><span class="op">(</span><span class="nam">self</span><span class="op">,</span> <span class="nam">u</span><span class="op">,</span> <span class="nam">axes</span><span class="op">,</span> <span class="nam">padding</span><span class="op">=</span><span class="key">None</span><span class="op">,</span> <span class="op">**</span><span class="nam">kwargs</span><span class="op">)</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1600" href="#t1600">1600</a></span><span class="t">        <span class="str">'''</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1601" href="#t1601">1601</a></span><span class="t"><span class="str">        This will only ever return real values!</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1602" href="#t1602">1602</a></span><span class="t"><span class="str">        '''</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1603" href="#t1603">1603</a></span><span class="t">        <span class="com"># TODO: clean up and try putting more of this in the 1D bases</span>&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t1604" href="#t1604">1604</a></span><span class="t">        <span class="key">if</span> <span class="nam">self</span><span class="op">.</span><span class="nam">debug</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t1605" href="#t1605">1605</a></span><span class="t">            <span class="key">assert</span> <span class="nam">self</span><span class="op">.</span><span class="nam">xp</span><span class="op">.</span><span class="nam">allclose</span><span class="op">(</span><span class="nam">u</span><span class="op">.</span><span class="nam">imag</span><span class="op">,</span> <span class="num">0</span><span class="op">)</span><span class="op">,</span> <span class="str">'This function can only handle real input.'</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1606" href="#t1606">1606</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1607" href="#t1607">1607</a></span><span class="t">        <span class="nam">v</span> <span class="op">=</span> <span class="nam">u</span><span class="op">.</span><span class="nam">copy</span><span class="op">(</span><span class="op">)</span><span class="op">.</span><span class="nam">astype</span><span class="op">(</span><span class="nam">complex</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1608" href="#t1608">1608</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1609" href="#t1609">1609</a></span><span class="t">        <span class="key">if</span> <span class="nam">len</span><span class="op">(</span><span class="nam">axes</span><span class="op">)</span> <span class="op">></span> <span class="num">1</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1610" href="#t1610">1610</a></span><span class="t">            <span class="nam">v</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">_transform_idct</span><span class="op">(</span><span class="nam">self</span><span class="op">.</span><span class="nam">_transform_idct</span><span class="op">(</span><span class="nam">u</span><span class="op">,</span> <span class="nam">axes</span><span class="op">[</span><span class="num">1</span><span class="op">:</span><span class="op">]</span><span class="op">)</span><span class="op">,</span> <span class="op">(</span><span class="nam">axes</span><span class="op">[</span><span class="num">0</span><span class="op">]</span><span class="op">,</span><span class="op">)</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1611" href="#t1611">1611</a></span><span class="t">        <span class="key">else</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1612" href="#t1612">1612</a></span><span class="t">            <span class="nam">axis</span> <span class="op">=</span> <span class="nam">axes</span><span class="op">[</span><span class="num">0</span><span class="op">]</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1613" href="#t1613">1613</a></span><span class="t">            <span class="nam">base</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">axes</span><span class="op">[</span><span class="nam">axis</span><span class="op">]</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1614" href="#t1614">1614</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1615" href="#t1615">1615</a></span><span class="t">            <span class="key">if</span> <span class="nam">padding</span> <span class="key">is</span> <span class="key">not</span> <span class="key">None</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1616" href="#t1616">1616</a></span><span class="t">                <span class="key">if</span> <span class="nam">padding</span><span class="op">[</span><span class="nam">axis</span><span class="op">]</span> <span class="op">!=</span> <span class="num">1</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1617" href="#t1617">1617</a></span><span class="t">                    <span class="nam">N_pad</span> <span class="op">=</span> <span class="nam">int</span><span class="op">(</span><span class="nam">np</span><span class="op">.</span><span class="nam">ceil</span><span class="op">(</span><span class="nam">v</span><span class="op">.</span><span class="nam">shape</span><span class="op">[</span><span class="nam">axis</span><span class="op">]</span> <span class="op">*</span> <span class="nam">padding</span><span class="op">[</span><span class="nam">axis</span><span class="op">]</span><span class="op">)</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1618" href="#t1618">1618</a></span><span class="t">                    <span class="nam">_pad</span> <span class="op">=</span> <span class="op">[</span><span class="op">[</span><span class="num">0</span><span class="op">,</span> <span class="num">0</span><span class="op">]</span> <span class="key">for</span> <span class="nam">_</span> <span class="key">in</span> <span class="nam">v</span><span class="op">.</span><span class="nam">shape</span><span class="op">]</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1619" href="#t1619">1619</a></span><span class="t">                    <span class="nam">_pad</span><span class="op">[</span><span class="nam">axis</span><span class="op">]</span> <span class="op">=</span> <span class="op">[</span><span class="num">0</span><span class="op">,</span> <span class="nam">N_pad</span> <span class="op">-</span> <span class="nam">base</span><span class="op">.</span><span class="nam">N</span><span class="op">]</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1620" href="#t1620">1620</a></span><span class="t">                    <span class="nam">v</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">xp</span><span class="op">.</span><span class="nam">pad</span><span class="op">(</span><span class="nam">v</span><span class="op">,</span> <span class="nam">_pad</span><span class="op">,</span> <span class="str">'constant'</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1621" href="#t1621">1621</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1622" href="#t1622">1622</a></span><span class="t">                    <span class="nam">shift</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">xp</span><span class="op">.</span><span class="nam">exp</span><span class="op">(</span><span class="num">1j</span> <span class="op">*</span> <span class="nam">np</span><span class="op">.</span><span class="nam">pi</span> <span class="op">*</span> <span class="nam">self</span><span class="op">.</span><span class="nam">xp</span><span class="op">.</span><span class="nam">arange</span><span class="op">(</span><span class="nam">N_pad</span><span class="op">)</span> <span class="op">/</span> <span class="op">(</span><span class="num">2</span> <span class="op">*</span> <span class="nam">N_pad</span><span class="op">)</span><span class="op">)</span> <span class="op">*</span> <span class="nam">base</span><span class="op">.</span><span class="nam">N</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1623" href="#t1623">1623</a></span><span class="t">                <span class="key">else</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1624" href="#t1624">1624</a></span><span class="t">                    <span class="nam">shift</span> <span class="op">=</span> <span class="nam">base</span><span class="op">.</span><span class="nam">fft_utils</span><span class="op">[</span><span class="str">'bck'</span><span class="op">]</span><span class="op">[</span><span class="str">'shift'</span><span class="op">]</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1625" href="#t1625">1625</a></span><span class="t">            <span class="key">else</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1626" href="#t1626">1626</a></span><span class="t">                <span class="nam">shift</span> <span class="op">=</span> <span class="nam">base</span><span class="op">.</span><span class="nam">fft_utils</span><span class="op">[</span><span class="str">'bck'</span><span class="op">]</span><span class="op">[</span><span class="str">'shift'</span><span class="op">]</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1627" href="#t1627">1627</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1628" href="#t1628">1628</a></span><span class="t">            <span class="nam">expansion</span> <span class="op">=</span> <span class="op">[</span><span class="nam">np</span><span class="op">.</span><span class="nam">newaxis</span> <span class="key">for</span> <span class="nam">_</span> <span class="key">in</span> <span class="nam">u</span><span class="op">.</span><span class="nam">shape</span><span class="op">]</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1629" href="#t1629">1629</a></span><span class="t">            <span class="nam">expansion</span><span class="op">[</span><span class="nam">axis</span><span class="op">]</span> <span class="op">=</span> <span class="nam">slice</span><span class="op">(</span><span class="num">0</span><span class="op">,</span> <span class="nam">v</span><span class="op">.</span><span class="nam">shape</span><span class="op">[</span><span class="nam">axis</span><span class="op">]</span><span class="op">,</span> <span class="num">1</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1630" href="#t1630">1630</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1631" href="#t1631">1631</a></span><span class="t">            <span class="nam">v</span> <span class="op">*=</span> <span class="nam">shift</span><span class="op">[</span><span class="op">(</span><span class="op">*</span><span class="nam">expansion</span><span class="op">,</span><span class="op">)</span><span class="op">]</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1632" href="#t1632">1632</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1633" href="#t1633">1633</a></span><span class="t">            <span class="key">if</span> <span class="nam">padding</span> <span class="key">is</span> <span class="key">not</span> <span class="key">None</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1634" href="#t1634">1634</a></span><span class="t">                <span class="key">if</span> <span class="nam">padding</span><span class="op">[</span><span class="nam">axis</span><span class="op">]</span> <span class="op">!=</span> <span class="num">1</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1635" href="#t1635">1635</a></span><span class="t">                    <span class="nam">shape</span> <span class="op">=</span> <span class="nam">list</span><span class="op">(</span><span class="nam">v</span><span class="op">.</span><span class="nam">shape</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1636" href="#t1636">1636</a></span><span class="t">                    <span class="key">if</span> <span class="nam">self</span><span class="op">.</span><span class="nam">comm</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1637" href="#t1637">1637</a></span><span class="t">                        <span class="nam">shape</span><span class="op">[</span><span class="num">0</span><span class="op">]</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">comm</span><span class="op">.</span><span class="nam">allreduce</span><span class="op">(</span><span class="nam">v</span><span class="op">.</span><span class="nam">shape</span><span class="op">[</span><span class="num">0</span><span class="op">]</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1638" href="#t1638">1638</a></span><span class="t">                    <span class="nam">ifft</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">get_fft</span><span class="op">(</span><span class="nam">axes</span><span class="op">,</span> <span class="str">'backward'</span><span class="op">,</span> <span class="nam">shape</span><span class="op">=</span><span class="nam">shape</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1639" href="#t1639">1639</a></span><span class="t">                <span class="key">else</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1640" href="#t1640">1640</a></span><span class="t">                    <span class="nam">ifft</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">get_fft</span><span class="op">(</span><span class="nam">axes</span><span class="op">,</span> <span class="str">'backward'</span><span class="op">,</span> <span class="nam">padding</span><span class="op">=</span><span class="nam">padding</span><span class="op">,</span> <span class="op">**</span><span class="nam">kwargs</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1641" href="#t1641">1641</a></span><span class="t">            <span class="key">else</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1642" href="#t1642">1642</a></span><span class="t">                <span class="nam">ifft</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">get_fft</span><span class="op">(</span><span class="nam">axes</span><span class="op">,</span> <span class="str">'backward'</span><span class="op">,</span> <span class="nam">padding</span><span class="op">=</span><span class="nam">padding</span><span class="op">,</span> <span class="op">**</span><span class="nam">kwargs</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1643" href="#t1643">1643</a></span><span class="t">            <span class="nam">v</span> <span class="op">=</span> <span class="nam">ifft</span><span class="op">(</span><span class="nam">v</span><span class="op">,</span> <span class="nam">axes</span><span class="op">=</span><span class="nam">axes</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1644" href="#t1644">1644</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1645" href="#t1645">1645</a></span><span class="t">            <span class="nam">shuffle</span> <span class="op">=</span> <span class="op">[</span><span class="nam">slice</span><span class="op">(</span><span class="num">0</span><span class="op">,</span> <span class="nam">s</span><span class="op">,</span> <span class="num">1</span><span class="op">)</span> <span class="key">for</span> <span class="nam">s</span> <span class="key">in</span> <span class="nam">v</span><span class="op">.</span><span class="nam">shape</span><span class="op">]</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1646" href="#t1646">1646</a></span><span class="t">            <span class="nam">shuffle</span><span class="op">[</span><span class="nam">axis</span><span class="op">]</span> <span class="op">=</span> <span class="nam">base</span><span class="op">.</span><span class="nam">get_fft_shuffle</span><span class="op">(</span><span class="key">False</span><span class="op">,</span> <span class="nam">N</span><span class="op">=</span><span class="nam">v</span><span class="op">.</span><span class="nam">shape</span><span class="op">[</span><span class="nam">axis</span><span class="op">]</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1647" href="#t1647">1647</a></span><span class="t">            <span class="nam">v</span> <span class="op">=</span> <span class="nam">v</span><span class="op">[</span><span class="op">(</span><span class="op">*</span><span class="nam">shuffle</span><span class="op">,</span><span class="op">)</span><span class="op">]</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1648" href="#t1648">1648</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1649" href="#t1649">1649</a></span><span class="t">        <span class="key">return</span> <span class="nam">v</span><span class="op">.</span><span class="nam">real</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1650" href="#t1650">1650</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1651" href="#t1651">1651</a></span><span class="t">    <span class="key">def</span> <span class="nam">itransform_single_component</span><span class="op">(</span><span class="nam">self</span><span class="op">,</span> <span class="nam">u</span><span class="op">,</span> <span class="nam">axes</span><span class="op">=</span><span class="key">None</span><span class="op">,</span> <span class="nam">padding</span><span class="op">=</span><span class="key">None</span><span class="op">)</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1652" href="#t1652">1652</a></span><span class="t">        <span class="str">"""</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1653" href="#t1653">1653</a></span><span class="t"><span class="str">        Inverse transform over single component of the solution</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1654" href="#t1654">1654</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1655" href="#t1655">1655</a></span><span class="t"><span class="str">        Args:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1656" href="#t1656">1656</a></span><span class="t"><span class="str">            u data to transform:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1657" href="#t1657">1657</a></span><span class="t"><span class="str">            axes (tuple): Axes over which to transform</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1658" href="#t1658">1658</a></span><span class="t"><span class="str">            padding (list): Padding factor for transform. E.g. a padding factor of 2 will add as many zeros as there were modes before before transforming</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1659" href="#t1659">1659</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1660" href="#t1660">1660</a></span><span class="t"><span class="str">        Returns:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1661" href="#t1661">1661</a></span><span class="t"><span class="str">            Transformed data</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1662" href="#t1662">1662</a></span><span class="t"><span class="str">        """</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1663" href="#t1663">1663</a></span><span class="t">        <span class="com"># TODO: clean up and try putting more of this in the 1D bases</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1664" href="#t1664">1664</a></span><span class="t">        <span class="nam">trfs</span> <span class="op">=</span> <span class="op">{</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1665" href="#t1665">1665</a></span><span class="t">            <span class="nam">FFTHelper</span><span class="op">:</span> <span class="nam">self</span><span class="op">.</span><span class="nam">_transform_ifft</span><span class="op">,</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1666" href="#t1666">1666</a></span><span class="t">            <span class="nam">ChebychevHelper</span><span class="op">:</span> <span class="nam">self</span><span class="op">.</span><span class="nam">_transform_idct</span><span class="op">,</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1667" href="#t1667">1667</a></span><span class="t">            <span class="nam">UltrasphericalHelper</span><span class="op">:</span> <span class="nam">self</span><span class="op">.</span><span class="nam">_transform_idct</span><span class="op">,</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1668" href="#t1668">1668</a></span><span class="t">        <span class="op">}</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1669" href="#t1669">1669</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1670" href="#t1670">1670</a></span><span class="t">        <span class="nam">axes</span> <span class="op">=</span> <span class="nam">tuple</span><span class="op">(</span><span class="op">-</span><span class="nam">i</span> <span class="op">-</span> <span class="num">1</span> <span class="key">for</span> <span class="nam">i</span> <span class="key">in</span> <span class="nam">range</span><span class="op">(</span><span class="nam">self</span><span class="op">.</span><span class="nam">ndim</span><span class="op">)</span><span class="op">[</span><span class="op">:</span><span class="op">:</span><span class="op">-</span><span class="num">1</span><span class="op">]</span><span class="op">)</span> <span class="key">if</span> <span class="nam">axes</span> <span class="key">is</span> <span class="key">None</span> <span class="key">else</span> <span class="nam">axes</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1671" href="#t1671">1671</a></span><span class="t">        <span class="nam">padding</span> <span class="op">=</span> <span class="op">(</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1672" href="#t1672">1672</a></span><span class="t">            <span class="op">[</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1673" href="#t1673">1673</a></span><span class="t">                <span class="num">1</span><span class="op">,</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1674" href="#t1674">1674</a></span><span class="t">            <span class="op">]</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1675" href="#t1675">1675</a></span><span class="t">            <span class="op">*</span> <span class="nam">self</span><span class="op">.</span><span class="nam">ndim</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1676" href="#t1676">1676</a></span><span class="t">            <span class="key">if</span> <span class="nam">padding</span> <span class="key">is</span> <span class="key">None</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1677" href="#t1677">1677</a></span><span class="t">            <span class="key">else</span> <span class="nam">padding</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1678" href="#t1678">1678</a></span><span class="t">        <span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1679" href="#t1679">1679</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1680" href="#t1680">1680</a></span><span class="t">        <span class="nam">result</span> <span class="op">=</span> <span class="nam">u</span><span class="op">.</span><span class="nam">copy</span><span class="op">(</span><span class="op">)</span><span class="op">.</span><span class="nam">astype</span><span class="op">(</span><span class="nam">complex</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1681" href="#t1681">1681</a></span><span class="t">        <span class="nam">alignment</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">ndim</span> <span class="op">-</span> <span class="num">1</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1682" href="#t1682">1682</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1683" href="#t1683">1683</a></span><span class="t">        <span class="nam">axes_collapsed</span> <span class="op">=</span> <span class="op">[</span><span class="nam">tuple</span><span class="op">(</span><span class="nam">sorted</span><span class="op">(</span><span class="nam">me</span> <span class="key">for</span> <span class="nam">me</span> <span class="key">in</span> <span class="nam">axes</span> <span class="key">if</span> <span class="nam">type</span><span class="op">(</span><span class="nam">self</span><span class="op">.</span><span class="nam">axes</span><span class="op">[</span><span class="nam">me</span><span class="op">]</span><span class="op">)</span> <span class="op">==</span> <span class="nam">base</span><span class="op">)</span><span class="op">)</span> <span class="key">for</span> <span class="nam">base</span> <span class="key">in</span> <span class="nam">trfs</span><span class="op">.</span><span class="nam">keys</span><span class="op">(</span><span class="op">)</span><span class="op">]</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1684" href="#t1684">1684</a></span><span class="t">        <span class="nam">bases</span> <span class="op">=</span> <span class="op">[</span><span class="nam">list</span><span class="op">(</span><span class="nam">trfs</span><span class="op">.</span><span class="nam">keys</span><span class="op">(</span><span class="op">)</span><span class="op">)</span><span class="op">[</span><span class="nam">i</span><span class="op">]</span> <span class="key">for</span> <span class="nam">i</span> <span class="key">in</span> <span class="nam">range</span><span class="op">(</span><span class="nam">len</span><span class="op">(</span><span class="nam">axes_collapsed</span><span class="op">)</span><span class="op">)</span> <span class="key">if</span> <span class="nam">len</span><span class="op">(</span><span class="nam">axes_collapsed</span><span class="op">[</span><span class="nam">i</span><span class="op">]</span><span class="op">)</span> <span class="op">></span> <span class="num">0</span><span class="op">]</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1685" href="#t1685">1685</a></span><span class="t">        <span class="nam">axes_collapsed</span> <span class="op">=</span> <span class="op">[</span><span class="nam">me</span> <span class="key">for</span> <span class="nam">me</span> <span class="key">in</span> <span class="nam">axes_collapsed</span> <span class="key">if</span> <span class="nam">len</span><span class="op">(</span><span class="nam">me</span><span class="op">)</span> <span class="op">></span> <span class="num">0</span><span class="op">]</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1686" href="#t1686">1686</a></span><span class="t">        <span class="nam">shape</span> <span class="op">=</span> <span class="nam">list</span><span class="op">(</span><span class="nam">self</span><span class="op">.</span><span class="nam">global_shape</span><span class="op">[</span><span class="num">1</span><span class="op">:</span><span class="op">]</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1687" href="#t1687">1687</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1688" href="#t1688">1688</a></span><span class="t">        <span class="key">for</span> <span class="nam">trf</span> <span class="key">in</span> <span class="nam">range</span><span class="op">(</span><span class="nam">len</span><span class="op">(</span><span class="nam">axes_collapsed</span><span class="op">)</span><span class="op">)</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1689" href="#t1689">1689</a></span><span class="t">            <span class="nam">_axes</span> <span class="op">=</span> <span class="nam">axes_collapsed</span><span class="op">[</span><span class="nam">trf</span><span class="op">]</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1690" href="#t1690">1690</a></span><span class="t">            <span class="nam">base</span> <span class="op">=</span> <span class="nam">bases</span><span class="op">[</span><span class="nam">trf</span><span class="op">]</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1691" href="#t1691">1691</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1692" href="#t1692">1692</a></span><span class="t">            <span class="key">if</span> <span class="nam">len</span><span class="op">(</span><span class="nam">_axes</span><span class="op">)</span> <span class="op">==</span> <span class="num">0</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t1693" href="#t1693">1693</a></span><span class="t">                <span class="key">continue</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1694" href="#t1694">1694</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1695" href="#t1695">1695</a></span><span class="t">            <span class="nam">fft</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">get_fft</span><span class="op">(</span><span class="nam">_axes</span><span class="op">,</span> <span class="str">'object'</span><span class="op">,</span> <span class="nam">padding</span><span class="op">=</span><span class="nam">padding</span><span class="op">,</span> <span class="nam">shape</span><span class="op">=</span><span class="nam">shape</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1696" href="#t1696">1696</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1697" href="#t1697">1697</a></span><span class="t">            <span class="nam">_in</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">get_aligned</span><span class="op">(</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1698" href="#t1698">1698</a></span><span class="t">                <span class="nam">result</span><span class="op">,</span> <span class="nam">axis_in</span><span class="op">=</span><span class="nam">alignment</span><span class="op">,</span> <span class="nam">axis_out</span><span class="op">=</span><span class="nam">self</span><span class="op">.</span><span class="nam">ndim</span> <span class="op">+</span> <span class="nam">_axes</span><span class="op">[</span><span class="num">0</span><span class="op">]</span><span class="op">,</span> <span class="nam">forward</span><span class="op">=</span><span class="key">True</span><span class="op">,</span> <span class="nam">fft</span><span class="op">=</span><span class="nam">fft</span><span class="op">,</span> <span class="nam">shape</span><span class="op">=</span><span class="nam">shape</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1699" href="#t1699">1699</a></span><span class="t">            <span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1700" href="#t1700">1700</a></span><span class="t">            <span class="key">if</span> <span class="nam">self</span><span class="op">.</span><span class="nam">comm</span> <span class="key">is</span> <span class="key">not</span> <span class="key">None</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1701" href="#t1701">1701</a></span><span class="t">                <span class="nam">_in</span> <span class="op">/=</span> <span class="nam">np</span><span class="op">.</span><span class="nam">prod</span><span class="op">(</span><span class="op">[</span><span class="nam">self</span><span class="op">.</span><span class="nam">axes</span><span class="op">[</span><span class="nam">i</span><span class="op">]</span><span class="op">.</span><span class="nam">N</span> <span class="key">for</span> <span class="nam">i</span> <span class="key">in</span> <span class="nam">_axes</span><span class="op">]</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1702" href="#t1702">1702</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1703" href="#t1703">1703</a></span><span class="t">            <span class="nam">alignment</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">ndim</span> <span class="op">+</span> <span class="nam">_axes</span><span class="op">[</span><span class="num">0</span><span class="op">]</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1704" href="#t1704">1704</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1705" href="#t1705">1705</a></span><span class="t">            <span class="nam">_out</span> <span class="op">=</span> <span class="nam">trfs</span><span class="op">[</span><span class="nam">base</span><span class="op">]</span><span class="op">(</span><span class="nam">_in</span><span class="op">,</span> <span class="nam">axes</span><span class="op">=</span><span class="nam">_axes</span><span class="op">,</span> <span class="nam">padding</span><span class="op">=</span><span class="nam">padding</span><span class="op">,</span> <span class="nam">shape</span><span class="op">=</span><span class="nam">shape</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1706" href="#t1706">1706</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1707" href="#t1707">1707</a></span><span class="t">            <span class="key">for</span> <span class="nam">_ax</span> <span class="key">in</span> <span class="nam">_axes</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1708" href="#t1708">1708</a></span><span class="t">                <span class="key">if</span> <span class="nam">fft</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1709" href="#t1709">1709</a></span><span class="t">                    <span class="nam">shape</span><span class="op">[</span><span class="nam">_ax</span><span class="op">]</span> <span class="op">=</span> <span class="nam">fft</span><span class="op">.</span><span class="nam">_input_shape</span><span class="op">[</span><span class="nam">_ax</span><span class="op">]</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1710" href="#t1710">1710</a></span><span class="t">                <span class="key">else</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1711" href="#t1711">1711</a></span><span class="t">                    <span class="nam">shape</span><span class="op">[</span><span class="nam">_ax</span><span class="op">]</span> <span class="op">=</span> <span class="nam">_out</span><span class="op">.</span><span class="nam">shape</span><span class="op">[</span><span class="nam">_ax</span><span class="op">]</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1712" href="#t1712">1712</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1713" href="#t1713">1713</a></span><span class="t">            <span class="nam">axes_next_base</span> <span class="op">=</span> <span class="nam">axes_collapsed</span><span class="op">[</span><span class="op">(</span><span class="nam">trf</span> <span class="op">+</span> <span class="num">1</span><span class="op">)</span> <span class="op">%</span> <span class="nam">len</span><span class="op">(</span><span class="nam">axes_collapsed</span><span class="op">)</span><span class="op">]</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1714" href="#t1714">1714</a></span><span class="t">            <span class="nam">alignment</span> <span class="op">=</span> <span class="nam">alignment</span> <span class="key">if</span> <span class="nam">len</span><span class="op">(</span><span class="nam">axes_next_base</span><span class="op">)</span> <span class="op">==</span> <span class="num">0</span> <span class="key">else</span> <span class="nam">self</span><span class="op">.</span><span class="nam">ndim</span> <span class="op">+</span> <span class="nam">axes_next_base</span><span class="op">[</span><span class="num">0</span><span class="op">]</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1715" href="#t1715">1715</a></span><span class="t">            <span class="nam">result</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">get_aligned</span><span class="op">(</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1716" href="#t1716">1716</a></span><span class="t">                <span class="nam">_out</span><span class="op">,</span> <span class="nam">axis_in</span><span class="op">=</span><span class="nam">self</span><span class="op">.</span><span class="nam">ndim</span> <span class="op">+</span> <span class="nam">_axes</span><span class="op">[</span><span class="op">-</span><span class="num">1</span><span class="op">]</span><span class="op">,</span> <span class="nam">axis_out</span><span class="op">=</span><span class="nam">alignment</span><span class="op">,</span> <span class="nam">fft</span><span class="op">=</span><span class="nam">fft</span><span class="op">,</span> <span class="nam">forward</span><span class="op">=</span><span class="key">False</span><span class="op">,</span> <span class="nam">shape</span><span class="op">=</span><span class="nam">shape</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1717" href="#t1717">1717</a></span><span class="t">            <span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1718" href="#t1718">1718</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1719" href="#t1719">1719</a></span><span class="t">        <span class="nam">fft</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">get_fft</span><span class="op">(</span><span class="nam">axes</span><span class="op">=</span><span class="nam">axes</span><span class="op">,</span> <span class="nam">padding</span><span class="op">=</span><span class="nam">padding</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1720" href="#t1720">1720</a></span><span class="t">        <span class="key">return</span> <span class="nam">self</span><span class="op">.</span><span class="nam">get_aligned</span><span class="op">(</span><span class="nam">result</span><span class="op">,</span> <span class="nam">axis_in</span><span class="op">=</span><span class="nam">alignment</span><span class="op">,</span> <span class="nam">axis_out</span><span class="op">=</span><span class="nam">self</span><span class="op">.</span><span class="nam">ndim</span> <span class="op">-</span> <span class="num">1</span><span class="op">,</span> <span class="nam">fft</span><span class="op">=</span><span class="nam">fft</span><span class="op">,</span> <span class="nam">shape</span><span class="op">=</span><span class="nam">shape</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1721" href="#t1721">1721</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1722" href="#t1722">1722</a></span><span class="t">    <span class="key">def</span> <span class="nam">get_aligned</span><span class="op">(</span><span class="nam">self</span><span class="op">,</span> <span class="nam">u</span><span class="op">,</span> <span class="nam">axis_in</span><span class="op">,</span> <span class="nam">axis_out</span><span class="op">,</span> <span class="nam">fft</span><span class="op">=</span><span class="key">None</span><span class="op">,</span> <span class="nam">forward</span><span class="op">=</span><span class="key">False</span><span class="op">,</span> <span class="op">**</span><span class="nam">kwargs</span><span class="op">)</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1723" href="#t1723">1723</a></span><span class="t">        <span class="str">"""</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1724" href="#t1724">1724</a></span><span class="t"><span class="str">        Realign the data along the axis when using distributed FFTs. `kwargs` will be used to get the correct PFFT</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1725" href="#t1725">1725</a></span><span class="t"><span class="str">        object from `mpi4py-fft`, which has suitable transfer classes for the shape of data. Hence, they should include</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1726" href="#t1726">1726</a></span><span class="t"><span class="str">        shape especially, if applicable.</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1727" href="#t1727">1727</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1728" href="#t1728">1728</a></span><span class="t"><span class="str">        Args:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1729" href="#t1729">1729</a></span><span class="t"><span class="str">            u: The solution</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1730" href="#t1730">1730</a></span><span class="t"><span class="str">            axis_in (int): Current alignment</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1731" href="#t1731">1731</a></span><span class="t"><span class="str">            axis_out (int): New alignment</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1732" href="#t1732">1732</a></span><span class="t"><span class="str">            fft (mpi4py_fft.PFFT), optional: parallel FFT object</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1733" href="#t1733">1733</a></span><span class="t"><span class="str">            forward (bool): Whether the input is in spectral space or not</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1734" href="#t1734">1734</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1735" href="#t1735">1735</a></span><span class="t"><span class="str">        Returns:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1736" href="#t1736">1736</a></span><span class="t"><span class="str">            solution aligned on `axis_in`</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1737" href="#t1737">1737</a></span><span class="t"><span class="str">        """</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1738" href="#t1738">1738</a></span><span class="t">        <span class="key">if</span> <span class="nam">self</span><span class="op">.</span><span class="nam">comm</span> <span class="key">is</span> <span class="key">None</span> <span class="key">or</span> <span class="nam">axis_in</span> <span class="op">==</span> <span class="nam">axis_out</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1739" href="#t1739">1739</a></span><span class="t">            <span class="key">return</span> <span class="nam">u</span><span class="op">.</span><span class="nam">copy</span><span class="op">(</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1740" href="#t1740">1740</a></span><span class="t">        <span class="key">if</span> <span class="nam">self</span><span class="op">.</span><span class="nam">comm</span><span class="op">.</span><span class="nam">size</span> <span class="op">==</span> <span class="num">1</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1741" href="#t1741">1741</a></span><span class="t">            <span class="key">return</span> <span class="nam">u</span><span class="op">.</span><span class="nam">copy</span><span class="op">(</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1742" href="#t1742">1742</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1743" href="#t1743">1743</a></span><span class="t">        <span class="nam">fft</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">get_fft</span><span class="op">(</span><span class="op">**</span><span class="nam">kwargs</span><span class="op">)</span> <span class="key">if</span> <span class="nam">fft</span> <span class="key">is</span> <span class="key">None</span> <span class="key">else</span> <span class="nam">fft</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1744" href="#t1744">1744</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1745" href="#t1745">1745</a></span><span class="t">        <span class="nam">global_fft</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">get_fft</span><span class="op">(</span><span class="op">**</span><span class="nam">kwargs</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1746" href="#t1746">1746</a></span><span class="t">        <span class="nam">axisA</span> <span class="op">=</span> <span class="op">[</span><span class="nam">me</span><span class="op">.</span><span class="nam">axisA</span> <span class="key">for</span> <span class="nam">me</span> <span class="key">in</span> <span class="nam">global_fft</span><span class="op">.</span><span class="nam">transfer</span><span class="op">]</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1747" href="#t1747">1747</a></span><span class="t">        <span class="nam">axisB</span> <span class="op">=</span> <span class="op">[</span><span class="nam">me</span><span class="op">.</span><span class="nam">axisB</span> <span class="key">for</span> <span class="nam">me</span> <span class="key">in</span> <span class="nam">global_fft</span><span class="op">.</span><span class="nam">transfer</span><span class="op">]</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1748" href="#t1748">1748</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1749" href="#t1749">1749</a></span><span class="t">        <span class="nam">current_axis</span> <span class="op">=</span> <span class="nam">axis_in</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1750" href="#t1750">1750</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1751" href="#t1751">1751</a></span><span class="t">        <span class="key">if</span> <span class="nam">axis_in</span> <span class="key">in</span> <span class="nam">axisA</span> <span class="key">and</span> <span class="nam">axis_out</span> <span class="key">in</span> <span class="nam">axisB</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1752" href="#t1752">1752</a></span><span class="t">            <span class="key">while</span> <span class="nam">current_axis</span> <span class="op">!=</span> <span class="nam">axis_out</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1753" href="#t1753">1753</a></span><span class="t">                <span class="nam">transfer</span> <span class="op">=</span> <span class="nam">global_fft</span><span class="op">.</span><span class="nam">transfer</span><span class="op">[</span><span class="nam">axisA</span><span class="op">.</span><span class="nam">index</span><span class="op">(</span><span class="nam">current_axis</span><span class="op">)</span><span class="op">]</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1754" href="#t1754">1754</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1755" href="#t1755">1755</a></span><span class="t">                <span class="nam">arrayB</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">xp</span><span class="op">.</span><span class="nam">empty</span><span class="op">(</span><span class="nam">shape</span><span class="op">=</span><span class="nam">transfer</span><span class="op">.</span><span class="nam">subshapeB</span><span class="op">,</span> <span class="nam">dtype</span><span class="op">=</span><span class="nam">transfer</span><span class="op">.</span><span class="nam">dtype</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1756" href="#t1756">1756</a></span><span class="t">                <span class="nam">arrayA</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">xp</span><span class="op">.</span><span class="nam">empty</span><span class="op">(</span><span class="nam">shape</span><span class="op">=</span><span class="nam">transfer</span><span class="op">.</span><span class="nam">subshapeA</span><span class="op">,</span> <span class="nam">dtype</span><span class="op">=</span><span class="nam">transfer</span><span class="op">.</span><span class="nam">dtype</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1757" href="#t1757">1757</a></span><span class="t">                <span class="nam">arrayA</span><span class="op">[</span><span class="op">:</span><span class="op">]</span> <span class="op">=</span> <span class="nam">u</span><span class="op">[</span><span class="op">:</span><span class="op">]</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1758" href="#t1758">1758</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1759" href="#t1759">1759</a></span><span class="t">                <span class="nam">transfer</span><span class="op">.</span><span class="nam">forward</span><span class="op">(</span><span class="nam">arrayA</span><span class="op">=</span><span class="nam">arrayA</span><span class="op">,</span> <span class="nam">arrayB</span><span class="op">=</span><span class="nam">arrayB</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1760" href="#t1760">1760</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1761" href="#t1761">1761</a></span><span class="t">                <span class="nam">current_axis</span> <span class="op">=</span> <span class="nam">transfer</span><span class="op">.</span><span class="nam">axisB</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1762" href="#t1762">1762</a></span><span class="t">                <span class="nam">u</span> <span class="op">=</span> <span class="nam">arrayB</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1763" href="#t1763">1763</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1764" href="#t1764">1764</a></span><span class="t">            <span class="key">return</span> <span class="nam">u</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1765" href="#t1765">1765</a></span><span class="t">        <span class="key">elif</span> <span class="nam">axis_in</span> <span class="key">in</span> <span class="nam">axisB</span> <span class="key">and</span> <span class="nam">axis_out</span> <span class="key">in</span> <span class="nam">axisA</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1766" href="#t1766">1766</a></span><span class="t">            <span class="key">while</span> <span class="nam">current_axis</span> <span class="op">!=</span> <span class="nam">axis_out</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1767" href="#t1767">1767</a></span><span class="t">                <span class="nam">transfer</span> <span class="op">=</span> <span class="nam">global_fft</span><span class="op">.</span><span class="nam">transfer</span><span class="op">[</span><span class="nam">axisB</span><span class="op">.</span><span class="nam">index</span><span class="op">(</span><span class="nam">current_axis</span><span class="op">)</span><span class="op">]</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1768" href="#t1768">1768</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1769" href="#t1769">1769</a></span><span class="t">                <span class="nam">arrayB</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">xp</span><span class="op">.</span><span class="nam">empty</span><span class="op">(</span><span class="nam">shape</span><span class="op">=</span><span class="nam">transfer</span><span class="op">.</span><span class="nam">subshapeB</span><span class="op">,</span> <span class="nam">dtype</span><span class="op">=</span><span class="nam">transfer</span><span class="op">.</span><span class="nam">dtype</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1770" href="#t1770">1770</a></span><span class="t">                <span class="nam">arrayA</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">xp</span><span class="op">.</span><span class="nam">empty</span><span class="op">(</span><span class="nam">shape</span><span class="op">=</span><span class="nam">transfer</span><span class="op">.</span><span class="nam">subshapeA</span><span class="op">,</span> <span class="nam">dtype</span><span class="op">=</span><span class="nam">transfer</span><span class="op">.</span><span class="nam">dtype</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1771" href="#t1771">1771</a></span><span class="t">                <span class="nam">arrayB</span><span class="op">[</span><span class="op">:</span><span class="op">]</span> <span class="op">=</span> <span class="nam">u</span><span class="op">[</span><span class="op">:</span><span class="op">]</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1772" href="#t1772">1772</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1773" href="#t1773">1773</a></span><span class="t">                <span class="nam">transfer</span><span class="op">.</span><span class="nam">backward</span><span class="op">(</span><span class="nam">arrayA</span><span class="op">=</span><span class="nam">arrayA</span><span class="op">,</span> <span class="nam">arrayB</span><span class="op">=</span><span class="nam">arrayB</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1774" href="#t1774">1774</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1775" href="#t1775">1775</a></span><span class="t">                <span class="nam">current_axis</span> <span class="op">=</span> <span class="nam">transfer</span><span class="op">.</span><span class="nam">axisA</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1776" href="#t1776">1776</a></span><span class="t">                <span class="nam">u</span> <span class="op">=</span> <span class="nam">arrayA</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1777" href="#t1777">1777</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1778" href="#t1778">1778</a></span><span class="t">            <span class="key">return</span> <span class="nam">u</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1779" href="#t1779">1779</a></span><span class="t">        <span class="key">else</span><span class="op">:</span>  <span class="com"># go the potentially slower route of not reusing transfer classes</span>&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t1780" href="#t1780">1780</a></span><span class="t">            <span class="key">from</span> <span class="nam">mpi4py_fft</span> <span class="key">import</span> <span class="nam">newDistArray</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1781" href="#t1781">1781</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t1782" href="#t1782">1782</a></span><span class="t">            <span class="nam">_in</span> <span class="op">=</span> <span class="nam">newDistArray</span><span class="op">(</span><span class="nam">fft</span><span class="op">,</span> <span class="nam">forward</span><span class="op">)</span><span class="op">.</span><span class="nam">redistribute</span><span class="op">(</span><span class="nam">axis_in</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t1783" href="#t1783">1783</a></span><span class="t">            <span class="nam">_in</span><span class="op">[</span><span class="op">...</span><span class="op">]</span> <span class="op">=</span> <span class="nam">u</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1784" href="#t1784">1784</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t1785" href="#t1785">1785</a></span><span class="t">            <span class="key">return</span> <span class="nam">_in</span><span class="op">.</span><span class="nam">redistribute</span><span class="op">(</span><span class="nam">axis_out</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1786" href="#t1786">1786</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1787" href="#t1787">1787</a></span><span class="t">    <span class="key">def</span> <span class="nam">itransform</span><span class="op">(</span><span class="nam">self</span><span class="op">,</span> <span class="nam">u</span><span class="op">,</span> <span class="nam">axes</span><span class="op">=</span><span class="key">None</span><span class="op">,</span> <span class="nam">padding</span><span class="op">=</span><span class="key">None</span><span class="op">)</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1788" href="#t1788">1788</a></span><span class="t">        <span class="str">"""</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1789" href="#t1789">1789</a></span><span class="t"><span class="str">        Inverse transform over all components of the solution</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1790" href="#t1790">1790</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1791" href="#t1791">1791</a></span><span class="t"><span class="str">        Args:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1792" href="#t1792">1792</a></span><span class="t"><span class="str">            u data to transform:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1793" href="#t1793">1793</a></span><span class="t"><span class="str">            axes (tuple): Axes over which to transform</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1794" href="#t1794">1794</a></span><span class="t"><span class="str">            padding (list): Padding factor for transform. E.g. a padding factor of 2 will add as many zeros as there were modes before before transforming</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1795" href="#t1795">1795</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1796" href="#t1796">1796</a></span><span class="t"><span class="str">        Returns:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1797" href="#t1797">1797</a></span><span class="t"><span class="str">            Transformed data</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1798" href="#t1798">1798</a></span><span class="t"><span class="str">        """</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1799" href="#t1799">1799</a></span><span class="t">        <span class="nam">axes</span> <span class="op">=</span> <span class="nam">tuple</span><span class="op">(</span><span class="op">-</span><span class="nam">i</span> <span class="op">-</span> <span class="num">1</span> <span class="key">for</span> <span class="nam">i</span> <span class="key">in</span> <span class="nam">range</span><span class="op">(</span><span class="nam">self</span><span class="op">.</span><span class="nam">ndim</span><span class="op">)</span><span class="op">[</span><span class="op">:</span><span class="op">:</span><span class="op">-</span><span class="num">1</span><span class="op">]</span><span class="op">)</span> <span class="key">if</span> <span class="nam">axes</span> <span class="key">is</span> <span class="key">None</span> <span class="key">else</span> <span class="nam">axes</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1800" href="#t1800">1800</a></span><span class="t">        <span class="nam">padding</span> <span class="op">=</span> <span class="op">(</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1801" href="#t1801">1801</a></span><span class="t">            <span class="op">[</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1802" href="#t1802">1802</a></span><span class="t">                <span class="num">1</span><span class="op">,</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1803" href="#t1803">1803</a></span><span class="t">            <span class="op">]</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1804" href="#t1804">1804</a></span><span class="t">            <span class="op">*</span> <span class="nam">self</span><span class="op">.</span><span class="nam">ndim</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1805" href="#t1805">1805</a></span><span class="t">            <span class="key">if</span> <span class="nam">padding</span> <span class="key">is</span> <span class="key">None</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1806" href="#t1806">1806</a></span><span class="t">            <span class="key">else</span> <span class="nam">padding</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1807" href="#t1807">1807</a></span><span class="t">        <span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1808" href="#t1808">1808</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1809" href="#t1809">1809</a></span><span class="t">        <span class="nam">result</span> <span class="op">=</span> <span class="op">[</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1810" href="#t1810">1810</a></span><span class="t">            <span class="key">None</span><span class="op">,</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1811" href="#t1811">1811</a></span><span class="t">        <span class="op">]</span> <span class="op">*</span> <span class="nam">self</span><span class="op">.</span><span class="nam">ncomponents</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1812" href="#t1812">1812</a></span><span class="t">        <span class="key">for</span> <span class="nam">comp</span> <span class="key">in</span> <span class="nam">self</span><span class="op">.</span><span class="nam">components</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1813" href="#t1813">1813</a></span><span class="t">            <span class="nam">i</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">index</span><span class="op">(</span><span class="nam">comp</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1814" href="#t1814">1814</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1815" href="#t1815">1815</a></span><span class="t">            <span class="nam">result</span><span class="op">[</span><span class="nam">i</span><span class="op">]</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">itransform_single_component</span><span class="op">(</span><span class="nam">u</span><span class="op">[</span><span class="nam">i</span><span class="op">]</span><span class="op">,</span> <span class="nam">axes</span><span class="op">=</span><span class="nam">axes</span><span class="op">,</span> <span class="nam">padding</span><span class="op">=</span><span class="nam">padding</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1816" href="#t1816">1816</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1817" href="#t1817">1817</a></span><span class="t">        <span class="key">return</span> <span class="nam">self</span><span class="op">.</span><span class="nam">xp</span><span class="op">.</span><span class="nam">stack</span><span class="op">(</span><span class="nam">result</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1818" href="#t1818">1818</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1819" href="#t1819">1819</a></span><span class="t">    <span class="key">def</span> <span class="nam">get_local_slice_of_1D_matrix</span><span class="op">(</span><span class="nam">self</span><span class="op">,</span> <span class="nam">M</span><span class="op">,</span> <span class="nam">axis</span><span class="op">)</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1820" href="#t1820">1820</a></span><span class="t">        <span class="str">"""</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1821" href="#t1821">1821</a></span><span class="t"><span class="str">        Get the local version of a 1D matrix. When using distributed FFTs, each rank will carry only a subset of modes,</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1822" href="#t1822">1822</a></span><span class="t"><span class="str">        which you can sort out via the `SpectralHelper.local_slice` attribute. When constructing a 1D matrix, you can</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1823" href="#t1823">1823</a></span><span class="t"><span class="str">        use this method to get the part corresponding to the modes carried by this rank.</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1824" href="#t1824">1824</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1825" href="#t1825">1825</a></span><span class="t"><span class="str">        Args:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1826" href="#t1826">1826</a></span><span class="t"><span class="str">            M (sparse matrix): Global 1D matrix you want to get the local version of</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1827" href="#t1827">1827</a></span><span class="t"><span class="str">            axis (int): Direction in which you want the local version. You will get the global matrix in other directions. This means slab decomposition only.</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1828" href="#t1828">1828</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1829" href="#t1829">1829</a></span><span class="t"><span class="str">        Returns:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1830" href="#t1830">1830</a></span><span class="t"><span class="str">            sparse local matrix</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1831" href="#t1831">1831</a></span><span class="t"><span class="str">        """</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1832" href="#t1832">1832</a></span><span class="t">        <span class="key">return</span> <span class="nam">M</span><span class="op">.</span><span class="nam">tocsc</span><span class="op">(</span><span class="op">)</span><span class="op">[</span><span class="nam">self</span><span class="op">.</span><span class="nam">local_slice</span><span class="op">[</span><span class="nam">axis</span><span class="op">]</span><span class="op">,</span> <span class="nam">self</span><span class="op">.</span><span class="nam">local_slice</span><span class="op">[</span><span class="nam">axis</span><span class="op">]</span><span class="op">]</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1833" href="#t1833">1833</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1834" href="#t1834">1834</a></span><span class="t">    <span class="key">def</span> <span class="nam">get_filter_matrix</span><span class="op">(</span><span class="nam">self</span><span class="op">,</span> <span class="nam">axis</span><span class="op">,</span> <span class="op">**</span><span class="nam">kwargs</span><span class="op">)</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t1835" href="#t1835">1835</a></span><span class="t">        <span class="str">"""</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1836" href="#t1836">1836</a></span><span class="t"><span class="str">        Get bandpass filter along `axis`. See the documentation `get_filter_matrix` in the 1D bases for what kwargs are</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1837" href="#t1837">1837</a></span><span class="t"><span class="str">        admissible.</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1838" href="#t1838">1838</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1839" href="#t1839">1839</a></span><span class="t"><span class="str">        Returns:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1840" href="#t1840">1840</a></span><span class="t"><span class="str">            sparse bandpass matrix</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1841" href="#t1841">1841</a></span><span class="t"><span class="str">        """</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1842" href="#t1842">1842</a></span><span class="t">        <span class="key">if</span> <span class="nam">self</span><span class="op">.</span><span class="nam">ndim</span> <span class="op">==</span> <span class="num">1</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1843" href="#t1843">1843</a></span><span class="t">            <span class="key">return</span> <span class="nam">self</span><span class="op">.</span><span class="nam">axes</span><span class="op">[</span><span class="num">0</span><span class="op">]</span><span class="op">.</span><span class="nam">get_filter_matrix</span><span class="op">(</span><span class="op">**</span><span class="nam">kwargs</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1844" href="#t1844">1844</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t1845" href="#t1845">1845</a></span><span class="t">        <span class="nam">mats</span> <span class="op">=</span> <span class="op">[</span><span class="nam">base</span><span class="op">.</span><span class="nam">get_Id</span><span class="op">(</span><span class="op">)</span> <span class="key">for</span> <span class="nam">base</span> <span class="key">in</span> <span class="nam">self</span><span class="op">.</span><span class="nam">axes</span><span class="op">]</span>&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t1846" href="#t1846">1846</a></span><span class="t">        <span class="nam">mats</span><span class="op">[</span><span class="nam">axis</span><span class="op">]</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">axes</span><span class="op">[</span><span class="nam">axis</span><span class="op">]</span><span class="op">.</span><span class="nam">get_filter_matrix</span><span class="op">(</span><span class="op">**</span><span class="nam">kwargs</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t1847" href="#t1847">1847</a></span><span class="t">        <span class="key">return</span> <span class="nam">self</span><span class="op">.</span><span class="nam">sparse_lib</span><span class="op">.</span><span class="nam">kron</span><span class="op">(</span><span class="op">*</span><span class="nam">mats</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1848" href="#t1848">1848</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1849" href="#t1849">1849</a></span><span class="t">    <span class="key">def</span> <span class="nam">get_differentiation_matrix</span><span class="op">(</span><span class="nam">self</span><span class="op">,</span> <span class="nam">axes</span><span class="op">,</span> <span class="op">**</span><span class="nam">kwargs</span><span class="op">)</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1850" href="#t1850">1850</a></span><span class="t">        <span class="str">"""</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1851" href="#t1851">1851</a></span><span class="t"><span class="str">        Get differentiation matrix along specified axis. `kwargs` are forwarded to the 1D base implementation.</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1852" href="#t1852">1852</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1853" href="#t1853">1853</a></span><span class="t"><span class="str">        Args:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1854" href="#t1854">1854</a></span><span class="t"><span class="str">            axes (tuple): Axes along which to differentiate.</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1855" href="#t1855">1855</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1856" href="#t1856">1856</a></span><span class="t"><span class="str">        Returns:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1857" href="#t1857">1857</a></span><span class="t"><span class="str">            sparse differentiation matrix</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1858" href="#t1858">1858</a></span><span class="t"><span class="str">        """</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1859" href="#t1859">1859</a></span><span class="t">        <span class="nam">sp</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">sparse_lib</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1860" href="#t1860">1860</a></span><span class="t">        <span class="nam">ndim</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">ndim</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1861" href="#t1861">1861</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1862" href="#t1862">1862</a></span><span class="t">        <span class="key">if</span> <span class="nam">ndim</span> <span class="op">==</span> <span class="num">1</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1863" href="#t1863">1863</a></span><span class="t">            <span class="nam">D</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">axes</span><span class="op">[</span><span class="num">0</span><span class="op">]</span><span class="op">.</span><span class="nam">get_differentiation_matrix</span><span class="op">(</span><span class="op">**</span><span class="nam">kwargs</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1864" href="#t1864">1864</a></span><span class="t">        <span class="key">elif</span> <span class="nam">ndim</span> <span class="op">==</span> <span class="num">2</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1865" href="#t1865">1865</a></span><span class="t">            <span class="key">for</span> <span class="nam">axis</span> <span class="key">in</span> <span class="nam">axes</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1866" href="#t1866">1866</a></span><span class="t">                <span class="nam">axis2</span> <span class="op">=</span> <span class="op">(</span><span class="nam">axis</span> <span class="op">+</span> <span class="num">1</span><span class="op">)</span> <span class="op">%</span> <span class="nam">ndim</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1867" href="#t1867">1867</a></span><span class="t">                <span class="nam">D1D</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">axes</span><span class="op">[</span><span class="nam">axis</span><span class="op">]</span><span class="op">.</span><span class="nam">get_differentiation_matrix</span><span class="op">(</span><span class="op">**</span><span class="nam">kwargs</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1868" href="#t1868">1868</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1869" href="#t1869">1869</a></span><span class="t">                <span class="key">if</span> <span class="nam">len</span><span class="op">(</span><span class="nam">axes</span><span class="op">)</span> <span class="op">></span> <span class="num">1</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1870" href="#t1870">1870</a></span><span class="t">                    <span class="nam">I1D</span> <span class="op">=</span> <span class="nam">sp</span><span class="op">.</span><span class="nam">eye</span><span class="op">(</span><span class="nam">self</span><span class="op">.</span><span class="nam">axes</span><span class="op">[</span><span class="nam">axis2</span><span class="op">]</span><span class="op">.</span><span class="nam">N</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1871" href="#t1871">1871</a></span><span class="t">                <span class="key">else</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1872" href="#t1872">1872</a></span><span class="t">                    <span class="nam">I1D</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">axes</span><span class="op">[</span><span class="nam">axis2</span><span class="op">]</span><span class="op">.</span><span class="nam">get_Id</span><span class="op">(</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1873" href="#t1873">1873</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1874" href="#t1874">1874</a></span><span class="t">                <span class="nam">mats</span> <span class="op">=</span> <span class="op">[</span><span class="key">None</span><span class="op">]</span> <span class="op">*</span> <span class="nam">ndim</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1875" href="#t1875">1875</a></span><span class="t">                <span class="nam">mats</span><span class="op">[</span><span class="nam">axis</span><span class="op">]</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">get_local_slice_of_1D_matrix</span><span class="op">(</span><span class="nam">D1D</span><span class="op">,</span> <span class="nam">axis</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1876" href="#t1876">1876</a></span><span class="t">                <span class="nam">mats</span><span class="op">[</span><span class="nam">axis2</span><span class="op">]</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">get_local_slice_of_1D_matrix</span><span class="op">(</span><span class="nam">I1D</span><span class="op">,</span> <span class="nam">axis2</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1877" href="#t1877">1877</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1878" href="#t1878">1878</a></span><span class="t">                <span class="key">if</span> <span class="nam">axis</span> <span class="op">==</span> <span class="nam">axes</span><span class="op">[</span><span class="num">0</span><span class="op">]</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1879" href="#t1879">1879</a></span><span class="t">                    <span class="nam">D</span> <span class="op">=</span> <span class="nam">sp</span><span class="op">.</span><span class="nam">kron</span><span class="op">(</span><span class="op">*</span><span class="nam">mats</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1880" href="#t1880">1880</a></span><span class="t">                <span class="key">else</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1881" href="#t1881">1881</a></span><span class="t">                    <span class="nam">D</span> <span class="op">=</span> <span class="nam">D</span> <span class="op">@</span> <span class="nam">sp</span><span class="op">.</span><span class="nam">kron</span><span class="op">(</span><span class="op">*</span><span class="nam">mats</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1882" href="#t1882">1882</a></span><span class="t">        <span class="key">else</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t1883" href="#t1883">1883</a></span><span class="t">            <span class="key">raise</span> <span class="nam">NotImplementedError</span><span class="op">(</span><span class="fst">f'</span><span class="fst">Differentiation matrix not implemented for </span><span class="op">{</span><span class="nam">ndim</span><span class="op">}</span><span class="fst"> dimension!</span><span class="fst">'</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1884" href="#t1884">1884</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1885" href="#t1885">1885</a></span><span class="t">        <span class="key">return</span> <span class="nam">D</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1886" href="#t1886">1886</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1887" href="#t1887">1887</a></span><span class="t">    <span class="key">def</span> <span class="nam">get_integration_matrix</span><span class="op">(</span><span class="nam">self</span><span class="op">,</span> <span class="nam">axes</span><span class="op">)</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1888" href="#t1888">1888</a></span><span class="t">        <span class="str">"""</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1889" href="#t1889">1889</a></span><span class="t"><span class="str">        Get integration matrix to integrate along specified axis.</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1890" href="#t1890">1890</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1891" href="#t1891">1891</a></span><span class="t"><span class="str">        Args:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1892" href="#t1892">1892</a></span><span class="t"><span class="str">            axes (tuple): Axes along which to integrate over.</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1893" href="#t1893">1893</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1894" href="#t1894">1894</a></span><span class="t"><span class="str">        Returns:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1895" href="#t1895">1895</a></span><span class="t"><span class="str">            sparse integration matrix</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1896" href="#t1896">1896</a></span><span class="t"><span class="str">        """</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1897" href="#t1897">1897</a></span><span class="t">        <span class="nam">sp</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">sparse_lib</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1898" href="#t1898">1898</a></span><span class="t">        <span class="nam">ndim</span> <span class="op">=</span> <span class="nam">len</span><span class="op">(</span><span class="nam">self</span><span class="op">.</span><span class="nam">axes</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1899" href="#t1899">1899</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1900" href="#t1900">1900</a></span><span class="t">        <span class="key">if</span> <span class="nam">ndim</span> <span class="op">==</span> <span class="num">1</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1901" href="#t1901">1901</a></span><span class="t">            <span class="nam">S</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">axes</span><span class="op">[</span><span class="num">0</span><span class="op">]</span><span class="op">.</span><span class="nam">get_integration_matrix</span><span class="op">(</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1902" href="#t1902">1902</a></span><span class="t">        <span class="key">elif</span> <span class="nam">ndim</span> <span class="op">==</span> <span class="num">2</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1903" href="#t1903">1903</a></span><span class="t">            <span class="key">for</span> <span class="nam">axis</span> <span class="key">in</span> <span class="nam">axes</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1904" href="#t1904">1904</a></span><span class="t">                <span class="nam">axis2</span> <span class="op">=</span> <span class="op">(</span><span class="nam">axis</span> <span class="op">+</span> <span class="num">1</span><span class="op">)</span> <span class="op">%</span> <span class="nam">ndim</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1905" href="#t1905">1905</a></span><span class="t">                <span class="nam">S1D</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">axes</span><span class="op">[</span><span class="nam">axis</span><span class="op">]</span><span class="op">.</span><span class="nam">get_integration_matrix</span><span class="op">(</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1906" href="#t1906">1906</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1907" href="#t1907">1907</a></span><span class="t">                <span class="key">if</span> <span class="nam">len</span><span class="op">(</span><span class="nam">axes</span><span class="op">)</span> <span class="op">></span> <span class="num">1</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1908" href="#t1908">1908</a></span><span class="t">                    <span class="nam">I1D</span> <span class="op">=</span> <span class="nam">sp</span><span class="op">.</span><span class="nam">eye</span><span class="op">(</span><span class="nam">self</span><span class="op">.</span><span class="nam">axes</span><span class="op">[</span><span class="nam">axis2</span><span class="op">]</span><span class="op">.</span><span class="nam">N</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1909" href="#t1909">1909</a></span><span class="t">                <span class="key">else</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1910" href="#t1910">1910</a></span><span class="t">                    <span class="nam">I1D</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">axes</span><span class="op">[</span><span class="nam">axis2</span><span class="op">]</span><span class="op">.</span><span class="nam">get_Id</span><span class="op">(</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1911" href="#t1911">1911</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1912" href="#t1912">1912</a></span><span class="t">                <span class="nam">mats</span> <span class="op">=</span> <span class="op">[</span><span class="key">None</span><span class="op">]</span> <span class="op">*</span> <span class="nam">ndim</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1913" href="#t1913">1913</a></span><span class="t">                <span class="nam">mats</span><span class="op">[</span><span class="nam">axis</span><span class="op">]</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">get_local_slice_of_1D_matrix</span><span class="op">(</span><span class="nam">S1D</span><span class="op">,</span> <span class="nam">axis</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1914" href="#t1914">1914</a></span><span class="t">                <span class="nam">mats</span><span class="op">[</span><span class="nam">axis2</span><span class="op">]</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">get_local_slice_of_1D_matrix</span><span class="op">(</span><span class="nam">I1D</span><span class="op">,</span> <span class="nam">axis2</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1915" href="#t1915">1915</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1916" href="#t1916">1916</a></span><span class="t">                <span class="key">if</span> <span class="nam">axis</span> <span class="op">==</span> <span class="nam">axes</span><span class="op">[</span><span class="num">0</span><span class="op">]</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1917" href="#t1917">1917</a></span><span class="t">                    <span class="nam">S</span> <span class="op">=</span> <span class="nam">sp</span><span class="op">.</span><span class="nam">kron</span><span class="op">(</span><span class="op">*</span><span class="nam">mats</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1918" href="#t1918">1918</a></span><span class="t">                <span class="key">else</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1919" href="#t1919">1919</a></span><span class="t">                    <span class="nam">S</span> <span class="op">=</span> <span class="nam">S</span> <span class="op">@</span> <span class="nam">sp</span><span class="op">.</span><span class="nam">kron</span><span class="op">(</span><span class="op">*</span><span class="nam">mats</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1920" href="#t1920">1920</a></span><span class="t">        <span class="key">else</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t1921" href="#t1921">1921</a></span><span class="t">            <span class="key">raise</span> <span class="nam">NotImplementedError</span><span class="op">(</span><span class="fst">f'</span><span class="fst">Integration matrix not implemented for </span><span class="op">{</span><span class="nam">ndim</span><span class="op">}</span><span class="fst"> dimension!</span><span class="fst">'</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1922" href="#t1922">1922</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1923" href="#t1923">1923</a></span><span class="t">        <span class="key">return</span> <span class="nam">S</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1924" href="#t1924">1924</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1925" href="#t1925">1925</a></span><span class="t">    <span class="key">def</span> <span class="nam">get_Id</span><span class="op">(</span><span class="nam">self</span><span class="op">)</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1926" href="#t1926">1926</a></span><span class="t">        <span class="str">"""</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1927" href="#t1927">1927</a></span><span class="t"><span class="str">        Get identity matrix</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1928" href="#t1928">1928</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1929" href="#t1929">1929</a></span><span class="t"><span class="str">        Returns:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1930" href="#t1930">1930</a></span><span class="t"><span class="str">            sparse identity matrix</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1931" href="#t1931">1931</a></span><span class="t"><span class="str">        """</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1932" href="#t1932">1932</a></span><span class="t">        <span class="nam">sp</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">sparse_lib</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1933" href="#t1933">1933</a></span><span class="t">        <span class="nam">ndim</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">ndim</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1934" href="#t1934">1934</a></span><span class="t">        <span class="nam">I</span> <span class="op">=</span> <span class="nam">sp</span><span class="op">.</span><span class="nam">eye</span><span class="op">(</span><span class="nam">np</span><span class="op">.</span><span class="nam">prod</span><span class="op">(</span><span class="nam">self</span><span class="op">.</span><span class="nam">init</span><span class="op">[</span><span class="num">0</span><span class="op">]</span><span class="op">[</span><span class="num">1</span><span class="op">:</span><span class="op">]</span><span class="op">)</span><span class="op">,</span> <span class="nam">dtype</span><span class="op">=</span><span class="nam">complex</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1935" href="#t1935">1935</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1936" href="#t1936">1936</a></span><span class="t">        <span class="key">if</span> <span class="nam">ndim</span> <span class="op">==</span> <span class="num">1</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1937" href="#t1937">1937</a></span><span class="t">            <span class="nam">I</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">axes</span><span class="op">[</span><span class="num">0</span><span class="op">]</span><span class="op">.</span><span class="nam">get_Id</span><span class="op">(</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1938" href="#t1938">1938</a></span><span class="t">        <span class="key">elif</span> <span class="nam">ndim</span> <span class="op">==</span> <span class="num">2</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1939" href="#t1939">1939</a></span><span class="t">            <span class="key">for</span> <span class="nam">axis</span> <span class="key">in</span> <span class="nam">range</span><span class="op">(</span><span class="nam">ndim</span><span class="op">)</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1940" href="#t1940">1940</a></span><span class="t">                <span class="nam">axis2</span> <span class="op">=</span> <span class="op">(</span><span class="nam">axis</span> <span class="op">+</span> <span class="num">1</span><span class="op">)</span> <span class="op">%</span> <span class="nam">ndim</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1941" href="#t1941">1941</a></span><span class="t">                <span class="nam">I1D</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">axes</span><span class="op">[</span><span class="nam">axis</span><span class="op">]</span><span class="op">.</span><span class="nam">get_Id</span><span class="op">(</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1942" href="#t1942">1942</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1943" href="#t1943">1943</a></span><span class="t">                <span class="nam">I1D2</span> <span class="op">=</span> <span class="nam">sp</span><span class="op">.</span><span class="nam">eye</span><span class="op">(</span><span class="nam">self</span><span class="op">.</span><span class="nam">axes</span><span class="op">[</span><span class="nam">axis2</span><span class="op">]</span><span class="op">.</span><span class="nam">N</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1944" href="#t1944">1944</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1945" href="#t1945">1945</a></span><span class="t">                <span class="nam">mats</span> <span class="op">=</span> <span class="op">[</span><span class="key">None</span><span class="op">]</span> <span class="op">*</span> <span class="nam">ndim</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1946" href="#t1946">1946</a></span><span class="t">                <span class="nam">mats</span><span class="op">[</span><span class="nam">axis</span><span class="op">]</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">get_local_slice_of_1D_matrix</span><span class="op">(</span><span class="nam">I1D</span><span class="op">,</span> <span class="nam">axis</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1947" href="#t1947">1947</a></span><span class="t">                <span class="nam">mats</span><span class="op">[</span><span class="nam">axis2</span><span class="op">]</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">get_local_slice_of_1D_matrix</span><span class="op">(</span><span class="nam">I1D2</span><span class="op">,</span> <span class="nam">axis2</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1948" href="#t1948">1948</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1949" href="#t1949">1949</a></span><span class="t">                <span class="nam">I</span> <span class="op">=</span> <span class="nam">I</span> <span class="op">@</span> <span class="nam">sp</span><span class="op">.</span><span class="nam">kron</span><span class="op">(</span><span class="op">*</span><span class="nam">mats</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1950" href="#t1950">1950</a></span><span class="t">        <span class="key">else</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t1951" href="#t1951">1951</a></span><span class="t">            <span class="key">raise</span> <span class="nam">NotImplementedError</span><span class="op">(</span><span class="fst">f'</span><span class="fst">Identity matrix not implemented for </span><span class="op">{</span><span class="nam">ndim</span><span class="op">}</span><span class="fst"> dimension!</span><span class="fst">'</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1952" href="#t1952">1952</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1953" href="#t1953">1953</a></span><span class="t">        <span class="key">return</span> <span class="nam">I</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1954" href="#t1954">1954</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1955" href="#t1955">1955</a></span><span class="t">    <span class="key">def</span> <span class="nam">get_Dirichlet_recombination_matrix</span><span class="op">(</span><span class="nam">self</span><span class="op">,</span> <span class="nam">axis</span><span class="op">=</span><span class="op">-</span><span class="num">1</span><span class="op">)</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1956" href="#t1956">1956</a></span><span class="t">        <span class="str">"""</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1957" href="#t1957">1957</a></span><span class="t"><span class="str">        Get Dirichlet recombination matrix along axis. Not that it only makes sense in directions discretized with variations of Chebychev bases.</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1958" href="#t1958">1958</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1959" href="#t1959">1959</a></span><span class="t"><span class="str">        Args:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1960" href="#t1960">1960</a></span><span class="t"><span class="str">            axis (int): Axis you discretized with Chebychev</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1961" href="#t1961">1961</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1962" href="#t1962">1962</a></span><span class="t"><span class="str">        Returns:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1963" href="#t1963">1963</a></span><span class="t"><span class="str">            sparse matrix</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1964" href="#t1964">1964</a></span><span class="t"><span class="str">        """</span>&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t1965" href="#t1965">1965</a></span><span class="t">        <span class="nam">sp</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">sparse_lib</span>&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t1966" href="#t1966">1966</a></span><span class="t">        <span class="nam">ndim</span> <span class="op">=</span> <span class="nam">len</span><span class="op">(</span><span class="nam">self</span><span class="op">.</span><span class="nam">axes</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1967" href="#t1967">1967</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t1968" href="#t1968">1968</a></span><span class="t">        <span class="key">if</span> <span class="nam">ndim</span> <span class="op">==</span> <span class="num">1</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t1969" href="#t1969">1969</a></span><span class="t">            <span class="nam">C</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">axes</span><span class="op">[</span><span class="num">0</span><span class="op">]</span><span class="op">.</span><span class="nam">get_Dirichlet_recombination_matrix</span><span class="op">(</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t1970" href="#t1970">1970</a></span><span class="t">        <span class="key">elif</span> <span class="nam">ndim</span> <span class="op">==</span> <span class="num">2</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t1971" href="#t1971">1971</a></span><span class="t">            <span class="nam">axis2</span> <span class="op">=</span> <span class="op">(</span><span class="nam">axis</span> <span class="op">+</span> <span class="num">1</span><span class="op">)</span> <span class="op">%</span> <span class="nam">ndim</span>&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t1972" href="#t1972">1972</a></span><span class="t">            <span class="nam">C1D</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">axes</span><span class="op">[</span><span class="nam">axis</span><span class="op">]</span><span class="op">.</span><span class="nam">get_Dirichlet_recombination_matrix</span><span class="op">(</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1973" href="#t1973">1973</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t1974" href="#t1974">1974</a></span><span class="t">            <span class="nam">I1D</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">axes</span><span class="op">[</span><span class="nam">axis2</span><span class="op">]</span><span class="op">.</span><span class="nam">get_Id</span><span class="op">(</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1975" href="#t1975">1975</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t1976" href="#t1976">1976</a></span><span class="t">            <span class="nam">mats</span> <span class="op">=</span> <span class="op">[</span><span class="key">None</span><span class="op">]</span> <span class="op">*</span> <span class="nam">ndim</span>&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t1977" href="#t1977">1977</a></span><span class="t">            <span class="nam">mats</span><span class="op">[</span><span class="nam">axis</span><span class="op">]</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">get_local_slice_of_1D_matrix</span><span class="op">(</span><span class="nam">C1D</span><span class="op">,</span> <span class="nam">axis</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t1978" href="#t1978">1978</a></span><span class="t">            <span class="nam">mats</span><span class="op">[</span><span class="nam">axis2</span><span class="op">]</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">get_local_slice_of_1D_matrix</span><span class="op">(</span><span class="nam">I1D</span><span class="op">,</span> <span class="nam">axis2</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1979" href="#t1979">1979</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t1980" href="#t1980">1980</a></span><span class="t">            <span class="nam">C</span> <span class="op">=</span> <span class="nam">sp</span><span class="op">.</span><span class="nam">kron</span><span class="op">(</span><span class="op">*</span><span class="nam">mats</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1981" href="#t1981">1981</a></span><span class="t">        <span class="key">else</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t1982" href="#t1982">1982</a></span><span class="t">            <span class="key">raise</span> <span class="nam">NotImplementedError</span><span class="op">(</span><span class="fst">f'</span><span class="fst">Basis change matrix not implemented for </span><span class="op">{</span><span class="nam">ndim</span><span class="op">}</span><span class="fst"> dimension!</span><span class="fst">'</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1983" href="#t1983">1983</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t1984" href="#t1984">1984</a></span><span class="t">        <span class="key">return</span> <span class="nam">C</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1985" href="#t1985">1985</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1986" href="#t1986">1986</a></span><span class="t">    <span class="key">def</span> <span class="nam">get_basis_change_matrix</span><span class="op">(</span><span class="nam">self</span><span class="op">,</span> <span class="nam">axes</span><span class="op">=</span><span class="key">None</span><span class="op">,</span> <span class="op">**</span><span class="nam">kwargs</span><span class="op">)</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t1987" href="#t1987">1987</a></span><span class="t">        <span class="str">"""</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1988" href="#t1988">1988</a></span><span class="t"><span class="str">        Some spectral bases do a change between bases while differentiating. This method returns matrices that changes the basis to whatever you want.</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1989" href="#t1989">1989</a></span><span class="t"><span class="str">        Refer to the methods of the same name of the 1D bases to learn what parameters you need to pass here as `kwargs`.</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1990" href="#t1990">1990</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1991" href="#t1991">1991</a></span><span class="t"><span class="str">        Args:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1992" href="#t1992">1992</a></span><span class="t"><span class="str">            axes (tuple): Axes along which to change basis.</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1993" href="#t1993">1993</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1994" href="#t1994">1994</a></span><span class="t"><span class="str">        Returns:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1995" href="#t1995">1995</a></span><span class="t"><span class="str">            sparse basis change matrix</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1996" href="#t1996">1996</a></span><span class="t"><span class="str">        """</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1997" href="#t1997">1997</a></span><span class="t">        <span class="nam">axes</span> <span class="op">=</span> <span class="nam">tuple</span><span class="op">(</span><span class="op">-</span><span class="nam">i</span> <span class="op">-</span> <span class="num">1</span> <span class="key">for</span> <span class="nam">i</span> <span class="key">in</span> <span class="nam">range</span><span class="op">(</span><span class="nam">self</span><span class="op">.</span><span class="nam">ndim</span><span class="op">)</span><span class="op">)</span> <span class="key">if</span> <span class="nam">axes</span> <span class="key">is</span> <span class="key">None</span> <span class="key">else</span> <span class="nam">axes</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t1998" href="#t1998">1998</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t1999" href="#t1999">1999</a></span><span class="t">        <span class="nam">sp</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">sparse_lib</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t2000" href="#t2000">2000</a></span><span class="t">        <span class="nam">ndim</span> <span class="op">=</span> <span class="nam">len</span><span class="op">(</span><span class="nam">self</span><span class="op">.</span><span class="nam">axes</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t2001" href="#t2001">2001</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t2002" href="#t2002">2002</a></span><span class="t">        <span class="key">if</span> <span class="nam">ndim</span> <span class="op">==</span> <span class="num">1</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t2003" href="#t2003">2003</a></span><span class="t">            <span class="nam">C</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">axes</span><span class="op">[</span><span class="num">0</span><span class="op">]</span><span class="op">.</span><span class="nam">get_basis_change_matrix</span><span class="op">(</span><span class="op">**</span><span class="nam">kwargs</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t2004" href="#t2004">2004</a></span><span class="t">        <span class="key">elif</span> <span class="nam">ndim</span> <span class="op">==</span> <span class="num">2</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t2005" href="#t2005">2005</a></span><span class="t">            <span class="key">for</span> <span class="nam">axis</span> <span class="key">in</span> <span class="nam">axes</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t2006" href="#t2006">2006</a></span><span class="t">                <span class="nam">axis2</span> <span class="op">=</span> <span class="op">(</span><span class="nam">axis</span> <span class="op">+</span> <span class="num">1</span><span class="op">)</span> <span class="op">%</span> <span class="nam">ndim</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t2007" href="#t2007">2007</a></span><span class="t">                <span class="nam">C1D</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">axes</span><span class="op">[</span><span class="nam">axis</span><span class="op">]</span><span class="op">.</span><span class="nam">get_basis_change_matrix</span><span class="op">(</span><span class="op">**</span><span class="nam">kwargs</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t2008" href="#t2008">2008</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t2009" href="#t2009">2009</a></span><span class="t">                <span class="key">if</span> <span class="nam">len</span><span class="op">(</span><span class="nam">axes</span><span class="op">)</span> <span class="op">></span> <span class="num">1</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t2010" href="#t2010">2010</a></span><span class="t">                    <span class="nam">I1D</span> <span class="op">=</span> <span class="nam">sp</span><span class="op">.</span><span class="nam">eye</span><span class="op">(</span><span class="nam">self</span><span class="op">.</span><span class="nam">axes</span><span class="op">[</span><span class="nam">axis2</span><span class="op">]</span><span class="op">.</span><span class="nam">N</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t2011" href="#t2011">2011</a></span><span class="t">                <span class="key">else</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t2012" href="#t2012">2012</a></span><span class="t">                    <span class="nam">I1D</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">axes</span><span class="op">[</span><span class="nam">axis2</span><span class="op">]</span><span class="op">.</span><span class="nam">get_Id</span><span class="op">(</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t2013" href="#t2013">2013</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t2014" href="#t2014">2014</a></span><span class="t">                <span class="nam">mats</span> <span class="op">=</span> <span class="op">[</span><span class="key">None</span><span class="op">]</span> <span class="op">*</span> <span class="nam">ndim</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t2015" href="#t2015">2015</a></span><span class="t">                <span class="nam">mats</span><span class="op">[</span><span class="nam">axis</span><span class="op">]</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">get_local_slice_of_1D_matrix</span><span class="op">(</span><span class="nam">C1D</span><span class="op">,</span> <span class="nam">axis</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t2016" href="#t2016">2016</a></span><span class="t">                <span class="nam">mats</span><span class="op">[</span><span class="nam">axis2</span><span class="op">]</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">get_local_slice_of_1D_matrix</span><span class="op">(</span><span class="nam">I1D</span><span class="op">,</span> <span class="nam">axis2</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t2017" href="#t2017">2017</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t2018" href="#t2018">2018</a></span><span class="t">                <span class="key">if</span> <span class="nam">axis</span> <span class="op">==</span> <span class="nam">axes</span><span class="op">[</span><span class="num">0</span><span class="op">]</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t2019" href="#t2019">2019</a></span><span class="t">                    <span class="nam">C</span> <span class="op">=</span> <span class="nam">sp</span><span class="op">.</span><span class="nam">kron</span><span class="op">(</span><span class="op">*</span><span class="nam">mats</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t2020" href="#t2020">2020</a></span><span class="t">                <span class="key">else</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t2021" href="#t2021">2021</a></span><span class="t">                    <span class="nam">C</span> <span class="op">=</span> <span class="nam">C</span> <span class="op">@</span> <span class="nam">sp</span><span class="op">.</span><span class="nam">kron</span><span class="op">(</span><span class="op">*</span><span class="nam">mats</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t2022" href="#t2022">2022</a></span><span class="t">        <span class="key">else</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t2023" href="#t2023">2023</a></span><span class="t">            <span class="key">raise</span> <span class="nam">NotImplementedError</span><span class="op">(</span><span class="fst">f'</span><span class="fst">Basis change matrix not implemented for </span><span class="op">{</span><span class="nam">ndim</span><span class="op">}</span><span class="fst"> dimension!</span><span class="fst">'</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t2024" href="#t2024">2024</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t2025" href="#t2025">2025</a></span><span class="t">        <span class="key">return</span> <span class="nam">C</span>&nbsp;</span><span class="r"></span></p>
+</main>
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+    <div class="content">
+        <p>
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index d9aa809f11..b5d24f83d8 100644
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@@ -60,12 +60,12 @@ <h2>
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     <p class="run"><span class="n"><a id="t61" href="#t61">61</a></span><span class="t">        <span class="nam">plt</span><span class="op">.</span><span class="nam">gca</span><span class="op">(</span><span class="op">)</span><span class="op">.</span><span class="nam">set</span><span class="op">(</span>&nbsp;</span><span class="r"></span></p>
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+    <p class="pln"><span class="n"><a id="t62" href="#t62">62</a></span><span class="t">            <span class="nam">xlabel</span><span class="op">=</span><span class="str">"Computation Time [s]"</span> <span class="key">if</span> <span class="str">"cost"</span> <span class="key">in</span> <span class="nam">figName</span> <span class="key">else</span> <span class="str">r"$\Delta {t}$"</span><span class="op">,</span>&nbsp;</span><span class="r"></span></p>
     <p class="pln"><span class="n"><a id="t63" href="#t63">63</a></span><span class="t">            <span class="nam">ylabel</span><span class="op">=</span><span class="str">r"$L_2$ error at $T$"</span><span class="op">,</span>&nbsp;</span><span class="r"></span></p>
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     <p class="run"><span class="n"><a id="t65" href="#t65">65</a></span><span class="t">        <span class="nam">plt</span><span class="op">.</span><span class="nam">legend</span><span class="op">(</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
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             &nbsp; &nbsp; &nbsp;
             <a class="nav" href="https://coverage.readthedocs.io/en/7.6.1">coverage.py v7.6.1</a>,
-            created at 2024-09-19 09:13 +0000
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+            created at 2024-09-20 16:55 +0000
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+            created at 2024-09-20 16:55 +0000
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diff --git a/coverage/z_5893b65af8042ae2_accuracy_check_py.html b/coverage/z_5893b65af8042ae2_accuracy_check_py.html
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+            created at 2024-09-20 16:55 +0000
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+            created at 2024-09-20 16:55 +0000
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+            created at 2024-09-20 16:55 +0000
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diff --git a/coverage/z_5893b65af8042ae2_collocation_adaptivity_py.html b/coverage/z_5893b65af8042ae2_collocation_adaptivity_py.html
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+            created at 2024-09-20 16:55 +0000
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+            created at 2024-09-20 16:55 +0000
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+            created at 2024-09-20 16:55 +0000
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             &nbsp; &nbsp; &nbsp;
             <a class="nav" href="https://coverage.readthedocs.io/en/7.6.1">coverage.py v7.6.1</a>,
-            created at 2024-09-19 09:13 +0000
+            created at 2024-09-20 16:55 +0000
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             &nbsp; &nbsp; &nbsp;
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-            created at 2024-09-19 09:13 +0000
+            created at 2024-09-20 16:55 +0000
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             &nbsp; &nbsp; &nbsp;
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-            created at 2024-09-19 09:13 +0000
+            created at 2024-09-20 16:55 +0000
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-            created at 2024-09-19 09:13 +0000
+            created at 2024-09-20 16:55 +0000
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             &nbsp; &nbsp; &nbsp;
             <a class="nav" href="https://coverage.readthedocs.io/en/7.6.1">coverage.py v7.6.1</a>,
-            created at 2024-09-19 09:13 +0000
+            created at 2024-09-20 16:55 +0000
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-            created at 2024-09-19 09:13 +0000
+            created at 2024-09-20 16:55 +0000
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index 2081486880..a92eef0f36 100644
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             &nbsp; &nbsp; &nbsp;
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-            created at 2024-09-19 09:13 +0000
+            created at 2024-09-20 16:55 +0000
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-            created at 2024-09-19 09:13 +0000
+            created at 2024-09-20 16:55 +0000
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diff --git a/coverage/z_850829697a149033_standard_integrators_py.html b/coverage/z_850829697a149033_standard_integrators_py.html
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             &nbsp; &nbsp; &nbsp;
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-            created at 2024-09-19 09:13 +0000
+            created at 2024-09-20 16:55 +0000
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             &nbsp; &nbsp; &nbsp;
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-            created at 2024-09-19 09:13 +0000
+            created at 2024-09-20 16:55 +0000
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diff --git a/coverage/z_850829697a149033_unflatten_py.html b/coverage/z_850829697a149033_unflatten_py.html
index 439322bf7e..da50229eeb 100644
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             &nbsp; &nbsp; &nbsp;
             <a class="nav" href="https://coverage.readthedocs.io/en/7.6.1">coverage.py v7.6.1</a>,
-            created at 2024-09-19 09:13 +0000
+            created at 2024-09-20 16:55 +0000
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@@ -98,7 +98,7 @@ <h2>
             <a class="nav" href="z_91faa57f8583c837_generic_MPIFFT_Laplacian_py.html">&#xbb; next</a>
             &nbsp; &nbsp; &nbsp;
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-            created at 2024-09-19 09:13 +0000
+            created at 2024-09-20 16:55 +0000
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diff --git a/coverage/z_8d0d803e4e125e5c_base_transfer_py.html b/coverage/z_8d0d803e4e125e5c_base_transfer_py.html
index ded19a16e8..ca882e03d2 100644
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@@ -65,7 +65,7 @@ <h2>
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             &nbsp; &nbsp; &nbsp;
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-            created at 2024-09-19 09:13 +0000
+            created at 2024-09-20 16:55 +0000
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             <a class="nav" href="z_8d0d803e4e125e5c_collocation_py.html">&#xbb; next</a>
             &nbsp; &nbsp; &nbsp;
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-            created at 2024-09-19 09:13 +0000
+            created at 2024-09-20 16:55 +0000
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diff --git a/coverage/z_8d0d803e4e125e5c_collocation_py.html b/coverage/z_8d0d803e4e125e5c_collocation_py.html
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@@ -65,7 +65,7 @@ <h2>
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             &nbsp; &nbsp; &nbsp;
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-            created at 2024-09-19 09:13 +0000
+            created at 2024-09-20 16:55 +0000
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-            created at 2024-09-19 09:13 +0000
+            created at 2024-09-20 16:55 +0000
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diff --git a/coverage/z_8d0d803e4e125e5c_common_py.html b/coverage/z_8d0d803e4e125e5c_common_py.html
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             &nbsp; &nbsp; &nbsp;
             <a class="nav" href="https://coverage.readthedocs.io/en/7.6.1">coverage.py v7.6.1</a>,
-            created at 2024-09-19 09:13 +0000
+            created at 2024-09-20 16:55 +0000
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             &nbsp; &nbsp; &nbsp;
             <a class="nav" href="https://coverage.readthedocs.io/en/7.6.1">coverage.py v7.6.1</a>,
-            created at 2024-09-19 09:13 +0000
+            created at 2024-09-20 16:55 +0000
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diff --git a/coverage/z_8d0d803e4e125e5c_controller_py.html b/coverage/z_8d0d803e4e125e5c_controller_py.html
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-            created at 2024-09-19 09:13 +0000
+            created at 2024-09-20 16:55 +0000
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-            created at 2024-09-19 09:13 +0000
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             <a class="nav" href="https://coverage.readthedocs.io/en/7.6.1">coverage.py v7.6.1</a>,
-            created at 2024-09-19 09:13 +0000
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-            created at 2024-09-19 09:13 +0000
+            created at 2024-09-20 16:55 +0000
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-            created at 2024-09-19 09:13 +0000
+            created at 2024-09-20 16:55 +0000
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-            created at 2024-09-19 09:13 +0000
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-            created at 2024-09-19 09:13 +0000
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-            created at 2024-09-19 09:13 +0000
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-            created at 2024-09-19 09:13 +0000
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index 0000000000..2cb5da1346
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+++ b/coverage/z_91faa57f8583c837_Burgers_py.html
@@ -0,0 +1,434 @@
+<!DOCTYPE html>
+<html lang="en">
+<head>
+    <meta http-equiv="Content-Type" content="text/html; charset=utf-8">
+    <title>Coverage for pySDC/implementations/problem_classes/Burgers.py: 95%</title>
+    <link rel="icon" sizes="32x32" href="favicon_32_cb_58284776.png">
+    <link rel="stylesheet" href="style_cb_8e611ae1.css" type="text/css">
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+<body class="pyfile">
+<header>
+    <div class="content">
+        <h1>
+            <span class="text">Coverage for </span><b>pySDC/implementations/problem_classes/Burgers.py</b>:
+            <span class="pc_cov">95%</span>
+        </h1>
+        <aside id="help_panel_wrapper">
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+            <span class="text">96 statements &nbsp;</span>
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+    </div>
+</header>
+<main id="source">
+    <p class="run"><span class="n"><a id="t1" href="#t1">1</a></span><span class="t"><span class="key">import</span> <span class="nam">numpy</span> <span class="key">as</span> <span class="nam">np</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t2" href="#t2">2</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t3" href="#t3">3</a></span><span class="t"><span class="key">from</span> <span class="nam">pySDC</span><span class="op">.</span><span class="nam">implementations</span><span class="op">.</span><span class="nam">datatype_classes</span><span class="op">.</span><span class="nam">mesh</span> <span class="key">import</span> <span class="nam">mesh</span><span class="op">,</span> <span class="nam">imex_mesh</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t4" href="#t4">4</a></span><span class="t"><span class="key">from</span> <span class="nam">pySDC</span><span class="op">.</span><span class="nam">implementations</span><span class="op">.</span><span class="nam">problem_classes</span><span class="op">.</span><span class="nam">generic_spectral</span> <span class="key">import</span> <span class="nam">GenericSpectralLinear</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t5" href="#t5">5</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t6" href="#t6">6</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t7" href="#t7">7</a></span><span class="t"><span class="key">class</span> <span class="nam">Burgers1D</span><span class="op">(</span><span class="nam">GenericSpectralLinear</span><span class="op">)</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t8" href="#t8">8</a></span><span class="t">    <span class="str">"""</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t9" href="#t9">9</a></span><span class="t"><span class="str">    See https://en.wikipedia.org/wiki/Burgers'_equation for the equation that is solved.</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t10" href="#t10">10</a></span><span class="t"><span class="str">    Discretization is done with a Chebychev method, which requires a first order derivative formulation.</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t11" href="#t11">11</a></span><span class="t"><span class="str">    Feel free to do a more efficient implementation using an ultraspherical method to avoid the first order business.</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t12" href="#t12">12</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t13" href="#t13">13</a></span><span class="t"><span class="str">    Parameters:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t14" href="#t14">14</a></span><span class="t"><span class="str">       N (int): Spatial resolution</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t15" href="#t15">15</a></span><span class="t"><span class="str">       epsilon (float): viscosity</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t16" href="#t16">16</a></span><span class="t"><span class="str">       BCl (float): Value at left boundary</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t17" href="#t17">17</a></span><span class="t"><span class="str">       BCr (float): Value at right boundary</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t18" href="#t18">18</a></span><span class="t"><span class="str">       f (int): Frequency of the initial conditions</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t19" href="#t19">19</a></span><span class="t"><span class="str">       mode (str): 'T2U' or 'T2T'. Use 'T2U' to get sparse differentiation matrices</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t20" href="#t20">20</a></span><span class="t"><span class="str">    """</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t21" href="#t21">21</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t22" href="#t22">22</a></span><span class="t">    <span class="nam">dtype_u</span> <span class="op">=</span> <span class="nam">mesh</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t23" href="#t23">23</a></span><span class="t">    <span class="nam">dtype_f</span> <span class="op">=</span> <span class="nam">imex_mesh</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t24" href="#t24">24</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t25" href="#t25">25</a></span><span class="t">    <span class="key">def</span> <span class="nam">__init__</span><span class="op">(</span><span class="nam">self</span><span class="op">,</span> <span class="nam">N</span><span class="op">=</span><span class="num">64</span><span class="op">,</span> <span class="nam">epsilon</span><span class="op">=</span><span class="num">0.1</span><span class="op">,</span> <span class="nam">BCl</span><span class="op">=</span><span class="num">1</span><span class="op">,</span> <span class="nam">BCr</span><span class="op">=</span><span class="op">-</span><span class="num">1</span><span class="op">,</span> <span class="nam">f</span><span class="op">=</span><span class="num">0</span><span class="op">,</span> <span class="nam">mode</span><span class="op">=</span><span class="str">'T2U'</span><span class="op">,</span> <span class="op">**</span><span class="nam">kwargs</span><span class="op">)</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t26" href="#t26">26</a></span><span class="t">        <span class="str">"""</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t27" href="#t27">27</a></span><span class="t"><span class="str">        Constructor. `kwargs` are forwarded to parent class constructor.</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t28" href="#t28">28</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t29" href="#t29">29</a></span><span class="t"><span class="str">        Args:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t30" href="#t30">30</a></span><span class="t"><span class="str">            N (int): Spatial resolution</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t31" href="#t31">31</a></span><span class="t"><span class="str">            epsilon (float): viscosity</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t32" href="#t32">32</a></span><span class="t"><span class="str">            BCl (float): Value at left boundary</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t33" href="#t33">33</a></span><span class="t"><span class="str">            BCr (float): Value at right boundary</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t34" href="#t34">34</a></span><span class="t"><span class="str">            f (int): Frequency of the initial conditions</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t35" href="#t35">35</a></span><span class="t"><span class="str">            mode (str): 'T2U' or 'T2T'. Use 'T2U' to get sparse differentiation matrices</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t36" href="#t36">36</a></span><span class="t"><span class="str">        """</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t37" href="#t37">37</a></span><span class="t">        <span class="nam">self</span><span class="op">.</span><span class="nam">_makeAttributeAndRegister</span><span class="op">(</span><span class="str">'N'</span><span class="op">,</span> <span class="str">'epsilon'</span><span class="op">,</span> <span class="str">'BCl'</span><span class="op">,</span> <span class="str">'BCr'</span><span class="op">,</span> <span class="str">'f'</span><span class="op">,</span> <span class="str">'mode'</span><span class="op">,</span> <span class="nam">localVars</span><span class="op">=</span><span class="nam">locals</span><span class="op">(</span><span class="op">)</span><span class="op">,</span> <span class="nam">readOnly</span><span class="op">=</span><span class="key">True</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t38" href="#t38">38</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t39" href="#t39">39</a></span><span class="t">        <span class="nam">bases</span> <span class="op">=</span> <span class="op">[</span><span class="op">{</span><span class="str">'base'</span><span class="op">:</span> <span class="str">'cheby'</span><span class="op">,</span> <span class="str">'N'</span><span class="op">:</span> <span class="nam">N</span><span class="op">}</span><span class="op">]</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t40" href="#t40">40</a></span><span class="t">        <span class="nam">components</span> <span class="op">=</span> <span class="op">[</span><span class="str">'u'</span><span class="op">,</span> <span class="str">'ux'</span><span class="op">]</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t41" href="#t41">41</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t42" href="#t42">42</a></span><span class="t">        <span class="nam">super</span><span class="op">(</span><span class="op">)</span><span class="op">.</span><span class="nam">__init__</span><span class="op">(</span><span class="nam">bases</span><span class="op">=</span><span class="nam">bases</span><span class="op">,</span> <span class="nam">components</span><span class="op">=</span><span class="nam">components</span><span class="op">,</span> <span class="nam">spectral_space</span><span class="op">=</span><span class="key">False</span><span class="op">,</span> <span class="op">**</span><span class="nam">kwargs</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t43" href="#t43">43</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t44" href="#t44">44</a></span><span class="t">        <span class="nam">self</span><span class="op">.</span><span class="nam">x</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">get_grid</span><span class="op">(</span><span class="op">)</span><span class="op">[</span><span class="num">0</span><span class="op">]</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t45" href="#t45">45</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t46" href="#t46">46</a></span><span class="t">        <span class="com"># prepare matrices</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t47" href="#t47">47</a></span><span class="t">        <span class="nam">Dx</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">get_differentiation_matrix</span><span class="op">(</span><span class="nam">axes</span><span class="op">=</span><span class="op">(</span><span class="num">0</span><span class="op">,</span><span class="op">)</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t48" href="#t48">48</a></span><span class="t">        <span class="nam">I</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">get_Id</span><span class="op">(</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t49" href="#t49">49</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t50" href="#t50">50</a></span><span class="t">        <span class="nam">T2U</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">get_basis_change_matrix</span><span class="op">(</span><span class="nam">conv</span><span class="op">=</span><span class="nam">mode</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t51" href="#t51">51</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t52" href="#t52">52</a></span><span class="t">        <span class="nam">self</span><span class="op">.</span><span class="nam">Dx</span> <span class="op">=</span> <span class="nam">Dx</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t53" href="#t53">53</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t54" href="#t54">54</a></span><span class="t">        <span class="com"># construct linear operator</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t55" href="#t55">55</a></span><span class="t">        <span class="nam">L_lhs</span> <span class="op">=</span> <span class="op">{</span><span class="str">'u'</span><span class="op">:</span> <span class="op">{</span><span class="str">'ux'</span><span class="op">:</span> <span class="op">-</span><span class="nam">epsilon</span> <span class="op">*</span> <span class="op">(</span><span class="nam">T2U</span> <span class="op">@</span> <span class="nam">Dx</span><span class="op">)</span><span class="op">}</span><span class="op">,</span> <span class="str">'ux'</span><span class="op">:</span> <span class="op">{</span><span class="str">'u'</span><span class="op">:</span> <span class="op">-</span><span class="nam">T2U</span> <span class="op">@</span> <span class="nam">Dx</span><span class="op">,</span> <span class="str">'ux'</span><span class="op">:</span> <span class="nam">T2U</span> <span class="op">@</span> <span class="nam">I</span><span class="op">}</span><span class="op">}</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t56" href="#t56">56</a></span><span class="t">        <span class="nam">self</span><span class="op">.</span><span class="nam">setup_L</span><span class="op">(</span><span class="nam">L_lhs</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t57" href="#t57">57</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t58" href="#t58">58</a></span><span class="t">        <span class="com"># construct mass matrix</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t59" href="#t59">59</a></span><span class="t">        <span class="nam">M_lhs</span> <span class="op">=</span> <span class="op">{</span><span class="str">'u'</span><span class="op">:</span> <span class="op">{</span><span class="str">'u'</span><span class="op">:</span> <span class="nam">T2U</span> <span class="op">@</span> <span class="nam">I</span><span class="op">}</span><span class="op">}</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t60" href="#t60">60</a></span><span class="t">        <span class="nam">self</span><span class="op">.</span><span class="nam">setup_M</span><span class="op">(</span><span class="nam">M_lhs</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t61" href="#t61">61</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t62" href="#t62">62</a></span><span class="t">        <span class="com"># boundary conditions</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t63" href="#t63">63</a></span><span class="t">        <span class="nam">self</span><span class="op">.</span><span class="nam">add_BC</span><span class="op">(</span><span class="nam">component</span><span class="op">=</span><span class="str">'u'</span><span class="op">,</span> <span class="nam">equation</span><span class="op">=</span><span class="str">'u'</span><span class="op">,</span> <span class="nam">axis</span><span class="op">=</span><span class="num">0</span><span class="op">,</span> <span class="nam">x</span><span class="op">=</span><span class="num">1</span><span class="op">,</span> <span class="nam">v</span><span class="op">=</span><span class="nam">BCr</span><span class="op">,</span> <span class="nam">kind</span><span class="op">=</span><span class="str">'Dirichlet'</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t64" href="#t64">64</a></span><span class="t">        <span class="nam">self</span><span class="op">.</span><span class="nam">add_BC</span><span class="op">(</span><span class="nam">component</span><span class="op">=</span><span class="str">'u'</span><span class="op">,</span> <span class="nam">equation</span><span class="op">=</span><span class="str">'ux'</span><span class="op">,</span> <span class="nam">axis</span><span class="op">=</span><span class="num">0</span><span class="op">,</span> <span class="nam">x</span><span class="op">=</span><span class="op">-</span><span class="num">1</span><span class="op">,</span> <span class="nam">v</span><span class="op">=</span><span class="nam">BCl</span><span class="op">,</span> <span class="nam">kind</span><span class="op">=</span><span class="str">'Dirichlet'</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t65" href="#t65">65</a></span><span class="t">        <span class="nam">self</span><span class="op">.</span><span class="nam">setup_BCs</span><span class="op">(</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t66" href="#t66">66</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t67" href="#t67">67</a></span><span class="t">    <span class="key">def</span> <span class="nam">u_exact</span><span class="op">(</span><span class="nam">self</span><span class="op">,</span> <span class="nam">t</span><span class="op">=</span><span class="num">0</span><span class="op">,</span> <span class="op">*</span><span class="nam">args</span><span class="op">,</span> <span class="op">**</span><span class="nam">kwargs</span><span class="op">)</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t68" href="#t68">68</a></span><span class="t">        <span class="nam">me</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">u_init</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t69" href="#t69">69</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t70" href="#t70">70</a></span><span class="t">        <span class="com"># x = (self.x + 1) / 2</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t71" href="#t71">71</a></span><span class="t">        <span class="com"># g = 4 * (1 + np.exp(-(4 * x + t)/self.epsilon/32))</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t72" href="#t72">72</a></span><span class="t">        <span class="com"># g_x = 4 * np.exp(-(4 * x + t)/self.epsilon/32) * (-4/self.epsilon/32)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t73" href="#t73">73</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t74" href="#t74">74</a></span><span class="t">        <span class="com"># me[0] = 3./4. - 1./g</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t75" href="#t75">75</a></span><span class="t">        <span class="com"># me[1] = 1/g**2 * g_x</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t76" href="#t76">76</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t77" href="#t77">77</a></span><span class="t">        <span class="com"># return me</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t78" href="#t78">78</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t79" href="#t79">79</a></span><span class="t">        <span class="key">if</span> <span class="nam">t</span> <span class="op">==</span> <span class="num">0</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t80" href="#t80">80</a></span><span class="t">            <span class="nam">me</span><span class="op">[</span><span class="nam">self</span><span class="op">.</span><span class="nam">index</span><span class="op">(</span><span class="str">'u'</span><span class="op">)</span><span class="op">]</span><span class="op">[</span><span class="op">:</span><span class="op">]</span> <span class="op">=</span> <span class="op">(</span><span class="op">(</span><span class="nam">self</span><span class="op">.</span><span class="nam">BCr</span> <span class="op">+</span> <span class="nam">self</span><span class="op">.</span><span class="nam">BCl</span><span class="op">)</span> <span class="op">/</span> <span class="num">2</span> <span class="op">+</span> <span class="op">(</span><span class="nam">self</span><span class="op">.</span><span class="nam">BCr</span> <span class="op">-</span> <span class="nam">self</span><span class="op">.</span><span class="nam">BCl</span><span class="op">)</span> <span class="op">/</span> <span class="num">2</span> <span class="op">*</span> <span class="nam">self</span><span class="op">.</span><span class="nam">x</span><span class="op">)</span> <span class="op">*</span> <span class="nam">np</span><span class="op">.</span><span class="nam">cos</span><span class="op">(</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t81" href="#t81">81</a></span><span class="t">                <span class="nam">self</span><span class="op">.</span><span class="nam">x</span> <span class="op">*</span> <span class="nam">np</span><span class="op">.</span><span class="nam">pi</span> <span class="op">*</span> <span class="nam">self</span><span class="op">.</span><span class="nam">f</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t82" href="#t82">82</a></span><span class="t">            <span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t83" href="#t83">83</a></span><span class="t">            <span class="nam">me</span><span class="op">[</span><span class="nam">self</span><span class="op">.</span><span class="nam">index</span><span class="op">(</span><span class="str">'ux'</span><span class="op">)</span><span class="op">]</span><span class="op">[</span><span class="op">:</span><span class="op">]</span> <span class="op">=</span> <span class="op">(</span><span class="nam">self</span><span class="op">.</span><span class="nam">BCr</span> <span class="op">-</span> <span class="nam">self</span><span class="op">.</span><span class="nam">BCl</span><span class="op">)</span> <span class="op">/</span> <span class="num">2</span> <span class="op">*</span> <span class="nam">np</span><span class="op">.</span><span class="nam">cos</span><span class="op">(</span><span class="nam">self</span><span class="op">.</span><span class="nam">x</span> <span class="op">*</span> <span class="nam">np</span><span class="op">.</span><span class="nam">pi</span> <span class="op">*</span> <span class="nam">self</span><span class="op">.</span><span class="nam">f</span><span class="op">)</span> <span class="op">+</span> <span class="op">(</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t84" href="#t84">84</a></span><span class="t">                <span class="op">(</span><span class="nam">self</span><span class="op">.</span><span class="nam">BCr</span> <span class="op">+</span> <span class="nam">self</span><span class="op">.</span><span class="nam">BCl</span><span class="op">)</span> <span class="op">/</span> <span class="num">2</span> <span class="op">+</span> <span class="op">(</span><span class="nam">self</span><span class="op">.</span><span class="nam">BCr</span> <span class="op">-</span> <span class="nam">self</span><span class="op">.</span><span class="nam">BCl</span><span class="op">)</span> <span class="op">/</span> <span class="num">2</span> <span class="op">*</span> <span class="nam">self</span><span class="op">.</span><span class="nam">x</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t85" href="#t85">85</a></span><span class="t">            <span class="op">)</span> <span class="op">*</span> <span class="nam">self</span><span class="op">.</span><span class="nam">f</span> <span class="op">*</span> <span class="nam">np</span><span class="op">.</span><span class="nam">pi</span> <span class="op">*</span> <span class="op">-</span><span class="nam">np</span><span class="op">.</span><span class="nam">sin</span><span class="op">(</span><span class="nam">self</span><span class="op">.</span><span class="nam">x</span> <span class="op">*</span> <span class="nam">np</span><span class="op">.</span><span class="nam">pi</span> <span class="op">*</span> <span class="nam">self</span><span class="op">.</span><span class="nam">f</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t86" href="#t86">86</a></span><span class="t">        <span class="key">elif</span> <span class="nam">t</span> <span class="op">==</span> <span class="nam">np</span><span class="op">.</span><span class="nam">inf</span> <span class="key">and</span> <span class="nam">self</span><span class="op">.</span><span class="nam">f</span> <span class="op">==</span> <span class="num">0</span> <span class="key">and</span> <span class="nam">self</span><span class="op">.</span><span class="nam">BCl</span> <span class="op">==</span> <span class="op">-</span><span class="nam">self</span><span class="op">.</span><span class="nam">BCr</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t87" href="#t87">87</a></span><span class="t">            <span class="nam">me</span><span class="op">[</span><span class="num">0</span><span class="op">]</span> <span class="op">=</span> <span class="op">(</span><span class="nam">self</span><span class="op">.</span><span class="nam">BCl</span> <span class="op">*</span> <span class="nam">np</span><span class="op">.</span><span class="nam">exp</span><span class="op">(</span><span class="op">(</span><span class="nam">self</span><span class="op">.</span><span class="nam">BCr</span> <span class="op">-</span> <span class="nam">self</span><span class="op">.</span><span class="nam">BCl</span><span class="op">)</span> <span class="op">/</span> <span class="op">(</span><span class="num">2</span> <span class="op">*</span> <span class="nam">self</span><span class="op">.</span><span class="nam">epsilon</span><span class="op">)</span> <span class="op">*</span> <span class="nam">self</span><span class="op">.</span><span class="nam">x</span><span class="op">)</span> <span class="op">+</span> <span class="nam">self</span><span class="op">.</span><span class="nam">BCr</span><span class="op">)</span> <span class="op">/</span> <span class="op">(</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t88" href="#t88">88</a></span><span class="t">                <span class="nam">np</span><span class="op">.</span><span class="nam">exp</span><span class="op">(</span><span class="op">(</span><span class="nam">self</span><span class="op">.</span><span class="nam">BCr</span> <span class="op">-</span> <span class="nam">self</span><span class="op">.</span><span class="nam">BCl</span><span class="op">)</span> <span class="op">/</span> <span class="op">(</span><span class="num">2</span> <span class="op">*</span> <span class="nam">self</span><span class="op">.</span><span class="nam">epsilon</span><span class="op">)</span> <span class="op">*</span> <span class="nam">self</span><span class="op">.</span><span class="nam">x</span><span class="op">)</span> <span class="op">+</span> <span class="num">1</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t89" href="#t89">89</a></span><span class="t">            <span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t90" href="#t90">90</a></span><span class="t">        <span class="key">else</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t91" href="#t91">91</a></span><span class="t">            <span class="key">raise</span> <span class="nam">NotImplementedError</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t92" href="#t92">92</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t93" href="#t93">93</a></span><span class="t">        <span class="key">return</span> <span class="nam">me</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t94" href="#t94">94</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t95" href="#t95">95</a></span><span class="t">    <span class="key">def</span> <span class="nam">eval_f</span><span class="op">(</span><span class="nam">self</span><span class="op">,</span> <span class="nam">u</span><span class="op">,</span> <span class="op">*</span><span class="nam">args</span><span class="op">,</span> <span class="op">**</span><span class="nam">kwargs</span><span class="op">)</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t96" href="#t96">96</a></span><span class="t">        <span class="nam">f</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">f_init</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t97" href="#t97">97</a></span><span class="t">        <span class="nam">iu</span><span class="op">,</span> <span class="nam">iux</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">index</span><span class="op">(</span><span class="str">'u'</span><span class="op">)</span><span class="op">,</span> <span class="nam">self</span><span class="op">.</span><span class="nam">index</span><span class="op">(</span><span class="str">'ux'</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t98" href="#t98">98</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t99" href="#t99">99</a></span><span class="t">        <span class="nam">u_hat</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">transform</span><span class="op">(</span><span class="nam">u</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t100" href="#t100">100</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t101" href="#t101">101</a></span><span class="t">        <span class="nam">Dx_u_hat</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">u_init_forward</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t102" href="#t102">102</a></span><span class="t">        <span class="nam">Dx_u_hat</span><span class="op">[</span><span class="nam">iux</span><span class="op">]</span> <span class="op">=</span> <span class="op">(</span><span class="nam">self</span><span class="op">.</span><span class="nam">Dx</span> <span class="op">@</span> <span class="nam">u_hat</span><span class="op">[</span><span class="nam">iux</span><span class="op">]</span><span class="op">.</span><span class="nam">flatten</span><span class="op">(</span><span class="op">)</span><span class="op">)</span><span class="op">.</span><span class="nam">reshape</span><span class="op">(</span><span class="nam">u_hat</span><span class="op">[</span><span class="nam">iu</span><span class="op">]</span><span class="op">.</span><span class="nam">shape</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t103" href="#t103">103</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t104" href="#t104">104</a></span><span class="t">        <span class="nam">f</span><span class="op">.</span><span class="nam">impl</span><span class="op">[</span><span class="nam">iu</span><span class="op">]</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">epsilon</span> <span class="op">*</span> <span class="nam">self</span><span class="op">.</span><span class="nam">itransform</span><span class="op">(</span><span class="nam">Dx_u_hat</span><span class="op">)</span><span class="op">[</span><span class="nam">iux</span><span class="op">]</span><span class="op">.</span><span class="nam">real</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t105" href="#t105">105</a></span><span class="t">        <span class="nam">f</span><span class="op">.</span><span class="nam">expl</span><span class="op">[</span><span class="nam">iu</span><span class="op">]</span> <span class="op">=</span> <span class="op">-</span><span class="nam">u</span><span class="op">[</span><span class="nam">iu</span><span class="op">]</span> <span class="op">*</span> <span class="nam">u</span><span class="op">[</span><span class="nam">iux</span><span class="op">]</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t106" href="#t106">106</a></span><span class="t">        <span class="key">return</span> <span class="nam">f</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t107" href="#t107">107</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t108" href="#t108">108</a></span><span class="t">    <span class="key">def</span> <span class="nam">get_fig</span><span class="op">(</span><span class="nam">self</span><span class="op">)</span><span class="op">:</span>  <span class="com"># pragma: no cover</span>&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t109" href="#t109">109</a></span><span class="t">        <span class="str">"""</span>&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t110" href="#t110">110</a></span><span class="t"><span class="str">        Get a figure suitable to plot the solution of this problem.</span>&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t111" href="#t111">111</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t112" href="#t112">112</a></span><span class="t"><span class="str">        Returns</span>&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t113" href="#t113">113</a></span><span class="t"><span class="str">        -------</span>&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t114" href="#t114">114</a></span><span class="t"><span class="str">        self.fig : matplotlib.pyplot.figure.Figure</span>&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t115" href="#t115">115</a></span><span class="t"><span class="str">        """</span>&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t116" href="#t116">116</a></span><span class="t">        <span class="key">import</span> <span class="nam">matplotlib</span><span class="op">.</span><span class="nam">pyplot</span> <span class="key">as</span> <span class="nam">plt</span>&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t117" href="#t117">117</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t118" href="#t118">118</a></span><span class="t">        <span class="nam">plt</span><span class="op">.</span><span class="nam">rcParams</span><span class="op">[</span><span class="str">'figure.constrained_layout.use'</span><span class="op">]</span> <span class="op">=</span> <span class="key">True</span>&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t119" href="#t119">119</a></span><span class="t">        <span class="nam">self</span><span class="op">.</span><span class="nam">fig</span><span class="op">,</span> <span class="nam">axs</span> <span class="op">=</span> <span class="nam">plt</span><span class="op">.</span><span class="nam">subplots</span><span class="op">(</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t120" href="#t120">120</a></span><span class="t">        <span class="key">return</span> <span class="nam">self</span><span class="op">.</span><span class="nam">fig</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t121" href="#t121">121</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t122" href="#t122">122</a></span><span class="t">    <span class="key">def</span> <span class="nam">plot</span><span class="op">(</span><span class="nam">self</span><span class="op">,</span> <span class="nam">u</span><span class="op">,</span> <span class="nam">t</span><span class="op">=</span><span class="key">None</span><span class="op">,</span> <span class="nam">fig</span><span class="op">=</span><span class="key">None</span><span class="op">,</span> <span class="nam">comp</span><span class="op">=</span><span class="str">'u'</span><span class="op">)</span><span class="op">:</span>  <span class="com"># pragma: no cover</span>&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t123" href="#t123">123</a></span><span class="t">        <span class="str">r"""</span>&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t124" href="#t124">124</a></span><span class="t"><span class="str">        Plot the solution.</span>&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t125" href="#t125">125</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t126" href="#t126">126</a></span><span class="t"><span class="str">        Parameters</span>&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t127" href="#t127">127</a></span><span class="t"><span class="str">        ----------</span>&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t128" href="#t128">128</a></span><span class="t"><span class="str">        u : dtype_u</span>&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t129" href="#t129">129</a></span><span class="t"><span class="str">            Solution to be plotted</span>&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t130" href="#t130">130</a></span><span class="t"><span class="str">        t : float</span>&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t131" href="#t131">131</a></span><span class="t"><span class="str">            Time to display at the top of the figure</span>&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t132" href="#t132">132</a></span><span class="t"><span class="str">        fig : matplotlib.pyplot.figure.Figure, optional</span>&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t133" href="#t133">133</a></span><span class="t"><span class="str">            Figure with the same structure as a figure generated by `self.get_fig`. If none is supplied, a new figure will be generated.</span>&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t134" href="#t134">134</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t135" href="#t135">135</a></span><span class="t"><span class="str">        Returns</span>&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t136" href="#t136">136</a></span><span class="t"><span class="str">        -------</span>&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t137" href="#t137">137</a></span><span class="t"><span class="str">        None</span>&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t138" href="#t138">138</a></span><span class="t"><span class="str">        """</span>&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t139" href="#t139">139</a></span><span class="t">        <span class="nam">fig</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">get_fig</span><span class="op">(</span><span class="op">)</span> <span class="key">if</span> <span class="nam">fig</span> <span class="key">is</span> <span class="key">None</span> <span class="key">else</span> <span class="nam">fig</span>&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t140" href="#t140">140</a></span><span class="t">        <span class="nam">ax</span> <span class="op">=</span> <span class="nam">fig</span><span class="op">.</span><span class="nam">axes</span><span class="op">[</span><span class="num">0</span><span class="op">]</span>&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t141" href="#t141">141</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t142" href="#t142">142</a></span><span class="t">        <span class="nam">ax</span><span class="op">.</span><span class="nam">plot</span><span class="op">(</span><span class="nam">self</span><span class="op">.</span><span class="nam">x</span><span class="op">,</span> <span class="nam">u</span><span class="op">[</span><span class="nam">self</span><span class="op">.</span><span class="nam">index</span><span class="op">(</span><span class="nam">comp</span><span class="op">)</span><span class="op">]</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t143" href="#t143">143</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t144" href="#t144">144</a></span><span class="t">        <span class="key">if</span> <span class="nam">t</span> <span class="key">is</span> <span class="key">not</span> <span class="key">None</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t145" href="#t145">145</a></span><span class="t">            <span class="nam">fig</span><span class="op">.</span><span class="nam">suptitle</span><span class="op">(</span><span class="fst">f'</span><span class="fst">t = </span><span class="op">{</span><span class="nam">t</span><span class="op">:</span><span class="fst">.2e</span><span class="op">}</span><span class="fst">'</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t146" href="#t146">146</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t147" href="#t147">147</a></span><span class="t">        <span class="nam">ax</span><span class="op">.</span><span class="nam">set_xlabel</span><span class="op">(</span><span class="str">r'$x$'</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t148" href="#t148">148</a></span><span class="t">        <span class="nam">ax</span><span class="op">.</span><span class="nam">set_ylabel</span><span class="op">(</span><span class="str">r'$u$'</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t149" href="#t149">149</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t150" href="#t150">150</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t151" href="#t151">151</a></span><span class="t"><span class="key">class</span> <span class="nam">Burgers2D</span><span class="op">(</span><span class="nam">GenericSpectralLinear</span><span class="op">)</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t152" href="#t152">152</a></span><span class="t">    <span class="str">"""</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t153" href="#t153">153</a></span><span class="t"><span class="str">    See https://en.wikipedia.org/wiki/Burgers'_equation for the equation that is solved.</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t154" href="#t154">154</a></span><span class="t"><span class="str">    This implementation is discretized with FFTs in x and Chebychev in z.</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t155" href="#t155">155</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t156" href="#t156">156</a></span><span class="t"><span class="str">    Parameters:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t157" href="#t157">157</a></span><span class="t"><span class="str">       nx (int): Spatial resolution in x direction</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t158" href="#t158">158</a></span><span class="t"><span class="str">       nz (int): Spatial resolution in z direction</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t159" href="#t159">159</a></span><span class="t"><span class="str">       epsilon (float): viscosity</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t160" href="#t160">160</a></span><span class="t"><span class="str">       BCl (float): Value at left boundary</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t161" href="#t161">161</a></span><span class="t"><span class="str">       BCr (float): Value at right boundary</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t162" href="#t162">162</a></span><span class="t"><span class="str">       fux (int): Frequency of the initial conditions in x-direction</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t163" href="#t163">163</a></span><span class="t"><span class="str">       fuz (int): Frequency of the initial conditions in z-direction</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t164" href="#t164">164</a></span><span class="t"><span class="str">       mode (str): 'T2U' or 'T2T'. Use 'T2U' to get sparse differentiation matrices</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t165" href="#t165">165</a></span><span class="t"><span class="str">    """</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t166" href="#t166">166</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t167" href="#t167">167</a></span><span class="t">    <span class="nam">dtype_u</span> <span class="op">=</span> <span class="nam">mesh</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t168" href="#t168">168</a></span><span class="t">    <span class="nam">dtype_f</span> <span class="op">=</span> <span class="nam">imex_mesh</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t169" href="#t169">169</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t170" href="#t170">170</a></span><span class="t">    <span class="key">def</span> <span class="nam">__init__</span><span class="op">(</span><span class="nam">self</span><span class="op">,</span> <span class="nam">nx</span><span class="op">=</span><span class="num">64</span><span class="op">,</span> <span class="nam">nz</span><span class="op">=</span><span class="num">64</span><span class="op">,</span> <span class="nam">epsilon</span><span class="op">=</span><span class="num">0.1</span><span class="op">,</span> <span class="nam">fux</span><span class="op">=</span><span class="num">2</span><span class="op">,</span> <span class="nam">fuz</span><span class="op">=</span><span class="num">1</span><span class="op">,</span> <span class="nam">mode</span><span class="op">=</span><span class="str">'T2U'</span><span class="op">,</span> <span class="op">**</span><span class="nam">kwargs</span><span class="op">)</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t171" href="#t171">171</a></span><span class="t">        <span class="str">"""</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t172" href="#t172">172</a></span><span class="t"><span class="str">         Constructor. `kwargs` are forwarded to parent class constructor.</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t173" href="#t173">173</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t174" href="#t174">174</a></span><span class="t"><span class="str">         Args:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t175" href="#t175">175</a></span><span class="t"><span class="str">        nx (int): Spatial resolution in x direction</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t176" href="#t176">176</a></span><span class="t"><span class="str">        nz (int): Spatial resolution in z direction</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t177" href="#t177">177</a></span><span class="t"><span class="str">        epsilon (float): viscosity</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t178" href="#t178">178</a></span><span class="t"><span class="str">        fux (int): Frequency of the initial conditions in x-direction</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t179" href="#t179">179</a></span><span class="t"><span class="str">        fuz (int): Frequency of the initial conditions in z-direction</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t180" href="#t180">180</a></span><span class="t"><span class="str">        mode (str): 'T2U' or 'T2T'. Use 'T2U' to get sparse differentiation matrices</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t181" href="#t181">181</a></span><span class="t"><span class="str">        """</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t182" href="#t182">182</a></span><span class="t">        <span class="nam">self</span><span class="op">.</span><span class="nam">_makeAttributeAndRegister</span><span class="op">(</span><span class="str">'nx'</span><span class="op">,</span> <span class="str">'nz'</span><span class="op">,</span> <span class="str">'epsilon'</span><span class="op">,</span> <span class="str">'fux'</span><span class="op">,</span> <span class="str">'fuz'</span><span class="op">,</span> <span class="str">'mode'</span><span class="op">,</span> <span class="nam">localVars</span><span class="op">=</span><span class="nam">locals</span><span class="op">(</span><span class="op">)</span><span class="op">,</span> <span class="nam">readOnly</span><span class="op">=</span><span class="key">True</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t183" href="#t183">183</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t184" href="#t184">184</a></span><span class="t">        <span class="nam">bases</span> <span class="op">=</span> <span class="op">[</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t185" href="#t185">185</a></span><span class="t">            <span class="op">{</span><span class="str">'base'</span><span class="op">:</span> <span class="str">'fft'</span><span class="op">,</span> <span class="str">'N'</span><span class="op">:</span> <span class="nam">nx</span><span class="op">}</span><span class="op">,</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t186" href="#t186">186</a></span><span class="t">            <span class="op">{</span><span class="str">'base'</span><span class="op">:</span> <span class="str">'cheby'</span><span class="op">,</span> <span class="str">'N'</span><span class="op">:</span> <span class="nam">nz</span><span class="op">}</span><span class="op">,</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t187" href="#t187">187</a></span><span class="t">        <span class="op">]</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t188" href="#t188">188</a></span><span class="t">        <span class="nam">components</span> <span class="op">=</span> <span class="op">[</span><span class="str">'u'</span><span class="op">,</span> <span class="str">'v'</span><span class="op">,</span> <span class="str">'ux'</span><span class="op">,</span> <span class="str">'uz'</span><span class="op">,</span> <span class="str">'vx'</span><span class="op">,</span> <span class="str">'vz'</span><span class="op">]</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t189" href="#t189">189</a></span><span class="t">        <span class="nam">super</span><span class="op">(</span><span class="op">)</span><span class="op">.</span><span class="nam">__init__</span><span class="op">(</span><span class="nam">bases</span><span class="op">=</span><span class="nam">bases</span><span class="op">,</span> <span class="nam">components</span><span class="op">=</span><span class="nam">components</span><span class="op">,</span> <span class="nam">spectral_space</span><span class="op">=</span><span class="key">False</span><span class="op">,</span> <span class="op">**</span><span class="nam">kwargs</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t190" href="#t190">190</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t191" href="#t191">191</a></span><span class="t">        <span class="nam">self</span><span class="op">.</span><span class="nam">Z</span><span class="op">,</span> <span class="nam">self</span><span class="op">.</span><span class="nam">X</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">get_grid</span><span class="op">(</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t192" href="#t192">192</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t193" href="#t193">193</a></span><span class="t">        <span class="com"># prepare matrices</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t194" href="#t194">194</a></span><span class="t">        <span class="nam">Dx</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">get_differentiation_matrix</span><span class="op">(</span><span class="nam">axes</span><span class="op">=</span><span class="op">(</span><span class="num">0</span><span class="op">,</span><span class="op">)</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t195" href="#t195">195</a></span><span class="t">        <span class="nam">Dz</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">get_differentiation_matrix</span><span class="op">(</span><span class="nam">axes</span><span class="op">=</span><span class="op">(</span><span class="num">1</span><span class="op">,</span><span class="op">)</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t196" href="#t196">196</a></span><span class="t">        <span class="nam">I</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">get_Id</span><span class="op">(</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t197" href="#t197">197</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t198" href="#t198">198</a></span><span class="t">        <span class="nam">T2U</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">get_basis_change_matrix</span><span class="op">(</span><span class="nam">axes</span><span class="op">=</span><span class="op">(</span><span class="num">1</span><span class="op">,</span><span class="op">)</span><span class="op">,</span> <span class="nam">conv</span><span class="op">=</span><span class="nam">mode</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t199" href="#t199">199</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t200" href="#t200">200</a></span><span class="t">        <span class="nam">self</span><span class="op">.</span><span class="nam">Dx</span> <span class="op">=</span> <span class="nam">Dx</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t201" href="#t201">201</a></span><span class="t">        <span class="nam">self</span><span class="op">.</span><span class="nam">Dz</span> <span class="op">=</span> <span class="nam">Dz</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t202" href="#t202">202</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t203" href="#t203">203</a></span><span class="t">        <span class="com"># construct linear operator</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t204" href="#t204">204</a></span><span class="t">        <span class="nam">L_lhs</span> <span class="op">=</span> <span class="op">{</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t205" href="#t205">205</a></span><span class="t">            <span class="str">'u'</span><span class="op">:</span> <span class="op">{</span><span class="str">'ux'</span><span class="op">:</span> <span class="op">-</span><span class="nam">epsilon</span> <span class="op">*</span> <span class="nam">T2U</span> <span class="op">@</span> <span class="nam">Dx</span><span class="op">,</span> <span class="str">'uz'</span><span class="op">:</span> <span class="op">-</span><span class="nam">epsilon</span> <span class="op">*</span> <span class="nam">T2U</span> <span class="op">@</span> <span class="nam">Dz</span><span class="op">}</span><span class="op">,</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t206" href="#t206">206</a></span><span class="t">            <span class="str">'v'</span><span class="op">:</span> <span class="op">{</span><span class="str">'vx'</span><span class="op">:</span> <span class="op">-</span><span class="nam">epsilon</span> <span class="op">*</span> <span class="nam">T2U</span> <span class="op">@</span> <span class="nam">Dx</span><span class="op">,</span> <span class="str">'vz'</span><span class="op">:</span> <span class="op">-</span><span class="nam">epsilon</span> <span class="op">*</span> <span class="nam">T2U</span> <span class="op">@</span> <span class="nam">Dz</span><span class="op">}</span><span class="op">,</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t207" href="#t207">207</a></span><span class="t">            <span class="str">'ux'</span><span class="op">:</span> <span class="op">{</span><span class="str">'u'</span><span class="op">:</span> <span class="op">-</span><span class="nam">T2U</span> <span class="op">@</span> <span class="nam">Dx</span><span class="op">,</span> <span class="str">'ux'</span><span class="op">:</span> <span class="nam">T2U</span> <span class="op">@</span> <span class="nam">I</span><span class="op">}</span><span class="op">,</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t208" href="#t208">208</a></span><span class="t">            <span class="str">'uz'</span><span class="op">:</span> <span class="op">{</span><span class="str">'u'</span><span class="op">:</span> <span class="op">-</span><span class="nam">T2U</span> <span class="op">@</span> <span class="nam">Dz</span><span class="op">,</span> <span class="str">'uz'</span><span class="op">:</span> <span class="nam">T2U</span> <span class="op">@</span> <span class="nam">I</span><span class="op">}</span><span class="op">,</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t209" href="#t209">209</a></span><span class="t">            <span class="str">'vx'</span><span class="op">:</span> <span class="op">{</span><span class="str">'v'</span><span class="op">:</span> <span class="op">-</span><span class="nam">T2U</span> <span class="op">@</span> <span class="nam">Dx</span><span class="op">,</span> <span class="str">'vx'</span><span class="op">:</span> <span class="nam">T2U</span> <span class="op">@</span> <span class="nam">I</span><span class="op">}</span><span class="op">,</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t210" href="#t210">210</a></span><span class="t">            <span class="str">'vz'</span><span class="op">:</span> <span class="op">{</span><span class="str">'v'</span><span class="op">:</span> <span class="op">-</span><span class="nam">T2U</span> <span class="op">@</span> <span class="nam">Dz</span><span class="op">,</span> <span class="str">'vz'</span><span class="op">:</span> <span class="nam">T2U</span> <span class="op">@</span> <span class="nam">I</span><span class="op">}</span><span class="op">,</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t211" href="#t211">211</a></span><span class="t">        <span class="op">}</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t212" href="#t212">212</a></span><span class="t">        <span class="nam">self</span><span class="op">.</span><span class="nam">setup_L</span><span class="op">(</span><span class="nam">L_lhs</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t213" href="#t213">213</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t214" href="#t214">214</a></span><span class="t">        <span class="com"># construct mass matrix</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t215" href="#t215">215</a></span><span class="t">        <span class="nam">M_lhs</span> <span class="op">=</span> <span class="op">{</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t216" href="#t216">216</a></span><span class="t">            <span class="str">'u'</span><span class="op">:</span> <span class="op">{</span><span class="str">'u'</span><span class="op">:</span> <span class="nam">T2U</span> <span class="op">@</span> <span class="nam">I</span><span class="op">}</span><span class="op">,</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t217" href="#t217">217</a></span><span class="t">            <span class="str">'v'</span><span class="op">:</span> <span class="op">{</span><span class="str">'v'</span><span class="op">:</span> <span class="nam">T2U</span> <span class="op">@</span> <span class="nam">I</span><span class="op">}</span><span class="op">,</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t218" href="#t218">218</a></span><span class="t">        <span class="op">}</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t219" href="#t219">219</a></span><span class="t">        <span class="nam">self</span><span class="op">.</span><span class="nam">setup_M</span><span class="op">(</span><span class="nam">M_lhs</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t220" href="#t220">220</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t221" href="#t221">221</a></span><span class="t">        <span class="com"># boundary conditions</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t222" href="#t222">222</a></span><span class="t">        <span class="nam">self</span><span class="op">.</span><span class="nam">BCtop</span> <span class="op">=</span> <span class="num">1</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t223" href="#t223">223</a></span><span class="t">        <span class="nam">self</span><span class="op">.</span><span class="nam">BCbottom</span> <span class="op">=</span> <span class="op">-</span><span class="nam">self</span><span class="op">.</span><span class="nam">BCtop</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t224" href="#t224">224</a></span><span class="t">        <span class="nam">self</span><span class="op">.</span><span class="nam">BCtopu</span> <span class="op">=</span> <span class="num">0</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t225" href="#t225">225</a></span><span class="t">        <span class="nam">self</span><span class="op">.</span><span class="nam">add_BC</span><span class="op">(</span><span class="nam">component</span><span class="op">=</span><span class="str">'v'</span><span class="op">,</span> <span class="nam">equation</span><span class="op">=</span><span class="str">'v'</span><span class="op">,</span> <span class="nam">axis</span><span class="op">=</span><span class="num">1</span><span class="op">,</span> <span class="nam">v</span><span class="op">=</span><span class="nam">self</span><span class="op">.</span><span class="nam">BCtop</span><span class="op">,</span> <span class="nam">x</span><span class="op">=</span><span class="num">1</span><span class="op">,</span> <span class="nam">kind</span><span class="op">=</span><span class="str">'Dirichlet'</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t226" href="#t226">226</a></span><span class="t">        <span class="nam">self</span><span class="op">.</span><span class="nam">add_BC</span><span class="op">(</span><span class="nam">component</span><span class="op">=</span><span class="str">'v'</span><span class="op">,</span> <span class="nam">equation</span><span class="op">=</span><span class="str">'vz'</span><span class="op">,</span> <span class="nam">axis</span><span class="op">=</span><span class="num">1</span><span class="op">,</span> <span class="nam">v</span><span class="op">=</span><span class="nam">self</span><span class="op">.</span><span class="nam">BCbottom</span><span class="op">,</span> <span class="nam">x</span><span class="op">=</span><span class="op">-</span><span class="num">1</span><span class="op">,</span> <span class="nam">kind</span><span class="op">=</span><span class="str">'Dirichlet'</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t227" href="#t227">227</a></span><span class="t">        <span class="nam">self</span><span class="op">.</span><span class="nam">add_BC</span><span class="op">(</span><span class="nam">component</span><span class="op">=</span><span class="str">'u'</span><span class="op">,</span> <span class="nam">equation</span><span class="op">=</span><span class="str">'uz'</span><span class="op">,</span> <span class="nam">axis</span><span class="op">=</span><span class="num">1</span><span class="op">,</span> <span class="nam">v</span><span class="op">=</span><span class="nam">self</span><span class="op">.</span><span class="nam">BCtopu</span><span class="op">,</span> <span class="nam">x</span><span class="op">=</span><span class="num">1</span><span class="op">,</span> <span class="nam">kind</span><span class="op">=</span><span class="str">'Dirichlet'</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t228" href="#t228">228</a></span><span class="t">        <span class="nam">self</span><span class="op">.</span><span class="nam">add_BC</span><span class="op">(</span><span class="nam">component</span><span class="op">=</span><span class="str">'u'</span><span class="op">,</span> <span class="nam">equation</span><span class="op">=</span><span class="str">'u'</span><span class="op">,</span> <span class="nam">axis</span><span class="op">=</span><span class="num">1</span><span class="op">,</span> <span class="nam">v</span><span class="op">=</span><span class="nam">self</span><span class="op">.</span><span class="nam">BCtopu</span><span class="op">,</span> <span class="nam">x</span><span class="op">=</span><span class="op">-</span><span class="num">1</span><span class="op">,</span> <span class="nam">kind</span><span class="op">=</span><span class="str">'Dirichlet'</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t229" href="#t229">229</a></span><span class="t">        <span class="nam">self</span><span class="op">.</span><span class="nam">setup_BCs</span><span class="op">(</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t230" href="#t230">230</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t231" href="#t231">231</a></span><span class="t">    <span class="key">def</span> <span class="nam">u_exact</span><span class="op">(</span><span class="nam">self</span><span class="op">,</span> <span class="nam">t</span><span class="op">=</span><span class="num">0</span><span class="op">,</span> <span class="op">*</span><span class="nam">args</span><span class="op">,</span> <span class="nam">noise_level</span><span class="op">=</span><span class="num">0</span><span class="op">,</span> <span class="op">**</span><span class="nam">kwargs</span><span class="op">)</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t232" href="#t232">232</a></span><span class="t">        <span class="nam">me</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">u_init</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t233" href="#t233">233</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t234" href="#t234">234</a></span><span class="t">        <span class="nam">iu</span><span class="op">,</span> <span class="nam">iv</span><span class="op">,</span> <span class="nam">iux</span><span class="op">,</span> <span class="nam">iuz</span><span class="op">,</span> <span class="nam">ivx</span><span class="op">,</span> <span class="nam">ivz</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">index</span><span class="op">(</span><span class="nam">self</span><span class="op">.</span><span class="nam">components</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t235" href="#t235">235</a></span><span class="t">        <span class="key">if</span> <span class="nam">t</span> <span class="op">==</span> <span class="num">0</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t236" href="#t236">236</a></span><span class="t">            <span class="nam">me</span><span class="op">[</span><span class="nam">iu</span><span class="op">]</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">xp</span><span class="op">.</span><span class="nam">cos</span><span class="op">(</span><span class="nam">self</span><span class="op">.</span><span class="nam">X</span> <span class="op">*</span> <span class="nam">self</span><span class="op">.</span><span class="nam">fux</span><span class="op">)</span> <span class="op">*</span> <span class="nam">self</span><span class="op">.</span><span class="nam">xp</span><span class="op">.</span><span class="nam">sin</span><span class="op">(</span><span class="nam">self</span><span class="op">.</span><span class="nam">Z</span> <span class="op">*</span> <span class="nam">np</span><span class="op">.</span><span class="nam">pi</span> <span class="op">*</span> <span class="nam">self</span><span class="op">.</span><span class="nam">fuz</span><span class="op">)</span> <span class="op">+</span> <span class="nam">self</span><span class="op">.</span><span class="nam">BCtopu</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t237" href="#t237">237</a></span><span class="t">            <span class="nam">me</span><span class="op">[</span><span class="nam">iux</span><span class="op">]</span> <span class="op">=</span> <span class="op">-</span><span class="nam">self</span><span class="op">.</span><span class="nam">xp</span><span class="op">.</span><span class="nam">sin</span><span class="op">(</span><span class="nam">self</span><span class="op">.</span><span class="nam">X</span> <span class="op">*</span> <span class="nam">self</span><span class="op">.</span><span class="nam">fux</span><span class="op">)</span> <span class="op">*</span> <span class="nam">self</span><span class="op">.</span><span class="nam">fux</span> <span class="op">*</span> <span class="nam">self</span><span class="op">.</span><span class="nam">xp</span><span class="op">.</span><span class="nam">sin</span><span class="op">(</span><span class="nam">self</span><span class="op">.</span><span class="nam">Z</span> <span class="op">*</span> <span class="nam">np</span><span class="op">.</span><span class="nam">pi</span> <span class="op">*</span> <span class="nam">self</span><span class="op">.</span><span class="nam">fuz</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t238" href="#t238">238</a></span><span class="t">            <span class="nam">me</span><span class="op">[</span><span class="nam">iuz</span><span class="op">]</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">xp</span><span class="op">.</span><span class="nam">cos</span><span class="op">(</span><span class="nam">self</span><span class="op">.</span><span class="nam">X</span> <span class="op">*</span> <span class="nam">self</span><span class="op">.</span><span class="nam">fux</span><span class="op">)</span> <span class="op">*</span> <span class="nam">self</span><span class="op">.</span><span class="nam">xp</span><span class="op">.</span><span class="nam">cos</span><span class="op">(</span><span class="nam">self</span><span class="op">.</span><span class="nam">Z</span> <span class="op">*</span> <span class="nam">np</span><span class="op">.</span><span class="nam">pi</span> <span class="op">*</span> <span class="nam">self</span><span class="op">.</span><span class="nam">fuz</span><span class="op">)</span> <span class="op">*</span> <span class="nam">np</span><span class="op">.</span><span class="nam">pi</span> <span class="op">*</span> <span class="nam">self</span><span class="op">.</span><span class="nam">fuz</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t239" href="#t239">239</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t240" href="#t240">240</a></span><span class="t">            <span class="nam">me</span><span class="op">[</span><span class="nam">iv</span><span class="op">]</span> <span class="op">=</span> <span class="op">(</span><span class="nam">self</span><span class="op">.</span><span class="nam">BCtop</span> <span class="op">+</span> <span class="nam">self</span><span class="op">.</span><span class="nam">BCbottom</span><span class="op">)</span> <span class="op">/</span> <span class="num">2</span> <span class="op">+</span> <span class="op">(</span><span class="nam">self</span><span class="op">.</span><span class="nam">BCtop</span> <span class="op">-</span> <span class="nam">self</span><span class="op">.</span><span class="nam">BCbottom</span><span class="op">)</span> <span class="op">/</span> <span class="num">2</span> <span class="op">*</span> <span class="nam">self</span><span class="op">.</span><span class="nam">Z</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t241" href="#t241">241</a></span><span class="t">            <span class="nam">me</span><span class="op">[</span><span class="nam">ivz</span><span class="op">]</span><span class="op">[</span><span class="op">:</span><span class="op">]</span> <span class="op">=</span> <span class="op">(</span><span class="nam">self</span><span class="op">.</span><span class="nam">BCtop</span> <span class="op">-</span> <span class="nam">self</span><span class="op">.</span><span class="nam">BCbottom</span><span class="op">)</span> <span class="op">/</span> <span class="num">2</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t242" href="#t242">242</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t243" href="#t243">243</a></span><span class="t">            <span class="com"># add noise</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t244" href="#t244">244</a></span><span class="t">            <span class="nam">rng</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">xp</span><span class="op">.</span><span class="nam">random</span><span class="op">.</span><span class="nam">default_rng</span><span class="op">(</span><span class="nam">seed</span><span class="op">=</span><span class="num">99</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t245" href="#t245">245</a></span><span class="t">            <span class="nam">me</span><span class="op">[</span><span class="nam">iv</span><span class="op">]</span><span class="op">.</span><span class="nam">real</span> <span class="op">+=</span> <span class="nam">rng</span><span class="op">.</span><span class="nam">normal</span><span class="op">(</span><span class="nam">size</span><span class="op">=</span><span class="nam">me</span><span class="op">[</span><span class="nam">iv</span><span class="op">]</span><span class="op">.</span><span class="nam">shape</span><span class="op">)</span> <span class="op">*</span> <span class="op">(</span><span class="nam">self</span><span class="op">.</span><span class="nam">Z</span> <span class="op">-</span> <span class="num">1</span><span class="op">)</span> <span class="op">*</span> <span class="op">(</span><span class="nam">self</span><span class="op">.</span><span class="nam">Z</span> <span class="op">+</span> <span class="num">1</span><span class="op">)</span> <span class="op">*</span> <span class="nam">noise_level</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t246" href="#t246">246</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t247" href="#t247">247</a></span><span class="t">        <span class="key">else</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t248" href="#t248">248</a></span><span class="t">            <span class="key">raise</span> <span class="nam">NotImplementedError</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t249" href="#t249">249</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t250" href="#t250">250</a></span><span class="t">        <span class="key">return</span> <span class="nam">me</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t251" href="#t251">251</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t252" href="#t252">252</a></span><span class="t">    <span class="key">def</span> <span class="nam">eval_f</span><span class="op">(</span><span class="nam">self</span><span class="op">,</span> <span class="nam">u</span><span class="op">,</span> <span class="op">*</span><span class="nam">args</span><span class="op">,</span> <span class="op">**</span><span class="nam">kwargs</span><span class="op">)</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t253" href="#t253">253</a></span><span class="t">        <span class="nam">f</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">f_init</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t254" href="#t254">254</a></span><span class="t">        <span class="nam">iu</span><span class="op">,</span> <span class="nam">iv</span><span class="op">,</span> <span class="nam">iux</span><span class="op">,</span> <span class="nam">iuz</span><span class="op">,</span> <span class="nam">ivx</span><span class="op">,</span> <span class="nam">ivz</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">index</span><span class="op">(</span><span class="nam">self</span><span class="op">.</span><span class="nam">components</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t255" href="#t255">255</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t256" href="#t256">256</a></span><span class="t">        <span class="nam">u_hat</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">transform</span><span class="op">(</span><span class="nam">u</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t257" href="#t257">257</a></span><span class="t">        <span class="nam">f_hat</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">u_init_forward</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t258" href="#t258">258</a></span><span class="t">        <span class="nam">f_hat</span><span class="op">[</span><span class="nam">iu</span><span class="op">]</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">epsilon</span> <span class="op">*</span> <span class="op">(</span><span class="op">(</span><span class="nam">self</span><span class="op">.</span><span class="nam">Dx</span> <span class="op">@</span> <span class="nam">u_hat</span><span class="op">[</span><span class="nam">iux</span><span class="op">]</span><span class="op">.</span><span class="nam">flatten</span><span class="op">(</span><span class="op">)</span> <span class="op">+</span> <span class="nam">self</span><span class="op">.</span><span class="nam">Dz</span> <span class="op">@</span> <span class="nam">u_hat</span><span class="op">[</span><span class="nam">iuz</span><span class="op">]</span><span class="op">.</span><span class="nam">flatten</span><span class="op">(</span><span class="op">)</span><span class="op">)</span><span class="op">)</span><span class="op">.</span><span class="nam">reshape</span><span class="op">(</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t259" href="#t259">259</a></span><span class="t">            <span class="nam">u_hat</span><span class="op">[</span><span class="nam">iux</span><span class="op">]</span><span class="op">.</span><span class="nam">shape</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t260" href="#t260">260</a></span><span class="t">        <span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t261" href="#t261">261</a></span><span class="t">        <span class="nam">f_hat</span><span class="op">[</span><span class="nam">iv</span><span class="op">]</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">epsilon</span> <span class="op">*</span> <span class="op">(</span><span class="op">(</span><span class="nam">self</span><span class="op">.</span><span class="nam">Dx</span> <span class="op">@</span> <span class="nam">u_hat</span><span class="op">[</span><span class="nam">ivx</span><span class="op">]</span><span class="op">.</span><span class="nam">flatten</span><span class="op">(</span><span class="op">)</span> <span class="op">+</span> <span class="nam">self</span><span class="op">.</span><span class="nam">Dz</span> <span class="op">@</span> <span class="nam">u_hat</span><span class="op">[</span><span class="nam">ivz</span><span class="op">]</span><span class="op">.</span><span class="nam">flatten</span><span class="op">(</span><span class="op">)</span><span class="op">)</span><span class="op">)</span><span class="op">.</span><span class="nam">reshape</span><span class="op">(</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t262" href="#t262">262</a></span><span class="t">            <span class="nam">u_hat</span><span class="op">[</span><span class="nam">iux</span><span class="op">]</span><span class="op">.</span><span class="nam">shape</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t263" href="#t263">263</a></span><span class="t">        <span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t264" href="#t264">264</a></span><span class="t">        <span class="nam">f</span><span class="op">.</span><span class="nam">impl</span><span class="op">[</span><span class="op">...</span><span class="op">]</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">itransform</span><span class="op">(</span><span class="nam">f_hat</span><span class="op">)</span><span class="op">.</span><span class="nam">real</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t265" href="#t265">265</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t266" href="#t266">266</a></span><span class="t">        <span class="nam">f</span><span class="op">.</span><span class="nam">expl</span><span class="op">[</span><span class="nam">iu</span><span class="op">]</span> <span class="op">=</span> <span class="op">-</span><span class="op">(</span><span class="nam">u</span><span class="op">[</span><span class="nam">iu</span><span class="op">]</span> <span class="op">*</span> <span class="nam">u</span><span class="op">[</span><span class="nam">iux</span><span class="op">]</span> <span class="op">+</span> <span class="nam">u</span><span class="op">[</span><span class="nam">iv</span><span class="op">]</span> <span class="op">*</span> <span class="nam">u</span><span class="op">[</span><span class="nam">iuz</span><span class="op">]</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t267" href="#t267">267</a></span><span class="t">        <span class="nam">f</span><span class="op">.</span><span class="nam">expl</span><span class="op">[</span><span class="nam">iv</span><span class="op">]</span> <span class="op">=</span> <span class="op">-</span><span class="op">(</span><span class="nam">u</span><span class="op">[</span><span class="nam">iu</span><span class="op">]</span> <span class="op">*</span> <span class="nam">u</span><span class="op">[</span><span class="nam">ivx</span><span class="op">]</span> <span class="op">+</span> <span class="nam">u</span><span class="op">[</span><span class="nam">iv</span><span class="op">]</span> <span class="op">*</span> <span class="nam">u</span><span class="op">[</span><span class="nam">ivz</span><span class="op">]</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t268" href="#t268">268</a></span><span class="t">        <span class="key">return</span> <span class="nam">f</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t269" href="#t269">269</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t270" href="#t270">270</a></span><span class="t">    <span class="key">def</span> <span class="nam">compute_vorticity</span><span class="op">(</span><span class="nam">self</span><span class="op">,</span> <span class="nam">u</span><span class="op">)</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t271" href="#t271">271</a></span><span class="t">        <span class="nam">me</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">u_init_forward</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t272" href="#t272">272</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t273" href="#t273">273</a></span><span class="t">        <span class="nam">u_hat</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">transform</span><span class="op">(</span><span class="nam">u</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t274" href="#t274">274</a></span><span class="t">        <span class="nam">iu</span><span class="op">,</span> <span class="nam">iv</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">index</span><span class="op">(</span><span class="op">[</span><span class="str">'u'</span><span class="op">,</span> <span class="str">'v'</span><span class="op">]</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t275" href="#t275">275</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t276" href="#t276">276</a></span><span class="t">        <span class="nam">me</span><span class="op">[</span><span class="nam">iu</span><span class="op">]</span> <span class="op">=</span> <span class="op">(</span><span class="nam">self</span><span class="op">.</span><span class="nam">Dx</span> <span class="op">@</span> <span class="nam">u_hat</span><span class="op">[</span><span class="nam">iv</span><span class="op">]</span><span class="op">.</span><span class="nam">flatten</span><span class="op">(</span><span class="op">)</span> <span class="op">+</span> <span class="nam">self</span><span class="op">.</span><span class="nam">Dz</span> <span class="op">@</span> <span class="nam">u_hat</span><span class="op">[</span><span class="nam">iu</span><span class="op">]</span><span class="op">.</span><span class="nam">flatten</span><span class="op">(</span><span class="op">)</span><span class="op">)</span><span class="op">.</span><span class="nam">reshape</span><span class="op">(</span><span class="nam">u</span><span class="op">[</span><span class="nam">iu</span><span class="op">]</span><span class="op">.</span><span class="nam">shape</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t277" href="#t277">277</a></span><span class="t">        <span class="key">return</span> <span class="nam">self</span><span class="op">.</span><span class="nam">itransform</span><span class="op">(</span><span class="nam">me</span><span class="op">)</span><span class="op">[</span><span class="nam">iu</span><span class="op">]</span><span class="op">.</span><span class="nam">real</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t278" href="#t278">278</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t279" href="#t279">279</a></span><span class="t">    <span class="key">def</span> <span class="nam">get_fig</span><span class="op">(</span><span class="nam">self</span><span class="op">)</span><span class="op">:</span>  <span class="com"># pragma: no cover</span>&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t280" href="#t280">280</a></span><span class="t">        <span class="str">"""</span>&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t281" href="#t281">281</a></span><span class="t"><span class="str">        Get a figure suitable to plot the solution of this problem</span>&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t282" href="#t282">282</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t283" href="#t283">283</a></span><span class="t"><span class="str">        Returns</span>&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t284" href="#t284">284</a></span><span class="t"><span class="str">        -------</span>&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t285" href="#t285">285</a></span><span class="t"><span class="str">        self.fig : matplotlib.pyplot.figure.Figure</span>&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t286" href="#t286">286</a></span><span class="t"><span class="str">        """</span>&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t287" href="#t287">287</a></span><span class="t">        <span class="key">import</span> <span class="nam">matplotlib</span><span class="op">.</span><span class="nam">pyplot</span> <span class="key">as</span> <span class="nam">plt</span>&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t288" href="#t288">288</a></span><span class="t">        <span class="key">from</span> <span class="nam">mpl_toolkits</span><span class="op">.</span><span class="nam">axes_grid1</span> <span class="key">import</span> <span class="nam">make_axes_locatable</span>&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t289" href="#t289">289</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t290" href="#t290">290</a></span><span class="t">        <span class="nam">plt</span><span class="op">.</span><span class="nam">rcParams</span><span class="op">[</span><span class="str">'figure.constrained_layout.use'</span><span class="op">]</span> <span class="op">=</span> <span class="key">True</span>&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t291" href="#t291">291</a></span><span class="t">        <span class="nam">self</span><span class="op">.</span><span class="nam">fig</span><span class="op">,</span> <span class="nam">axs</span> <span class="op">=</span> <span class="nam">plt</span><span class="op">.</span><span class="nam">subplots</span><span class="op">(</span><span class="num">3</span><span class="op">,</span> <span class="num">1</span><span class="op">,</span> <span class="nam">sharex</span><span class="op">=</span><span class="key">True</span><span class="op">,</span> <span class="nam">sharey</span><span class="op">=</span><span class="key">True</span><span class="op">,</span> <span class="nam">figsize</span><span class="op">=</span><span class="op">(</span><span class="op">(</span><span class="num">8</span><span class="op">,</span> <span class="num">7</span><span class="op">)</span><span class="op">)</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t292" href="#t292">292</a></span><span class="t">        <span class="nam">self</span><span class="op">.</span><span class="nam">cax</span> <span class="op">=</span> <span class="op">[</span><span class="op">]</span>&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t293" href="#t293">293</a></span><span class="t">        <span class="nam">divider</span> <span class="op">=</span> <span class="nam">make_axes_locatable</span><span class="op">(</span><span class="nam">axs</span><span class="op">[</span><span class="num">0</span><span class="op">]</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t294" href="#t294">294</a></span><span class="t">        <span class="nam">self</span><span class="op">.</span><span class="nam">cax</span> <span class="op">+=</span> <span class="op">[</span><span class="nam">divider</span><span class="op">.</span><span class="nam">append_axes</span><span class="op">(</span><span class="str">'right'</span><span class="op">,</span> <span class="nam">size</span><span class="op">=</span><span class="str">'3%'</span><span class="op">,</span> <span class="nam">pad</span><span class="op">=</span><span class="num">0.03</span><span class="op">)</span><span class="op">]</span>&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t295" href="#t295">295</a></span><span class="t">        <span class="nam">divider2</span> <span class="op">=</span> <span class="nam">make_axes_locatable</span><span class="op">(</span><span class="nam">axs</span><span class="op">[</span><span class="num">1</span><span class="op">]</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t296" href="#t296">296</a></span><span class="t">        <span class="nam">self</span><span class="op">.</span><span class="nam">cax</span> <span class="op">+=</span> <span class="op">[</span><span class="nam">divider2</span><span class="op">.</span><span class="nam">append_axes</span><span class="op">(</span><span class="str">'right'</span><span class="op">,</span> <span class="nam">size</span><span class="op">=</span><span class="str">'3%'</span><span class="op">,</span> <span class="nam">pad</span><span class="op">=</span><span class="num">0.03</span><span class="op">)</span><span class="op">]</span>&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t297" href="#t297">297</a></span><span class="t">        <span class="nam">divider3</span> <span class="op">=</span> <span class="nam">make_axes_locatable</span><span class="op">(</span><span class="nam">axs</span><span class="op">[</span><span class="num">2</span><span class="op">]</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t298" href="#t298">298</a></span><span class="t">        <span class="nam">self</span><span class="op">.</span><span class="nam">cax</span> <span class="op">+=</span> <span class="op">[</span><span class="nam">divider3</span><span class="op">.</span><span class="nam">append_axes</span><span class="op">(</span><span class="str">'right'</span><span class="op">,</span> <span class="nam">size</span><span class="op">=</span><span class="str">'3%'</span><span class="op">,</span> <span class="nam">pad</span><span class="op">=</span><span class="num">0.03</span><span class="op">)</span><span class="op">]</span>&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t299" href="#t299">299</a></span><span class="t">        <span class="key">return</span> <span class="nam">self</span><span class="op">.</span><span class="nam">fig</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t300" href="#t300">300</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t301" href="#t301">301</a></span><span class="t">    <span class="key">def</span> <span class="nam">plot</span><span class="op">(</span><span class="nam">self</span><span class="op">,</span> <span class="nam">u</span><span class="op">,</span> <span class="nam">t</span><span class="op">=</span><span class="key">None</span><span class="op">,</span> <span class="nam">fig</span><span class="op">=</span><span class="key">None</span><span class="op">,</span> <span class="nam">vmin</span><span class="op">=</span><span class="key">None</span><span class="op">,</span> <span class="nam">vmax</span><span class="op">=</span><span class="key">None</span><span class="op">)</span><span class="op">:</span>  <span class="com"># pragma: no cover</span>&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t302" href="#t302">302</a></span><span class="t">        <span class="str">r"""</span>&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t303" href="#t303">303</a></span><span class="t"><span class="str">        Plot the solution. Please supply a figure with the same structure as returned by ``self.get_fig``.</span>&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t304" href="#t304">304</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t305" href="#t305">305</a></span><span class="t"><span class="str">        Parameters</span>&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t306" href="#t306">306</a></span><span class="t"><span class="str">        ----------</span>&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t307" href="#t307">307</a></span><span class="t"><span class="str">        u : dtype_u</span>&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t308" href="#t308">308</a></span><span class="t"><span class="str">            Solution to be plotted</span>&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t309" href="#t309">309</a></span><span class="t"><span class="str">        t : float</span>&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t310" href="#t310">310</a></span><span class="t"><span class="str">            Time to display at the top of the figure</span>&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t311" href="#t311">311</a></span><span class="t"><span class="str">        fig : matplotlib.pyplot.figure.Figure</span>&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t312" href="#t312">312</a></span><span class="t"><span class="str">            Figure with the correct structure</span>&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t313" href="#t313">313</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t314" href="#t314">314</a></span><span class="t"><span class="str">        Returns</span>&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t315" href="#t315">315</a></span><span class="t"><span class="str">        -------</span>&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t316" href="#t316">316</a></span><span class="t"><span class="str">        None</span>&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t317" href="#t317">317</a></span><span class="t"><span class="str">        """</span>&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t318" href="#t318">318</a></span><span class="t">        <span class="nam">fig</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">get_fig</span><span class="op">(</span><span class="op">)</span> <span class="key">if</span> <span class="nam">fig</span> <span class="key">is</span> <span class="key">None</span> <span class="key">else</span> <span class="nam">fig</span>&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t319" href="#t319">319</a></span><span class="t">        <span class="nam">axs</span> <span class="op">=</span> <span class="nam">fig</span><span class="op">.</span><span class="nam">axes</span>&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t320" href="#t320">320</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t321" href="#t321">321</a></span><span class="t">        <span class="nam">iu</span><span class="op">,</span> <span class="nam">iv</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">index</span><span class="op">(</span><span class="op">[</span><span class="str">'u'</span><span class="op">,</span> <span class="str">'v'</span><span class="op">]</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t322" href="#t322">322</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t323" href="#t323">323</a></span><span class="t">        <span class="nam">imU</span> <span class="op">=</span> <span class="nam">axs</span><span class="op">[</span><span class="num">0</span><span class="op">]</span><span class="op">.</span><span class="nam">pcolormesh</span><span class="op">(</span><span class="nam">self</span><span class="op">.</span><span class="nam">X</span><span class="op">,</span> <span class="nam">self</span><span class="op">.</span><span class="nam">Z</span><span class="op">,</span> <span class="nam">u</span><span class="op">[</span><span class="nam">iu</span><span class="op">]</span><span class="op">.</span><span class="nam">real</span><span class="op">,</span> <span class="nam">vmin</span><span class="op">=</span><span class="nam">vmin</span><span class="op">,</span> <span class="nam">vmax</span><span class="op">=</span><span class="nam">vmax</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t324" href="#t324">324</a></span><span class="t">        <span class="nam">imV</span> <span class="op">=</span> <span class="nam">axs</span><span class="op">[</span><span class="num">1</span><span class="op">]</span><span class="op">.</span><span class="nam">pcolormesh</span><span class="op">(</span><span class="nam">self</span><span class="op">.</span><span class="nam">X</span><span class="op">,</span> <span class="nam">self</span><span class="op">.</span><span class="nam">Z</span><span class="op">,</span> <span class="nam">u</span><span class="op">[</span><span class="nam">iv</span><span class="op">]</span><span class="op">.</span><span class="nam">real</span><span class="op">,</span> <span class="nam">vmin</span><span class="op">=</span><span class="nam">vmin</span><span class="op">,</span> <span class="nam">vmax</span><span class="op">=</span><span class="nam">vmax</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t325" href="#t325">325</a></span><span class="t">        <span class="nam">imVort</span> <span class="op">=</span> <span class="nam">axs</span><span class="op">[</span><span class="num">2</span><span class="op">]</span><span class="op">.</span><span class="nam">pcolormesh</span><span class="op">(</span><span class="nam">self</span><span class="op">.</span><span class="nam">X</span><span class="op">,</span> <span class="nam">self</span><span class="op">.</span><span class="nam">Z</span><span class="op">,</span> <span class="nam">self</span><span class="op">.</span><span class="nam">compute_vorticity</span><span class="op">(</span><span class="nam">u</span><span class="op">)</span><span class="op">.</span><span class="nam">real</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t326" href="#t326">326</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t327" href="#t327">327</a></span><span class="t">        <span class="key">for</span> <span class="nam">i</span><span class="op">,</span> <span class="nam">label</span> <span class="key">in</span> <span class="nam">zip</span><span class="op">(</span><span class="op">[</span><span class="num">0</span><span class="op">,</span> <span class="num">1</span><span class="op">,</span> <span class="num">2</span><span class="op">]</span><span class="op">,</span> <span class="op">[</span><span class="str">r'$u$'</span><span class="op">,</span> <span class="str">'$v$'</span><span class="op">,</span> <span class="str">'vorticity'</span><span class="op">]</span><span class="op">)</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t328" href="#t328">328</a></span><span class="t">            <span class="nam">axs</span><span class="op">[</span><span class="nam">i</span><span class="op">]</span><span class="op">.</span><span class="nam">set_aspect</span><span class="op">(</span><span class="num">1</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t329" href="#t329">329</a></span><span class="t">            <span class="nam">axs</span><span class="op">[</span><span class="nam">i</span><span class="op">]</span><span class="op">.</span><span class="nam">set_title</span><span class="op">(</span><span class="nam">label</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t330" href="#t330">330</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t331" href="#t331">331</a></span><span class="t">        <span class="key">if</span> <span class="nam">t</span> <span class="key">is</span> <span class="key">not</span> <span class="key">None</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t332" href="#t332">332</a></span><span class="t">            <span class="nam">fig</span><span class="op">.</span><span class="nam">suptitle</span><span class="op">(</span><span class="fst">f'</span><span class="fst">t = </span><span class="op">{</span><span class="nam">t</span><span class="op">:</span><span class="fst">.2e</span><span class="op">}</span><span class="fst">'</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t333" href="#t333">333</a></span><span class="t">        <span class="nam">axs</span><span class="op">[</span><span class="op">-</span><span class="num">1</span><span class="op">]</span><span class="op">.</span><span class="nam">set_xlabel</span><span class="op">(</span><span class="str">r'$x$'</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t334" href="#t334">334</a></span><span class="t">        <span class="nam">axs</span><span class="op">[</span><span class="op">-</span><span class="num">1</span><span class="op">]</span><span class="op">.</span><span class="nam">set_ylabel</span><span class="op">(</span><span class="str">r'$z$'</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t335" href="#t335">335</a></span><span class="t">        <span class="nam">fig</span><span class="op">.</span><span class="nam">colorbar</span><span class="op">(</span><span class="nam">imU</span><span class="op">,</span> <span class="nam">self</span><span class="op">.</span><span class="nam">cax</span><span class="op">[</span><span class="num">0</span><span class="op">]</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t336" href="#t336">336</a></span><span class="t">        <span class="nam">fig</span><span class="op">.</span><span class="nam">colorbar</span><span class="op">(</span><span class="nam">imV</span><span class="op">,</span> <span class="nam">self</span><span class="op">.</span><span class="nam">cax</span><span class="op">[</span><span class="num">1</span><span class="op">]</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t337" href="#t337">337</a></span><span class="t">        <span class="nam">fig</span><span class="op">.</span><span class="nam">colorbar</span><span class="op">(</span><span class="nam">imVort</span><span class="op">,</span> <span class="nam">self</span><span class="op">.</span><span class="nam">cax</span><span class="op">[</span><span class="num">2</span><span class="op">]</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+</main>
+<footer>
+    <div class="content">
+        <p>
+            <a class="nav" href="z_91faa57f8583c837_BuckConverter_py.html">&#xab; prev</a> &nbsp; &nbsp;
+            <a class="nav" href="index.html">&Hat; index</a> &nbsp; &nbsp;
+            <a class="nav" href="z_91faa57f8583c837_DiscontinuousTestODE_py.html">&#xbb; next</a>
+            &nbsp; &nbsp; &nbsp;
+            <a class="nav" href="https://coverage.readthedocs.io/en/7.6.1">coverage.py v7.6.1</a>,
+            created at 2024-09-20 16:55 +0000
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+    </div>
+</footer>
+</body>
+</html>
diff --git a/coverage/z_91faa57f8583c837_DiscontinuousTestODE_py.html b/coverage/z_91faa57f8583c837_DiscontinuousTestODE_py.html
index 6a2935288f..dbf9346648 100644
--- a/coverage/z_91faa57f8583c837_DiscontinuousTestODE_py.html
+++ b/coverage/z_91faa57f8583c837_DiscontinuousTestODE_py.html
@@ -60,12 +60,12 @@ <h2>
             <button type="button" class="exc show_exc button_toggle_exc" value="exc" data-shortcut="x" title="Toggle lines excluded">2<span class="text"> excluded</span></button>
         </h2>
         <p class="text">
-            <a id="prevFileLink" class="nav" href="z_91faa57f8583c837_BuckConverter_py.html">&#xab; prev</a> &nbsp; &nbsp;
+            <a id="prevFileLink" class="nav" href="z_91faa57f8583c837_Burgers_py.html">&#xab; prev</a> &nbsp; &nbsp;
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+            created at 2024-09-20 16:55 +0000
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         <aside class="hidden">
             <button type="button" class="button_next_chunk" data-shortcut="j"></button>
@@ -369,12 +369,12 @@ <h2>
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     <div class="content">
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+            created at 2024-09-20 16:55 +0000
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index 0781252391..06a847edc3 100644
--- a/coverage/z_91faa57f8583c837_FastWaveSlowWave_0D_py.html
+++ b/coverage/z_91faa57f8583c837_FastWaveSlowWave_0D_py.html
@@ -65,7 +65,7 @@ <h2>
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+            created at 2024-09-20 16:55 +0000
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+            created at 2024-09-20 16:55 +0000
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index 8c2da418a0..9af53fd421 100644
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+++ b/coverage/z_91faa57f8583c837_FermiPastaUlamTsingou_py.html
@@ -65,7 +65,7 @@ <h2>
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+            created at 2024-09-20 16:55 +0000
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+            created at 2024-09-20 16:55 +0000
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index c2fa3d8e61..127ed90291 100644
--- a/coverage/z_91faa57f8583c837_FullSolarSystem_py.html
+++ b/coverage/z_91faa57f8583c837_FullSolarSystem_py.html
@@ -65,7 +65,7 @@ <h2>
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+            created at 2024-09-20 16:55 +0000
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diff --git a/coverage/z_91faa57f8583c837_GeneralizedFisher_1D_FD_implicit_py.html b/coverage/z_91faa57f8583c837_GeneralizedFisher_1D_FD_implicit_py.html
index 866d0c70c4..7c8e223772 100644
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+++ b/coverage/z_91faa57f8583c837_GeneralizedFisher_1D_FD_implicit_py.html
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+            created at 2024-09-20 16:55 +0000
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+            created at 2024-09-20 16:55 +0000
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index 709c8433db..7eae4e083e 100644
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+++ b/coverage/z_91faa57f8583c837_GeneralizedFisher_1D_PETSc_py.html
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+            created at 2024-09-20 16:55 +0000
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index f5409f695a..0db8d659aa 100644
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+++ b/coverage/z_91faa57f8583c837_GrayScott_1D_FEniCS_implicit_py.html
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+            created at 2024-09-20 16:55 +0000
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index d673d5ae05..ca8fcc8954 100644
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+++ b/coverage/z_91faa57f8583c837_GrayScott_2D_PETSc_periodic_py.html
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+            created at 2024-09-20 16:55 +0000
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+            created at 2024-09-20 16:55 +0000
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+            created at 2024-09-20 16:55 +0000
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index ae3871ef4f..9d5e9f8080 100644
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+            created at 2024-09-20 16:55 +0000
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index fdc02649d8..bb83863e2c 100644
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+++ b/coverage/z_91faa57f8583c837_HeatEquation_1D_FEniCS_matrix_forced_py.html
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+            created at 2024-09-20 16:55 +0000
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index 1f54eeeab1..90a05dd263 100644
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+++ b/coverage/z_91faa57f8583c837_HeatEquation_2D_PETSc_forced_py.html
@@ -62,10 +62,10 @@ <h2>
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+            <a id="nextFileLink" class="nav" href="z_91faa57f8583c837_HeatEquation_Chebychev_py.html">&#xbb; next</a>
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+            created at 2024-09-20 16:55 +0000
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+            <a class="nav" href="z_91faa57f8583c837_HeatEquation_Chebychev_py.html">&#xbb; next</a>
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diff --git a/coverage/z_91faa57f8583c837_HeatEquation_Chebychev_py.html b/coverage/z_91faa57f8583c837_HeatEquation_Chebychev_py.html
new file mode 100644
index 0000000000..3facf0314b
--- /dev/null
+++ b/coverage/z_91faa57f8583c837_HeatEquation_Chebychev_py.html
@@ -0,0 +1,541 @@
+<!DOCTYPE html>
+<html lang="en">
+<head>
+    <meta http-equiv="Content-Type" content="text/html; charset=utf-8">
+    <title>Coverage for pySDC/implementations/problem_classes/HeatEquation_Chebychev.py: 100%</title>
+    <link rel="icon" sizes="32x32" href="favicon_32_cb_58284776.png">
+    <link rel="stylesheet" href="style_cb_8e611ae1.css" type="text/css">
+    <script src="coverage_html_cb_6fb7b396.js" defer></script>
+</head>
+<body class="pyfile">
+<header>
+    <div class="content">
+        <h1>
+            <span class="text">Coverage for </span><b>pySDC/implementations/problem_classes/HeatEquation_Chebychev.py</b>:
+            <span class="pc_cov">100%</span>
+        </h1>
+        <aside id="help_panel_wrapper">
+            <input id="help_panel_state" type="checkbox">
+            <label for="help_panel_state">
+                <img id="keyboard_icon" src="keybd_closed_cb_ce680311.png" alt="Show/hide keyboard shortcuts">
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+            <div id="help_panel">
+                <p class="legend">Shortcuts on this page</p>
+                <div class="keyhelp">
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+                        <kbd>0</kbd> &nbsp; (zero) top of page
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+        <h2>
+            <span class="text">220 statements &nbsp;</span>
+            <button type="button" class="run button_toggle_run" value="run" data-shortcut="r" title="Toggle lines run">220<span class="text"> run</span></button>
+            <button type="button" class="mis show_mis button_toggle_mis" value="mis" data-shortcut="m" title="Toggle lines missing">0<span class="text"> missing</span></button>
+            <button type="button" class="exc show_exc button_toggle_exc" value="exc" data-shortcut="x" title="Toggle lines excluded">4<span class="text"> excluded</span></button>
+        </h2>
+        <p class="text">
+            <a id="prevFileLink" class="nav" href="z_91faa57f8583c837_HeatEquation_2D_PETSc_forced_py.html">&#xab; prev</a> &nbsp; &nbsp;
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+        <aside class="hidden">
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+            <button type="button" class="button_show_hide_help" data-shortcut="?"></button>
+        </aside>
+    </div>
+</header>
+<main id="source">
+    <p class="run"><span class="n"><a id="t1" href="#t1">1</a></span><span class="t"><span class="key">import</span> <span class="nam">numpy</span> <span class="key">as</span> <span class="nam">np</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t2" href="#t2">2</a></span><span class="t"><span class="key">from</span> <span class="nam">scipy</span> <span class="key">import</span> <span class="nam">sparse</span> <span class="key">as</span> <span class="nam">sp</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t3" href="#t3">3</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t4" href="#t4">4</a></span><span class="t"><span class="key">from</span> <span class="nam">pySDC</span><span class="op">.</span><span class="nam">core</span><span class="op">.</span><span class="nam">problem</span> <span class="key">import</span> <span class="nam">Problem</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t5" href="#t5">5</a></span><span class="t"><span class="key">from</span> <span class="nam">pySDC</span><span class="op">.</span><span class="nam">implementations</span><span class="op">.</span><span class="nam">datatype_classes</span><span class="op">.</span><span class="nam">mesh</span> <span class="key">import</span> <span class="nam">mesh</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t6" href="#t6">6</a></span><span class="t"><span class="key">from</span> <span class="nam">pySDC</span><span class="op">.</span><span class="nam">implementations</span><span class="op">.</span><span class="nam">problem_classes</span><span class="op">.</span><span class="nam">generic_spectral</span> <span class="key">import</span> <span class="nam">GenericSpectralLinear</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t7" href="#t7">7</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t8" href="#t8">8</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t9" href="#t9">9</a></span><span class="t"><span class="key">class</span> <span class="nam">Heat1DChebychev</span><span class="op">(</span><span class="nam">GenericSpectralLinear</span><span class="op">)</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t10" href="#t10">10</a></span><span class="t">    <span class="str">"""</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t11" href="#t11">11</a></span><span class="t"><span class="str">    1D Heat equation with Dirichlet Boundary conditions discretized on (-1, 1) using a Chebychev spectral method.</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t12" href="#t12">12</a></span><span class="t"><span class="str">    """</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t13" href="#t13">13</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t14" href="#t14">14</a></span><span class="t">    <span class="nam">dtype_u</span> <span class="op">=</span> <span class="nam">mesh</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t15" href="#t15">15</a></span><span class="t">    <span class="nam">dtype_f</span> <span class="op">=</span> <span class="nam">mesh</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t16" href="#t16">16</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t17" href="#t17">17</a></span><span class="t">    <span class="key">def</span> <span class="nam">__init__</span><span class="op">(</span><span class="nam">self</span><span class="op">,</span> <span class="nam">nvars</span><span class="op">=</span><span class="num">128</span><span class="op">,</span> <span class="nam">a</span><span class="op">=</span><span class="num">0</span><span class="op">,</span> <span class="nam">b</span><span class="op">=</span><span class="num">0</span><span class="op">,</span> <span class="nam">f</span><span class="op">=</span><span class="num">1</span><span class="op">,</span> <span class="nam">nu</span><span class="op">=</span><span class="num">1.0</span><span class="op">,</span> <span class="nam">mode</span><span class="op">=</span><span class="str">'T2U'</span><span class="op">,</span> <span class="op">**</span><span class="nam">kwargs</span><span class="op">)</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t18" href="#t18">18</a></span><span class="t">        <span class="str">"""</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t19" href="#t19">19</a></span><span class="t"><span class="str">        Constructor. `kwargs` are forwarded to parent class constructor.</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t20" href="#t20">20</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t21" href="#t21">21</a></span><span class="t"><span class="str">        Args:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t22" href="#t22">22</a></span><span class="t"><span class="str">            nvars (int): Resolution</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t23" href="#t23">23</a></span><span class="t"><span class="str">            a (float): Left BC value</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t24" href="#t24">24</a></span><span class="t"><span class="str">            b (float): Right BC value</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t25" href="#t25">25</a></span><span class="t"><span class="str">            f (int): Frequency of the solution</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t26" href="#t26">26</a></span><span class="t"><span class="str">            nu (float): Diffusion parameter</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t27" href="#t27">27</a></span><span class="t"><span class="str">            mode ('T2T' or 'T2U'): Mode for Chebychev method.</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t28" href="#t28">28</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t29" href="#t29">29</a></span><span class="t"><span class="str">        """</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t30" href="#t30">30</a></span><span class="t">        <span class="nam">self</span><span class="op">.</span><span class="nam">_makeAttributeAndRegister</span><span class="op">(</span><span class="str">'nvars'</span><span class="op">,</span> <span class="str">'a'</span><span class="op">,</span> <span class="str">'b'</span><span class="op">,</span> <span class="str">'f'</span><span class="op">,</span> <span class="str">'nu'</span><span class="op">,</span> <span class="str">'mode'</span><span class="op">,</span> <span class="nam">localVars</span><span class="op">=</span><span class="nam">locals</span><span class="op">(</span><span class="op">)</span><span class="op">,</span> <span class="nam">readOnly</span><span class="op">=</span><span class="key">True</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t31" href="#t31">31</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t32" href="#t32">32</a></span><span class="t">        <span class="nam">bases</span> <span class="op">=</span> <span class="op">[</span><span class="op">{</span><span class="str">'base'</span><span class="op">:</span> <span class="str">'chebychev'</span><span class="op">,</span> <span class="str">'N'</span><span class="op">:</span> <span class="nam">nvars</span><span class="op">}</span><span class="op">]</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t33" href="#t33">33</a></span><span class="t">        <span class="nam">components</span> <span class="op">=</span> <span class="op">[</span><span class="str">'u'</span><span class="op">,</span> <span class="str">'ux'</span><span class="op">]</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t34" href="#t34">34</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t35" href="#t35">35</a></span><span class="t">        <span class="nam">super</span><span class="op">(</span><span class="op">)</span><span class="op">.</span><span class="nam">__init__</span><span class="op">(</span><span class="nam">bases</span><span class="op">,</span> <span class="nam">components</span><span class="op">,</span> <span class="nam">real_spectral_coefficients</span><span class="op">=</span><span class="key">True</span><span class="op">,</span> <span class="op">**</span><span class="nam">kwargs</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t36" href="#t36">36</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t37" href="#t37">37</a></span><span class="t">        <span class="nam">self</span><span class="op">.</span><span class="nam">x</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">get_grid</span><span class="op">(</span><span class="op">)</span><span class="op">[</span><span class="num">0</span><span class="op">]</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t38" href="#t38">38</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t39" href="#t39">39</a></span><span class="t">        <span class="nam">I</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">get_Id</span><span class="op">(</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t40" href="#t40">40</a></span><span class="t">        <span class="nam">Dx</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">get_differentiation_matrix</span><span class="op">(</span><span class="nam">axes</span><span class="op">=</span><span class="op">(</span><span class="num">0</span><span class="op">,</span><span class="op">)</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t41" href="#t41">41</a></span><span class="t">        <span class="nam">self</span><span class="op">.</span><span class="nam">Dx</span> <span class="op">=</span> <span class="nam">Dx</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t42" href="#t42">42</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t43" href="#t43">43</a></span><span class="t">        <span class="nam">self</span><span class="op">.</span><span class="nam">T2U</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">get_basis_change_matrix</span><span class="op">(</span><span class="nam">conv</span><span class="op">=</span><span class="nam">mode</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t44" href="#t44">44</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t45" href="#t45">45</a></span><span class="t">        <span class="nam">L_lhs</span> <span class="op">=</span> <span class="op">{</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t46" href="#t46">46</a></span><span class="t">            <span class="str">'ux'</span><span class="op">:</span> <span class="op">{</span><span class="str">'u'</span><span class="op">:</span> <span class="op">-</span><span class="nam">self</span><span class="op">.</span><span class="nam">T2U</span> <span class="op">@</span> <span class="nam">Dx</span><span class="op">,</span> <span class="str">'ux'</span><span class="op">:</span> <span class="nam">self</span><span class="op">.</span><span class="nam">T2U</span> <span class="op">@</span> <span class="nam">I</span><span class="op">}</span><span class="op">,</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t47" href="#t47">47</a></span><span class="t">            <span class="str">'u'</span><span class="op">:</span> <span class="op">{</span><span class="str">'ux'</span><span class="op">:</span> <span class="op">-</span><span class="nam">nu</span> <span class="op">*</span> <span class="op">(</span><span class="nam">self</span><span class="op">.</span><span class="nam">T2U</span> <span class="op">@</span> <span class="nam">Dx</span><span class="op">)</span><span class="op">}</span><span class="op">,</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t48" href="#t48">48</a></span><span class="t">        <span class="op">}</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t49" href="#t49">49</a></span><span class="t">        <span class="nam">self</span><span class="op">.</span><span class="nam">setup_L</span><span class="op">(</span><span class="nam">L_lhs</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t50" href="#t50">50</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t51" href="#t51">51</a></span><span class="t">        <span class="nam">M_lhs</span> <span class="op">=</span> <span class="op">{</span><span class="str">'u'</span><span class="op">:</span> <span class="op">{</span><span class="str">'u'</span><span class="op">:</span> <span class="nam">self</span><span class="op">.</span><span class="nam">T2U</span> <span class="op">@</span> <span class="nam">I</span><span class="op">}</span><span class="op">}</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t52" href="#t52">52</a></span><span class="t">        <span class="nam">self</span><span class="op">.</span><span class="nam">setup_M</span><span class="op">(</span><span class="nam">M_lhs</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t53" href="#t53">53</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t54" href="#t54">54</a></span><span class="t">        <span class="nam">self</span><span class="op">.</span><span class="nam">add_BC</span><span class="op">(</span><span class="nam">component</span><span class="op">=</span><span class="str">'u'</span><span class="op">,</span> <span class="nam">equation</span><span class="op">=</span><span class="str">'u'</span><span class="op">,</span> <span class="nam">axis</span><span class="op">=</span><span class="num">0</span><span class="op">,</span> <span class="nam">x</span><span class="op">=</span><span class="op">-</span><span class="num">1</span><span class="op">,</span> <span class="nam">v</span><span class="op">=</span><span class="nam">a</span><span class="op">,</span> <span class="nam">kind</span><span class="op">=</span><span class="str">"Dirichlet"</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t55" href="#t55">55</a></span><span class="t">        <span class="nam">self</span><span class="op">.</span><span class="nam">add_BC</span><span class="op">(</span><span class="nam">component</span><span class="op">=</span><span class="str">'u'</span><span class="op">,</span> <span class="nam">equation</span><span class="op">=</span><span class="str">'ux'</span><span class="op">,</span> <span class="nam">axis</span><span class="op">=</span><span class="num">0</span><span class="op">,</span> <span class="nam">x</span><span class="op">=</span><span class="num">1</span><span class="op">,</span> <span class="nam">v</span><span class="op">=</span><span class="nam">b</span><span class="op">,</span> <span class="nam">kind</span><span class="op">=</span><span class="str">"Dirichlet"</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t56" href="#t56">56</a></span><span class="t">        <span class="nam">self</span><span class="op">.</span><span class="nam">setup_BCs</span><span class="op">(</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t57" href="#t57">57</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t58" href="#t58">58</a></span><span class="t">    <span class="key">def</span> <span class="nam">eval_f</span><span class="op">(</span><span class="nam">self</span><span class="op">,</span> <span class="nam">u</span><span class="op">,</span> <span class="op">*</span><span class="nam">args</span><span class="op">,</span> <span class="op">**</span><span class="nam">kwargs</span><span class="op">)</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t59" href="#t59">59</a></span><span class="t">        <span class="nam">f</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">f_init</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t60" href="#t60">60</a></span><span class="t">        <span class="nam">iu</span><span class="op">,</span> <span class="nam">iux</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">index</span><span class="op">(</span><span class="nam">self</span><span class="op">.</span><span class="nam">components</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t61" href="#t61">61</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t62" href="#t62">62</a></span><span class="t">        <span class="key">if</span> <span class="nam">self</span><span class="op">.</span><span class="nam">spectral_space</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t63" href="#t63">63</a></span><span class="t">            <span class="nam">u_hat</span> <span class="op">=</span> <span class="nam">u</span><span class="op">.</span><span class="nam">copy</span><span class="op">(</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t64" href="#t64">64</a></span><span class="t">        <span class="key">else</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t65" href="#t65">65</a></span><span class="t">            <span class="nam">u_hat</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">transform</span><span class="op">(</span><span class="nam">u</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t66" href="#t66">66</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t67" href="#t67">67</a></span><span class="t">        <span class="nam">u_hat</span><span class="op">[</span><span class="nam">iu</span><span class="op">]</span> <span class="op">=</span> <span class="op">(</span><span class="nam">self</span><span class="op">.</span><span class="nam">nu</span> <span class="op">*</span> <span class="nam">self</span><span class="op">.</span><span class="nam">Dx</span> <span class="op">@</span> <span class="nam">u_hat</span><span class="op">[</span><span class="nam">iux</span><span class="op">]</span><span class="op">.</span><span class="nam">flatten</span><span class="op">(</span><span class="op">)</span><span class="op">)</span><span class="op">.</span><span class="nam">reshape</span><span class="op">(</span><span class="nam">u_hat</span><span class="op">[</span><span class="nam">iu</span><span class="op">]</span><span class="op">.</span><span class="nam">shape</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t68" href="#t68">68</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t69" href="#t69">69</a></span><span class="t">        <span class="key">if</span> <span class="nam">self</span><span class="op">.</span><span class="nam">spectral_space</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t70" href="#t70">70</a></span><span class="t">            <span class="nam">me</span> <span class="op">=</span> <span class="nam">u_hat</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t71" href="#t71">71</a></span><span class="t">        <span class="key">else</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t72" href="#t72">72</a></span><span class="t">            <span class="nam">me</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">itransform</span><span class="op">(</span><span class="nam">u_hat</span><span class="op">)</span><span class="op">.</span><span class="nam">real</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t73" href="#t73">73</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t74" href="#t74">74</a></span><span class="t">        <span class="nam">f</span><span class="op">[</span><span class="nam">iu</span><span class="op">]</span> <span class="op">=</span> <span class="nam">me</span><span class="op">[</span><span class="nam">iu</span><span class="op">]</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t75" href="#t75">75</a></span><span class="t">        <span class="key">return</span> <span class="nam">f</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t76" href="#t76">76</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t77" href="#t77">77</a></span><span class="t">    <span class="key">def</span> <span class="nam">u_exact</span><span class="op">(</span><span class="nam">self</span><span class="op">,</span> <span class="nam">t</span><span class="op">,</span> <span class="nam">noise</span><span class="op">=</span><span class="num">0</span><span class="op">,</span> <span class="nam">seed</span><span class="op">=</span><span class="num">666</span><span class="op">)</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t78" href="#t78">78</a></span><span class="t">        <span class="str">"""</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t79" href="#t79">79</a></span><span class="t"><span class="str">        Get exact solution at time `t`</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t80" href="#t80">80</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t81" href="#t81">81</a></span><span class="t"><span class="str">        Args:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t82" href="#t82">82</a></span><span class="t"><span class="str">            t (float): When you want the exact solution</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t83" href="#t83">83</a></span><span class="t"><span class="str">            noise (float): Add noise of this level</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t84" href="#t84">84</a></span><span class="t"><span class="str">            seed (int): Random seed for the noise</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t85" href="#t85">85</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t86" href="#t86">86</a></span><span class="t"><span class="str">        Returns:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t87" href="#t87">87</a></span><span class="t"><span class="str">            Heat1DChebychev.dtype_u: Exact solution</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t88" href="#t88">88</a></span><span class="t"><span class="str">        """</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t89" href="#t89">89</a></span><span class="t">        <span class="nam">xp</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">xp</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t90" href="#t90">90</a></span><span class="t">        <span class="nam">iu</span><span class="op">,</span> <span class="nam">iux</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">index</span><span class="op">(</span><span class="nam">self</span><span class="op">.</span><span class="nam">components</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t91" href="#t91">91</a></span><span class="t">        <span class="nam">u</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">spectral</span><span class="op">.</span><span class="nam">u_init</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t92" href="#t92">92</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t93" href="#t93">93</a></span><span class="t">        <span class="nam">u</span><span class="op">[</span><span class="nam">iu</span><span class="op">]</span> <span class="op">=</span> <span class="op">(</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t94" href="#t94">94</a></span><span class="t">            <span class="nam">xp</span><span class="op">.</span><span class="nam">sin</span><span class="op">(</span><span class="nam">self</span><span class="op">.</span><span class="nam">f</span> <span class="op">*</span> <span class="nam">np</span><span class="op">.</span><span class="nam">pi</span> <span class="op">*</span> <span class="nam">self</span><span class="op">.</span><span class="nam">x</span><span class="op">)</span> <span class="op">*</span> <span class="nam">xp</span><span class="op">.</span><span class="nam">exp</span><span class="op">(</span><span class="op">-</span><span class="nam">self</span><span class="op">.</span><span class="nam">nu</span> <span class="op">*</span> <span class="op">(</span><span class="nam">self</span><span class="op">.</span><span class="nam">f</span> <span class="op">*</span> <span class="nam">np</span><span class="op">.</span><span class="nam">pi</span><span class="op">)</span> <span class="op">**</span> <span class="num">2</span> <span class="op">*</span> <span class="nam">t</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t95" href="#t95">95</a></span><span class="t">            <span class="op">+</span> <span class="op">(</span><span class="nam">self</span><span class="op">.</span><span class="nam">b</span> <span class="op">-</span> <span class="nam">self</span><span class="op">.</span><span class="nam">a</span><span class="op">)</span> <span class="op">/</span> <span class="num">2</span> <span class="op">*</span> <span class="nam">self</span><span class="op">.</span><span class="nam">x</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t96" href="#t96">96</a></span><span class="t">            <span class="op">+</span> <span class="op">(</span><span class="nam">self</span><span class="op">.</span><span class="nam">b</span> <span class="op">+</span> <span class="nam">self</span><span class="op">.</span><span class="nam">a</span><span class="op">)</span> <span class="op">/</span> <span class="num">2</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t97" href="#t97">97</a></span><span class="t">        <span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t98" href="#t98">98</a></span><span class="t">        <span class="nam">u</span><span class="op">[</span><span class="nam">iux</span><span class="op">]</span> <span class="op">=</span> <span class="op">(</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t99" href="#t99">99</a></span><span class="t">            <span class="nam">self</span><span class="op">.</span><span class="nam">f</span> <span class="op">*</span> <span class="nam">np</span><span class="op">.</span><span class="nam">pi</span> <span class="op">*</span> <span class="nam">xp</span><span class="op">.</span><span class="nam">cos</span><span class="op">(</span><span class="nam">self</span><span class="op">.</span><span class="nam">f</span> <span class="op">*</span> <span class="nam">np</span><span class="op">.</span><span class="nam">pi</span> <span class="op">*</span> <span class="nam">self</span><span class="op">.</span><span class="nam">x</span><span class="op">)</span> <span class="op">*</span> <span class="nam">xp</span><span class="op">.</span><span class="nam">exp</span><span class="op">(</span><span class="op">-</span><span class="nam">self</span><span class="op">.</span><span class="nam">nu</span> <span class="op">*</span> <span class="op">(</span><span class="nam">self</span><span class="op">.</span><span class="nam">f</span> <span class="op">*</span> <span class="nam">np</span><span class="op">.</span><span class="nam">pi</span><span class="op">)</span> <span class="op">**</span> <span class="num">2</span> <span class="op">*</span> <span class="nam">t</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t100" href="#t100">100</a></span><span class="t">            <span class="op">+</span> <span class="op">(</span><span class="nam">self</span><span class="op">.</span><span class="nam">b</span> <span class="op">-</span> <span class="nam">self</span><span class="op">.</span><span class="nam">a</span><span class="op">)</span> <span class="op">/</span> <span class="num">2</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t101" href="#t101">101</a></span><span class="t">        <span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t102" href="#t102">102</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t103" href="#t103">103</a></span><span class="t">        <span class="key">if</span> <span class="nam">noise</span> <span class="op">></span> <span class="num">0</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t104" href="#t104">104</a></span><span class="t">            <span class="key">assert</span> <span class="nam">t</span> <span class="op">==</span> <span class="num">0</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t105" href="#t105">105</a></span><span class="t">            <span class="nam">_noise</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">u_init</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t106" href="#t106">106</a></span><span class="t">            <span class="nam">rng</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">xp</span><span class="op">.</span><span class="nam">random</span><span class="op">.</span><span class="nam">default_rng</span><span class="op">(</span><span class="nam">seed</span><span class="op">=</span><span class="nam">seed</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t107" href="#t107">107</a></span><span class="t">            <span class="nam">_noise</span><span class="op">[</span><span class="nam">iu</span><span class="op">]</span> <span class="op">=</span> <span class="nam">rng</span><span class="op">.</span><span class="nam">normal</span><span class="op">(</span><span class="nam">size</span><span class="op">=</span><span class="nam">u</span><span class="op">[</span><span class="nam">iu</span><span class="op">]</span><span class="op">.</span><span class="nam">shape</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t108" href="#t108">108</a></span><span class="t">            <span class="nam">noise_hat</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">transform</span><span class="op">(</span><span class="nam">_noise</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t109" href="#t109">109</a></span><span class="t">            <span class="nam">low_pass</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">get_filter_matrix</span><span class="op">(</span><span class="nam">axis</span><span class="op">=</span><span class="num">0</span><span class="op">,</span> <span class="nam">kmax</span><span class="op">=</span><span class="nam">self</span><span class="op">.</span><span class="nam">nvars</span> <span class="op">-</span> <span class="num">2</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t110" href="#t110">110</a></span><span class="t">            <span class="nam">noise_hat</span><span class="op">[</span><span class="nam">iu</span><span class="op">]</span> <span class="op">=</span> <span class="op">(</span><span class="nam">low_pass</span> <span class="op">@</span> <span class="nam">noise_hat</span><span class="op">[</span><span class="nam">iu</span><span class="op">]</span><span class="op">.</span><span class="nam">flatten</span><span class="op">(</span><span class="op">)</span><span class="op">)</span><span class="op">.</span><span class="nam">reshape</span><span class="op">(</span><span class="nam">noise_hat</span><span class="op">[</span><span class="nam">iu</span><span class="op">]</span><span class="op">.</span><span class="nam">shape</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t111" href="#t111">111</a></span><span class="t">            <span class="nam">_noise</span><span class="op">[</span><span class="op">:</span><span class="op">]</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">itransform</span><span class="op">(</span><span class="nam">noise_hat</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t112" href="#t112">112</a></span><span class="t">            <span class="nam">u</span> <span class="op">+=</span> <span class="nam">_noise</span> <span class="op">*</span> <span class="nam">noise</span> <span class="op">*</span> <span class="op">(</span><span class="nam">self</span><span class="op">.</span><span class="nam">x</span> <span class="op">-</span> <span class="num">1</span><span class="op">)</span> <span class="op">*</span> <span class="op">(</span><span class="nam">self</span><span class="op">.</span><span class="nam">x</span> <span class="op">+</span> <span class="num">1</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t113" href="#t113">113</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t114" href="#t114">114</a></span><span class="t">        <span class="nam">self</span><span class="op">.</span><span class="nam">check_BCs</span><span class="op">(</span><span class="nam">u</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t115" href="#t115">115</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t116" href="#t116">116</a></span><span class="t">        <span class="key">if</span> <span class="nam">self</span><span class="op">.</span><span class="nam">spectral_space</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t117" href="#t117">117</a></span><span class="t">            <span class="nam">u_hat</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">u_init</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t118" href="#t118">118</a></span><span class="t">            <span class="nam">u_hat</span><span class="op">[</span><span class="op">...</span><span class="op">]</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">transform</span><span class="op">(</span><span class="nam">u</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t119" href="#t119">119</a></span><span class="t">            <span class="key">return</span> <span class="nam">u_hat</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t120" href="#t120">120</a></span><span class="t">        <span class="key">else</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t121" href="#t121">121</a></span><span class="t">            <span class="key">return</span> <span class="nam">u</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t122" href="#t122">122</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t123" href="#t123">123</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t124" href="#t124">124</a></span><span class="t"><span class="key">class</span> <span class="nam">Heat1DUltraspherical</span><span class="op">(</span><span class="nam">GenericSpectralLinear</span><span class="op">)</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t125" href="#t125">125</a></span><span class="t">    <span class="str">"""</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t126" href="#t126">126</a></span><span class="t"><span class="str">    1D Heat equation with Dirichlet Boundary conditions discretized on (-1, 1) using an ultraspherical spectral method.</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t127" href="#t127">127</a></span><span class="t"><span class="str">    """</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t128" href="#t128">128</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t129" href="#t129">129</a></span><span class="t">    <span class="nam">dtype_u</span> <span class="op">=</span> <span class="nam">mesh</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t130" href="#t130">130</a></span><span class="t">    <span class="nam">dtype_f</span> <span class="op">=</span> <span class="nam">mesh</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t131" href="#t131">131</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t132" href="#t132">132</a></span><span class="t">    <span class="key">def</span> <span class="nam">__init__</span><span class="op">(</span><span class="nam">self</span><span class="op">,</span> <span class="nam">nvars</span><span class="op">=</span><span class="num">128</span><span class="op">,</span> <span class="nam">a</span><span class="op">=</span><span class="num">0</span><span class="op">,</span> <span class="nam">b</span><span class="op">=</span><span class="num">0</span><span class="op">,</span> <span class="nam">f</span><span class="op">=</span><span class="num">1</span><span class="op">,</span> <span class="nam">nu</span><span class="op">=</span><span class="num">1.0</span><span class="op">,</span> <span class="op">**</span><span class="nam">kwargs</span><span class="op">)</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t133" href="#t133">133</a></span><span class="t">        <span class="str">"""</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t134" href="#t134">134</a></span><span class="t"><span class="str">        Constructor. `kwargs` are forwarded to parent class constructor.</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t135" href="#t135">135</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t136" href="#t136">136</a></span><span class="t"><span class="str">        Args:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t137" href="#t137">137</a></span><span class="t"><span class="str">            nvars (int): Resolution</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t138" href="#t138">138</a></span><span class="t"><span class="str">            a (float): Left BC value</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t139" href="#t139">139</a></span><span class="t"><span class="str">            b (float): Right BC value</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t140" href="#t140">140</a></span><span class="t"><span class="str">            f (int): Frequency of the solution</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t141" href="#t141">141</a></span><span class="t"><span class="str">            nu (float): Diffusion parameter</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t142" href="#t142">142</a></span><span class="t"><span class="str">        """</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t143" href="#t143">143</a></span><span class="t">        <span class="nam">self</span><span class="op">.</span><span class="nam">_makeAttributeAndRegister</span><span class="op">(</span><span class="str">'nvars'</span><span class="op">,</span> <span class="str">'a'</span><span class="op">,</span> <span class="str">'b'</span><span class="op">,</span> <span class="str">'f'</span><span class="op">,</span> <span class="str">'nu'</span><span class="op">,</span> <span class="nam">localVars</span><span class="op">=</span><span class="nam">locals</span><span class="op">(</span><span class="op">)</span><span class="op">,</span> <span class="nam">readOnly</span><span class="op">=</span><span class="key">True</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t144" href="#t144">144</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t145" href="#t145">145</a></span><span class="t">        <span class="nam">bases</span> <span class="op">=</span> <span class="op">[</span><span class="op">{</span><span class="str">'base'</span><span class="op">:</span> <span class="str">'ultraspherical'</span><span class="op">,</span> <span class="str">'N'</span><span class="op">:</span> <span class="nam">nvars</span><span class="op">}</span><span class="op">]</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t146" href="#t146">146</a></span><span class="t">        <span class="nam">components</span> <span class="op">=</span> <span class="op">[</span><span class="str">'u'</span><span class="op">]</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t147" href="#t147">147</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t148" href="#t148">148</a></span><span class="t">        <span class="nam">GenericSpectralLinear</span><span class="op">.</span><span class="nam">__init__</span><span class="op">(</span><span class="nam">self</span><span class="op">,</span> <span class="nam">bases</span><span class="op">,</span> <span class="nam">components</span><span class="op">,</span> <span class="nam">real_spectral_coefficients</span><span class="op">=</span><span class="key">True</span><span class="op">,</span> <span class="op">**</span><span class="nam">kwargs</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t149" href="#t149">149</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t150" href="#t150">150</a></span><span class="t">        <span class="nam">self</span><span class="op">.</span><span class="nam">x</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">get_grid</span><span class="op">(</span><span class="op">)</span><span class="op">[</span><span class="num">0</span><span class="op">]</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t151" href="#t151">151</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t152" href="#t152">152</a></span><span class="t">        <span class="nam">I</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">get_Id</span><span class="op">(</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t153" href="#t153">153</a></span><span class="t">        <span class="nam">Dxx</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">get_differentiation_matrix</span><span class="op">(</span><span class="nam">axes</span><span class="op">=</span><span class="op">(</span><span class="num">0</span><span class="op">,</span><span class="op">)</span><span class="op">,</span> <span class="nam">p</span><span class="op">=</span><span class="num">2</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t154" href="#t154">154</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t155" href="#t155">155</a></span><span class="t">        <span class="nam">S2</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">get_basis_change_matrix</span><span class="op">(</span><span class="nam">p_in</span><span class="op">=</span><span class="num">2</span><span class="op">,</span> <span class="nam">p_out</span><span class="op">=</span><span class="num">0</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t156" href="#t156">156</a></span><span class="t">        <span class="nam">U2</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">get_basis_change_matrix</span><span class="op">(</span><span class="nam">p_in</span><span class="op">=</span><span class="num">0</span><span class="op">,</span> <span class="nam">p_out</span><span class="op">=</span><span class="num">2</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t157" href="#t157">157</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t158" href="#t158">158</a></span><span class="t">        <span class="nam">self</span><span class="op">.</span><span class="nam">Dxx</span> <span class="op">=</span> <span class="nam">S2</span> <span class="op">@</span> <span class="nam">Dxx</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t159" href="#t159">159</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t160" href="#t160">160</a></span><span class="t">        <span class="nam">L_lhs</span> <span class="op">=</span> <span class="op">{</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t161" href="#t161">161</a></span><span class="t">            <span class="str">'u'</span><span class="op">:</span> <span class="op">{</span><span class="str">'u'</span><span class="op">:</span> <span class="op">-</span><span class="nam">nu</span> <span class="op">*</span> <span class="nam">Dxx</span><span class="op">}</span><span class="op">,</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t162" href="#t162">162</a></span><span class="t">        <span class="op">}</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t163" href="#t163">163</a></span><span class="t">        <span class="nam">self</span><span class="op">.</span><span class="nam">setup_L</span><span class="op">(</span><span class="nam">L_lhs</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t164" href="#t164">164</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t165" href="#t165">165</a></span><span class="t">        <span class="nam">M_lhs</span> <span class="op">=</span> <span class="op">{</span><span class="str">'u'</span><span class="op">:</span> <span class="op">{</span><span class="str">'u'</span><span class="op">:</span> <span class="nam">U2</span> <span class="op">@</span> <span class="nam">I</span><span class="op">}</span><span class="op">}</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t166" href="#t166">166</a></span><span class="t">        <span class="nam">self</span><span class="op">.</span><span class="nam">setup_M</span><span class="op">(</span><span class="nam">M_lhs</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t167" href="#t167">167</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t168" href="#t168">168</a></span><span class="t">        <span class="nam">self</span><span class="op">.</span><span class="nam">add_BC</span><span class="op">(</span><span class="nam">component</span><span class="op">=</span><span class="str">'u'</span><span class="op">,</span> <span class="nam">equation</span><span class="op">=</span><span class="str">'u'</span><span class="op">,</span> <span class="nam">axis</span><span class="op">=</span><span class="num">0</span><span class="op">,</span> <span class="nam">x</span><span class="op">=</span><span class="op">-</span><span class="num">1</span><span class="op">,</span> <span class="nam">v</span><span class="op">=</span><span class="nam">a</span><span class="op">,</span> <span class="nam">kind</span><span class="op">=</span><span class="str">"Dirichlet"</span><span class="op">,</span> <span class="nam">line</span><span class="op">=</span><span class="op">-</span><span class="num">1</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t169" href="#t169">169</a></span><span class="t">        <span class="nam">self</span><span class="op">.</span><span class="nam">add_BC</span><span class="op">(</span><span class="nam">component</span><span class="op">=</span><span class="str">'u'</span><span class="op">,</span> <span class="nam">equation</span><span class="op">=</span><span class="str">'u'</span><span class="op">,</span> <span class="nam">axis</span><span class="op">=</span><span class="num">0</span><span class="op">,</span> <span class="nam">x</span><span class="op">=</span><span class="num">1</span><span class="op">,</span> <span class="nam">v</span><span class="op">=</span><span class="nam">b</span><span class="op">,</span> <span class="nam">kind</span><span class="op">=</span><span class="str">"Dirichlet"</span><span class="op">,</span> <span class="nam">line</span><span class="op">=</span><span class="op">-</span><span class="num">2</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t170" href="#t170">170</a></span><span class="t">        <span class="nam">self</span><span class="op">.</span><span class="nam">setup_BCs</span><span class="op">(</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t171" href="#t171">171</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t172" href="#t172">172</a></span><span class="t">    <span class="key">def</span> <span class="nam">eval_f</span><span class="op">(</span><span class="nam">self</span><span class="op">,</span> <span class="nam">u</span><span class="op">,</span> <span class="op">*</span><span class="nam">args</span><span class="op">,</span> <span class="op">**</span><span class="nam">kwargs</span><span class="op">)</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t173" href="#t173">173</a></span><span class="t">        <span class="nam">f</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">f_init</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t174" href="#t174">174</a></span><span class="t">        <span class="nam">iu</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">index</span><span class="op">(</span><span class="str">'u'</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t175" href="#t175">175</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t176" href="#t176">176</a></span><span class="t">        <span class="key">if</span> <span class="nam">self</span><span class="op">.</span><span class="nam">spectral_space</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t177" href="#t177">177</a></span><span class="t">            <span class="nam">u_hat</span> <span class="op">=</span> <span class="nam">u</span><span class="op">.</span><span class="nam">copy</span><span class="op">(</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t178" href="#t178">178</a></span><span class="t">        <span class="key">else</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t179" href="#t179">179</a></span><span class="t">            <span class="nam">u_hat</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">transform</span><span class="op">(</span><span class="nam">u</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t180" href="#t180">180</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t181" href="#t181">181</a></span><span class="t">        <span class="nam">u_hat</span><span class="op">[</span><span class="nam">iu</span><span class="op">]</span> <span class="op">=</span> <span class="op">(</span><span class="nam">self</span><span class="op">.</span><span class="nam">nu</span> <span class="op">*</span> <span class="op">(</span><span class="nam">self</span><span class="op">.</span><span class="nam">Dxx</span> <span class="op">@</span> <span class="nam">u_hat</span><span class="op">[</span><span class="nam">iu</span><span class="op">]</span><span class="op">.</span><span class="nam">flatten</span><span class="op">(</span><span class="op">)</span><span class="op">)</span><span class="op">)</span><span class="op">.</span><span class="nam">reshape</span><span class="op">(</span><span class="nam">u_hat</span><span class="op">[</span><span class="nam">iu</span><span class="op">]</span><span class="op">.</span><span class="nam">shape</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t182" href="#t182">182</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t183" href="#t183">183</a></span><span class="t">        <span class="key">if</span> <span class="nam">self</span><span class="op">.</span><span class="nam">spectral_space</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t184" href="#t184">184</a></span><span class="t">            <span class="nam">me</span> <span class="op">=</span> <span class="nam">u_hat</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t185" href="#t185">185</a></span><span class="t">        <span class="key">else</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t186" href="#t186">186</a></span><span class="t">            <span class="nam">me</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">itransform</span><span class="op">(</span><span class="nam">u_hat</span><span class="op">)</span><span class="op">.</span><span class="nam">real</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t187" href="#t187">187</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t188" href="#t188">188</a></span><span class="t">        <span class="nam">f</span><span class="op">[</span><span class="nam">iu</span><span class="op">]</span><span class="op">[</span><span class="op">...</span><span class="op">]</span> <span class="op">=</span> <span class="nam">me</span><span class="op">[</span><span class="nam">iu</span><span class="op">]</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t189" href="#t189">189</a></span><span class="t">        <span class="key">return</span> <span class="nam">f</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t190" href="#t190">190</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t191" href="#t191">191</a></span><span class="t">    <span class="key">def</span> <span class="nam">u_exact</span><span class="op">(</span><span class="nam">self</span><span class="op">,</span> <span class="nam">t</span><span class="op">,</span> <span class="nam">noise</span><span class="op">=</span><span class="num">0</span><span class="op">,</span> <span class="nam">seed</span><span class="op">=</span><span class="num">666</span><span class="op">)</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t192" href="#t192">192</a></span><span class="t">        <span class="str">"""</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t193" href="#t193">193</a></span><span class="t"><span class="str">        Get exact solution at time `t`</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t194" href="#t194">194</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t195" href="#t195">195</a></span><span class="t"><span class="str">        Args:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t196" href="#t196">196</a></span><span class="t"><span class="str">            t (float): When you want the exact solution</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t197" href="#t197">197</a></span><span class="t"><span class="str">            noise (float): Add noise of this level</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t198" href="#t198">198</a></span><span class="t"><span class="str">            seed (int): Random seed for the noise</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t199" href="#t199">199</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t200" href="#t200">200</a></span><span class="t"><span class="str">        Returns:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t201" href="#t201">201</a></span><span class="t"><span class="str">            Heat1DUltraspherical.dtype_u: Exact solution</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t202" href="#t202">202</a></span><span class="t"><span class="str">        """</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t203" href="#t203">203</a></span><span class="t">        <span class="nam">xp</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">xp</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t204" href="#t204">204</a></span><span class="t">        <span class="nam">iu</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">index</span><span class="op">(</span><span class="str">'u'</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t205" href="#t205">205</a></span><span class="t">        <span class="nam">u</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">spectral</span><span class="op">.</span><span class="nam">u_init</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t206" href="#t206">206</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t207" href="#t207">207</a></span><span class="t">        <span class="nam">u</span><span class="op">[</span><span class="nam">iu</span><span class="op">]</span> <span class="op">=</span> <span class="op">(</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t208" href="#t208">208</a></span><span class="t">            <span class="nam">xp</span><span class="op">.</span><span class="nam">sin</span><span class="op">(</span><span class="nam">self</span><span class="op">.</span><span class="nam">f</span> <span class="op">*</span> <span class="nam">np</span><span class="op">.</span><span class="nam">pi</span> <span class="op">*</span> <span class="nam">self</span><span class="op">.</span><span class="nam">x</span><span class="op">)</span> <span class="op">*</span> <span class="nam">xp</span><span class="op">.</span><span class="nam">exp</span><span class="op">(</span><span class="op">-</span><span class="nam">self</span><span class="op">.</span><span class="nam">nu</span> <span class="op">*</span> <span class="op">(</span><span class="nam">self</span><span class="op">.</span><span class="nam">f</span> <span class="op">*</span> <span class="nam">np</span><span class="op">.</span><span class="nam">pi</span><span class="op">)</span> <span class="op">**</span> <span class="num">2</span> <span class="op">*</span> <span class="nam">t</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t209" href="#t209">209</a></span><span class="t">            <span class="op">+</span> <span class="op">(</span><span class="nam">self</span><span class="op">.</span><span class="nam">b</span> <span class="op">-</span> <span class="nam">self</span><span class="op">.</span><span class="nam">a</span><span class="op">)</span> <span class="op">/</span> <span class="num">2</span> <span class="op">*</span> <span class="nam">self</span><span class="op">.</span><span class="nam">x</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t210" href="#t210">210</a></span><span class="t">            <span class="op">+</span> <span class="op">(</span><span class="nam">self</span><span class="op">.</span><span class="nam">b</span> <span class="op">+</span> <span class="nam">self</span><span class="op">.</span><span class="nam">a</span><span class="op">)</span> <span class="op">/</span> <span class="num">2</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t211" href="#t211">211</a></span><span class="t">        <span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t212" href="#t212">212</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t213" href="#t213">213</a></span><span class="t">        <span class="key">if</span> <span class="nam">noise</span> <span class="op">></span> <span class="num">0</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t214" href="#t214">214</a></span><span class="t">            <span class="key">assert</span> <span class="nam">t</span> <span class="op">==</span> <span class="num">0</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t215" href="#t215">215</a></span><span class="t">            <span class="nam">_noise</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">u_init</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t216" href="#t216">216</a></span><span class="t">            <span class="nam">rng</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">xp</span><span class="op">.</span><span class="nam">random</span><span class="op">.</span><span class="nam">default_rng</span><span class="op">(</span><span class="nam">seed</span><span class="op">=</span><span class="nam">seed</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t217" href="#t217">217</a></span><span class="t">            <span class="nam">_noise</span><span class="op">[</span><span class="nam">iu</span><span class="op">]</span> <span class="op">=</span> <span class="nam">rng</span><span class="op">.</span><span class="nam">normal</span><span class="op">(</span><span class="nam">size</span><span class="op">=</span><span class="nam">u</span><span class="op">[</span><span class="nam">iu</span><span class="op">]</span><span class="op">.</span><span class="nam">shape</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t218" href="#t218">218</a></span><span class="t">            <span class="nam">noise_hat</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">transform</span><span class="op">(</span><span class="nam">_noise</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t219" href="#t219">219</a></span><span class="t">            <span class="nam">low_pass</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">get_filter_matrix</span><span class="op">(</span><span class="nam">axis</span><span class="op">=</span><span class="num">0</span><span class="op">,</span> <span class="nam">kmax</span><span class="op">=</span><span class="nam">self</span><span class="op">.</span><span class="nam">nvars</span> <span class="op">-</span> <span class="num">2</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t220" href="#t220">220</a></span><span class="t">            <span class="nam">noise_hat</span><span class="op">[</span><span class="nam">iu</span><span class="op">]</span> <span class="op">=</span> <span class="op">(</span><span class="nam">low_pass</span> <span class="op">@</span> <span class="nam">noise_hat</span><span class="op">[</span><span class="nam">iu</span><span class="op">]</span><span class="op">.</span><span class="nam">flatten</span><span class="op">(</span><span class="op">)</span><span class="op">)</span><span class="op">.</span><span class="nam">reshape</span><span class="op">(</span><span class="nam">noise_hat</span><span class="op">[</span><span class="nam">iu</span><span class="op">]</span><span class="op">.</span><span class="nam">shape</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t221" href="#t221">221</a></span><span class="t">            <span class="nam">_noise</span><span class="op">[</span><span class="op">:</span><span class="op">]</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">itransform</span><span class="op">(</span><span class="nam">noise_hat</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t222" href="#t222">222</a></span><span class="t">            <span class="nam">u</span> <span class="op">+=</span> <span class="nam">_noise</span> <span class="op">*</span> <span class="nam">noise</span> <span class="op">*</span> <span class="op">(</span><span class="nam">self</span><span class="op">.</span><span class="nam">x</span> <span class="op">-</span> <span class="num">1</span><span class="op">)</span> <span class="op">*</span> <span class="op">(</span><span class="nam">self</span><span class="op">.</span><span class="nam">x</span> <span class="op">+</span> <span class="num">1</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t223" href="#t223">223</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t224" href="#t224">224</a></span><span class="t">        <span class="nam">self</span><span class="op">.</span><span class="nam">check_BCs</span><span class="op">(</span><span class="nam">u</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t225" href="#t225">225</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t226" href="#t226">226</a></span><span class="t">        <span class="key">if</span> <span class="nam">self</span><span class="op">.</span><span class="nam">spectral_space</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t227" href="#t227">227</a></span><span class="t">            <span class="key">return</span> <span class="nam">self</span><span class="op">.</span><span class="nam">transform</span><span class="op">(</span><span class="nam">u</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t228" href="#t228">228</a></span><span class="t">        <span class="key">else</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t229" href="#t229">229</a></span><span class="t">            <span class="key">return</span> <span class="nam">u</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t230" href="#t230">230</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t231" href="#t231">231</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t232" href="#t232">232</a></span><span class="t"><span class="key">class</span> <span class="nam">Heat2DChebychev</span><span class="op">(</span><span class="nam">GenericSpectralLinear</span><span class="op">)</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t233" href="#t233">233</a></span><span class="t">    <span class="str">"""</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t234" href="#t234">234</a></span><span class="t"><span class="str">    2D Heat equation with Dirichlet Boundary conditions discretized on (-1, 1)x(-1,1) using spectral methods based on FFT and Chebychev.</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t235" href="#t235">235</a></span><span class="t"><span class="str">    """</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t236" href="#t236">236</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t237" href="#t237">237</a></span><span class="t">    <span class="nam">dtype_u</span> <span class="op">=</span> <span class="nam">mesh</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t238" href="#t238">238</a></span><span class="t">    <span class="nam">dtype_f</span> <span class="op">=</span> <span class="nam">mesh</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t239" href="#t239">239</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t240" href="#t240">240</a></span><span class="t">    <span class="key">def</span> <span class="nam">__init__</span><span class="op">(</span><span class="nam">self</span><span class="op">,</span> <span class="nam">nx</span><span class="op">=</span><span class="num">128</span><span class="op">,</span> <span class="nam">ny</span><span class="op">=</span><span class="num">128</span><span class="op">,</span> <span class="nam">base_x</span><span class="op">=</span><span class="str">'fft'</span><span class="op">,</span> <span class="nam">base_y</span><span class="op">=</span><span class="str">'chebychev'</span><span class="op">,</span> <span class="nam">a</span><span class="op">=</span><span class="num">0</span><span class="op">,</span> <span class="nam">b</span><span class="op">=</span><span class="num">0</span><span class="op">,</span> <span class="nam">c</span><span class="op">=</span><span class="num">0</span><span class="op">,</span> <span class="nam">fx</span><span class="op">=</span><span class="num">1</span><span class="op">,</span> <span class="nam">fy</span><span class="op">=</span><span class="num">1</span><span class="op">,</span> <span class="nam">nu</span><span class="op">=</span><span class="num">1.0</span><span class="op">,</span> <span class="op">**</span><span class="nam">kwargs</span><span class="op">)</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t241" href="#t241">241</a></span><span class="t">        <span class="str">"""</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t242" href="#t242">242</a></span><span class="t"><span class="str">        Constructor. `kwargs` are forwarded to parent class constructor.</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t243" href="#t243">243</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t244" href="#t244">244</a></span><span class="t"><span class="str">        Args:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t245" href="#t245">245</a></span><span class="t"><span class="str">            nx (int): Resolution in x-direction</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t246" href="#t246">246</a></span><span class="t"><span class="str">            ny (int): Resolution in y-direction</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t247" href="#t247">247</a></span><span class="t"><span class="str">            base_x (str): Spectral base in x-direction</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t248" href="#t248">248</a></span><span class="t"><span class="str">            base_y (str): Spectral base in y-direction</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t249" href="#t249">249</a></span><span class="t"><span class="str">            a (float): BC at y=0 and x=-1</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t250" href="#t250">250</a></span><span class="t"><span class="str">            b (float): BC at y=0 and x=+1</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t251" href="#t251">251</a></span><span class="t"><span class="str">            c (float): BC at y=1 and x = -1</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t252" href="#t252">252</a></span><span class="t"><span class="str">            fx (int): Horizontal frequency of initial conditions</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t253" href="#t253">253</a></span><span class="t"><span class="str">            fy (int): Vertical frequency of initial conditions</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t254" href="#t254">254</a></span><span class="t"><span class="str">            nu (float): Diffusion parameter</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t255" href="#t255">255</a></span><span class="t"><span class="str">        """</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t256" href="#t256">256</a></span><span class="t">        <span class="key">assert</span> <span class="nam">nx</span> <span class="op">%</span> <span class="num">2</span> <span class="op">==</span> <span class="num">1</span> <span class="key">or</span> <span class="nam">base_x</span> <span class="op">==</span> <span class="str">'fft'</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t257" href="#t257">257</a></span><span class="t">        <span class="key">assert</span> <span class="nam">ny</span> <span class="op">%</span> <span class="num">2</span> <span class="op">==</span> <span class="num">1</span> <span class="key">or</span> <span class="nam">base_y</span> <span class="op">==</span> <span class="str">'fft'</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t258" href="#t258">258</a></span><span class="t">        <span class="nam">self</span><span class="op">.</span><span class="nam">_makeAttributeAndRegister</span><span class="op">(</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t259" href="#t259">259</a></span><span class="t">            <span class="str">'nx'</span><span class="op">,</span> <span class="str">'ny'</span><span class="op">,</span> <span class="str">'base_x'</span><span class="op">,</span> <span class="str">'base_y'</span><span class="op">,</span> <span class="str">'a'</span><span class="op">,</span> <span class="str">'b'</span><span class="op">,</span> <span class="str">'c'</span><span class="op">,</span> <span class="str">'fx'</span><span class="op">,</span> <span class="str">'fy'</span><span class="op">,</span> <span class="str">'nu'</span><span class="op">,</span> <span class="nam">localVars</span><span class="op">=</span><span class="nam">locals</span><span class="op">(</span><span class="op">)</span><span class="op">,</span> <span class="nam">readOnly</span><span class="op">=</span><span class="key">True</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t260" href="#t260">260</a></span><span class="t">        <span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t261" href="#t261">261</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t262" href="#t262">262</a></span><span class="t">        <span class="nam">bases</span> <span class="op">=</span> <span class="op">[</span><span class="op">{</span><span class="str">'base'</span><span class="op">:</span> <span class="nam">base_x</span><span class="op">,</span> <span class="str">'N'</span><span class="op">:</span> <span class="nam">nx</span><span class="op">}</span><span class="op">,</span> <span class="op">{</span><span class="str">'base'</span><span class="op">:</span> <span class="nam">base_y</span><span class="op">,</span> <span class="str">'N'</span><span class="op">:</span> <span class="nam">ny</span><span class="op">}</span><span class="op">]</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t263" href="#t263">263</a></span><span class="t">        <span class="nam">components</span> <span class="op">=</span> <span class="op">[</span><span class="str">'u'</span><span class="op">,</span> <span class="str">'ux'</span><span class="op">,</span> <span class="str">'uy'</span><span class="op">]</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t264" href="#t264">264</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t265" href="#t265">265</a></span><span class="t">        <span class="nam">super</span><span class="op">(</span><span class="op">)</span><span class="op">.</span><span class="nam">__init__</span><span class="op">(</span><span class="nam">bases</span><span class="op">,</span> <span class="nam">components</span><span class="op">,</span> <span class="nam">Dirichlet_recombination</span><span class="op">=</span><span class="key">False</span><span class="op">,</span> <span class="nam">spectral_space</span><span class="op">=</span><span class="key">False</span><span class="op">,</span> <span class="op">**</span><span class="nam">kwargs</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t266" href="#t266">266</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t267" href="#t267">267</a></span><span class="t">        <span class="nam">self</span><span class="op">.</span><span class="nam">Y</span><span class="op">,</span> <span class="nam">self</span><span class="op">.</span><span class="nam">X</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">get_grid</span><span class="op">(</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t268" href="#t268">268</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t269" href="#t269">269</a></span><span class="t">        <span class="nam">I</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">get_Id</span><span class="op">(</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t270" href="#t270">270</a></span><span class="t">        <span class="nam">self</span><span class="op">.</span><span class="nam">Dx</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">get_differentiation_matrix</span><span class="op">(</span><span class="nam">axes</span><span class="op">=</span><span class="op">(</span><span class="num">0</span><span class="op">,</span><span class="op">)</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t271" href="#t271">271</a></span><span class="t">        <span class="nam">self</span><span class="op">.</span><span class="nam">Dy</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">get_differentiation_matrix</span><span class="op">(</span><span class="nam">axes</span><span class="op">=</span><span class="op">(</span><span class="num">1</span><span class="op">,</span><span class="op">)</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t272" href="#t272">272</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t273" href="#t273">273</a></span><span class="t">        <span class="nam">L_lhs</span> <span class="op">=</span> <span class="op">{</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t274" href="#t274">274</a></span><span class="t">            <span class="str">'ux'</span><span class="op">:</span> <span class="op">{</span><span class="str">'u'</span><span class="op">:</span> <span class="op">-</span><span class="nam">self</span><span class="op">.</span><span class="nam">Dx</span><span class="op">,</span> <span class="str">'ux'</span><span class="op">:</span> <span class="nam">I</span><span class="op">}</span><span class="op">,</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t275" href="#t275">275</a></span><span class="t">            <span class="str">'uy'</span><span class="op">:</span> <span class="op">{</span><span class="str">'u'</span><span class="op">:</span> <span class="op">-</span><span class="nam">self</span><span class="op">.</span><span class="nam">Dy</span><span class="op">,</span> <span class="str">'uy'</span><span class="op">:</span> <span class="nam">I</span><span class="op">}</span><span class="op">,</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t276" href="#t276">276</a></span><span class="t">            <span class="str">'u'</span><span class="op">:</span> <span class="op">{</span><span class="str">'ux'</span><span class="op">:</span> <span class="op">-</span><span class="nam">nu</span> <span class="op">*</span> <span class="nam">self</span><span class="op">.</span><span class="nam">Dx</span><span class="op">,</span> <span class="str">'uy'</span><span class="op">:</span> <span class="op">-</span><span class="nam">nu</span> <span class="op">*</span> <span class="nam">self</span><span class="op">.</span><span class="nam">Dy</span><span class="op">}</span><span class="op">,</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t277" href="#t277">277</a></span><span class="t">        <span class="op">}</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t278" href="#t278">278</a></span><span class="t">        <span class="nam">self</span><span class="op">.</span><span class="nam">setup_L</span><span class="op">(</span><span class="nam">L_lhs</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t279" href="#t279">279</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t280" href="#t280">280</a></span><span class="t">        <span class="nam">M_lhs</span> <span class="op">=</span> <span class="op">{</span><span class="str">'u'</span><span class="op">:</span> <span class="op">{</span><span class="str">'u'</span><span class="op">:</span> <span class="nam">I</span><span class="op">}</span><span class="op">}</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t281" href="#t281">281</a></span><span class="t">        <span class="nam">self</span><span class="op">.</span><span class="nam">setup_M</span><span class="op">(</span><span class="nam">M_lhs</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t282" href="#t282">282</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t283" href="#t283">283</a></span><span class="t">        <span class="key">for</span> <span class="nam">base</span> <span class="key">in</span> <span class="op">[</span><span class="nam">base_x</span><span class="op">,</span> <span class="nam">base_y</span><span class="op">]</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t284" href="#t284">284</a></span><span class="t">            <span class="key">assert</span> <span class="nam">base</span> <span class="key">in</span> <span class="op">[</span><span class="str">'chebychev'</span><span class="op">,</span> <span class="str">'fft'</span><span class="op">]</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t285" href="#t285">285</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t286" href="#t286">286</a></span><span class="t">        <span class="nam">alpha</span> <span class="op">=</span> <span class="op">(</span><span class="nam">self</span><span class="op">.</span><span class="nam">b</span> <span class="op">-</span> <span class="nam">self</span><span class="op">.</span><span class="nam">a</span><span class="op">)</span> <span class="op">/</span> <span class="num">2.0</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t287" href="#t287">287</a></span><span class="t">        <span class="nam">beta</span> <span class="op">=</span> <span class="op">(</span><span class="nam">self</span><span class="op">.</span><span class="nam">c</span> <span class="op">-</span> <span class="nam">self</span><span class="op">.</span><span class="nam">b</span><span class="op">)</span> <span class="op">/</span> <span class="num">2.0</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t288" href="#t288">288</a></span><span class="t">        <span class="nam">gamma</span> <span class="op">=</span> <span class="op">(</span><span class="nam">self</span><span class="op">.</span><span class="nam">c</span> <span class="op">+</span> <span class="nam">self</span><span class="op">.</span><span class="nam">a</span><span class="op">)</span> <span class="op">/</span> <span class="num">2.0</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t289" href="#t289">289</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t290" href="#t290">290</a></span><span class="t">        <span class="key">if</span> <span class="nam">base_x</span> <span class="op">==</span> <span class="str">'chebychev'</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t291" href="#t291">291</a></span><span class="t">            <span class="nam">y</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">Y</span><span class="op">[</span><span class="num">0</span><span class="op">,</span> <span class="op">:</span><span class="op">]</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t292" href="#t292">292</a></span><span class="t">            <span class="key">if</span> <span class="nam">self</span><span class="op">.</span><span class="nam">base_y</span> <span class="op">==</span> <span class="str">'fft'</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t293" href="#t293">293</a></span><span class="t">                <span class="nam">self</span><span class="op">.</span><span class="nam">add_BC</span><span class="op">(</span><span class="nam">component</span><span class="op">=</span><span class="str">'u'</span><span class="op">,</span> <span class="nam">equation</span><span class="op">=</span><span class="str">'u'</span><span class="op">,</span> <span class="nam">axis</span><span class="op">=</span><span class="num">0</span><span class="op">,</span> <span class="nam">x</span><span class="op">=</span><span class="op">-</span><span class="num">1</span><span class="op">,</span> <span class="nam">v</span><span class="op">=</span><span class="nam">beta</span> <span class="op">*</span> <span class="nam">y</span> <span class="op">-</span> <span class="nam">alpha</span> <span class="op">+</span> <span class="nam">gamma</span><span class="op">,</span> <span class="nam">kind</span><span class="op">=</span><span class="str">'Dirichlet'</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t294" href="#t294">294</a></span><span class="t">            <span class="nam">self</span><span class="op">.</span><span class="nam">add_BC</span><span class="op">(</span><span class="nam">component</span><span class="op">=</span><span class="str">'ux'</span><span class="op">,</span> <span class="nam">equation</span><span class="op">=</span><span class="str">'ux'</span><span class="op">,</span> <span class="nam">axis</span><span class="op">=</span><span class="num">0</span><span class="op">,</span> <span class="nam">v</span><span class="op">=</span><span class="num">2</span> <span class="op">*</span> <span class="nam">alpha</span><span class="op">,</span> <span class="nam">kind</span><span class="op">=</span><span class="str">'integral'</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t295" href="#t295">295</a></span><span class="t">        <span class="key">else</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t296" href="#t296">296</a></span><span class="t">            <span class="key">assert</span> <span class="nam">a</span> <span class="op">==</span> <span class="nam">b</span><span class="op">,</span> <span class="fst">f'</span><span class="fst">Need periodic boundary conditions in x for </span><span class="op">{</span><span class="nam">base_x</span><span class="op">}</span><span class="fst"> method!</span><span class="fst">'</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t297" href="#t297">297</a></span><span class="t">        <span class="key">if</span> <span class="nam">base_y</span> <span class="op">==</span> <span class="str">'chebychev'</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t298" href="#t298">298</a></span><span class="t">            <span class="nam">x</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">X</span><span class="op">[</span><span class="op">:</span><span class="op">,</span> <span class="num">0</span><span class="op">]</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t299" href="#t299">299</a></span><span class="t">            <span class="nam">self</span><span class="op">.</span><span class="nam">add_BC</span><span class="op">(</span><span class="nam">component</span><span class="op">=</span><span class="str">'u'</span><span class="op">,</span> <span class="nam">equation</span><span class="op">=</span><span class="str">'u'</span><span class="op">,</span> <span class="nam">axis</span><span class="op">=</span><span class="num">1</span><span class="op">,</span> <span class="nam">x</span><span class="op">=</span><span class="op">-</span><span class="num">1</span><span class="op">,</span> <span class="nam">v</span><span class="op">=</span><span class="nam">alpha</span> <span class="op">*</span> <span class="nam">x</span> <span class="op">-</span> <span class="nam">beta</span> <span class="op">+</span> <span class="nam">gamma</span><span class="op">,</span> <span class="nam">kind</span><span class="op">=</span><span class="str">'Dirichlet'</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t300" href="#t300">300</a></span><span class="t">            <span class="nam">self</span><span class="op">.</span><span class="nam">add_BC</span><span class="op">(</span><span class="nam">component</span><span class="op">=</span><span class="str">'uy'</span><span class="op">,</span> <span class="nam">equation</span><span class="op">=</span><span class="str">'uy'</span><span class="op">,</span> <span class="nam">axis</span><span class="op">=</span><span class="num">1</span><span class="op">,</span> <span class="nam">v</span><span class="op">=</span><span class="num">2</span> <span class="op">*</span> <span class="nam">beta</span><span class="op">,</span> <span class="nam">kind</span><span class="op">=</span><span class="str">'integral'</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t301" href="#t301">301</a></span><span class="t">        <span class="key">else</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t302" href="#t302">302</a></span><span class="t">            <span class="key">assert</span> <span class="nam">c</span> <span class="op">==</span> <span class="nam">b</span><span class="op">,</span> <span class="fst">f'</span><span class="fst">Need periodic boundary conditions in y for </span><span class="op">{</span><span class="nam">base_y</span><span class="op">}</span><span class="fst"> method!</span><span class="fst">'</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t303" href="#t303">303</a></span><span class="t">        <span class="nam">self</span><span class="op">.</span><span class="nam">setup_BCs</span><span class="op">(</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t304" href="#t304">304</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t305" href="#t305">305</a></span><span class="t">    <span class="key">def</span> <span class="nam">eval_f</span><span class="op">(</span><span class="nam">self</span><span class="op">,</span> <span class="nam">u</span><span class="op">,</span> <span class="op">*</span><span class="nam">args</span><span class="op">,</span> <span class="op">**</span><span class="nam">kwargs</span><span class="op">)</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t306" href="#t306">306</a></span><span class="t">        <span class="nam">f</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">f_init</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t307" href="#t307">307</a></span><span class="t">        <span class="nam">iu</span><span class="op">,</span> <span class="nam">iux</span><span class="op">,</span> <span class="nam">iuy</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">index</span><span class="op">(</span><span class="nam">self</span><span class="op">.</span><span class="nam">components</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t308" href="#t308">308</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t309" href="#t309">309</a></span><span class="t">        <span class="nam">me_hat</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">u_init_forward</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t310" href="#t310">310</a></span><span class="t">        <span class="nam">me_hat</span><span class="op">[</span><span class="op">:</span><span class="op">]</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">transform</span><span class="op">(</span><span class="nam">u</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t311" href="#t311">311</a></span><span class="t">        <span class="nam">me_hat</span><span class="op">[</span><span class="nam">iu</span><span class="op">]</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">nu</span> <span class="op">*</span> <span class="op">(</span><span class="nam">self</span><span class="op">.</span><span class="nam">Dx</span> <span class="op">@</span> <span class="nam">me_hat</span><span class="op">[</span><span class="nam">iux</span><span class="op">]</span><span class="op">.</span><span class="nam">flatten</span><span class="op">(</span><span class="op">)</span> <span class="op">+</span> <span class="nam">self</span><span class="op">.</span><span class="nam">Dy</span> <span class="op">@</span> <span class="nam">me_hat</span><span class="op">[</span><span class="nam">iuy</span><span class="op">]</span><span class="op">.</span><span class="nam">flatten</span><span class="op">(</span><span class="op">)</span><span class="op">)</span><span class="op">.</span><span class="nam">reshape</span><span class="op">(</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t312" href="#t312">312</a></span><span class="t">            <span class="nam">me_hat</span><span class="op">[</span><span class="nam">iu</span><span class="op">]</span><span class="op">.</span><span class="nam">shape</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t313" href="#t313">313</a></span><span class="t">        <span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t314" href="#t314">314</a></span><span class="t">        <span class="nam">me</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">itransform</span><span class="op">(</span><span class="nam">me_hat</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t315" href="#t315">315</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t316" href="#t316">316</a></span><span class="t">        <span class="nam">f</span><span class="op">[</span><span class="nam">self</span><span class="op">.</span><span class="nam">index</span><span class="op">(</span><span class="str">"u"</span><span class="op">)</span><span class="op">]</span> <span class="op">=</span> <span class="nam">me</span><span class="op">[</span><span class="nam">iu</span><span class="op">]</span><span class="op">.</span><span class="nam">real</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t317" href="#t317">317</a></span><span class="t">        <span class="key">return</span> <span class="nam">f</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t318" href="#t318">318</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t319" href="#t319">319</a></span><span class="t">    <span class="key">def</span> <span class="nam">u_exact</span><span class="op">(</span><span class="nam">self</span><span class="op">,</span> <span class="nam">t</span><span class="op">)</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t320" href="#t320">320</a></span><span class="t">        <span class="nam">xp</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">xp</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t321" href="#t321">321</a></span><span class="t">        <span class="nam">iu</span><span class="op">,</span> <span class="nam">iux</span><span class="op">,</span> <span class="nam">iuy</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">index</span><span class="op">(</span><span class="nam">self</span><span class="op">.</span><span class="nam">components</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t322" href="#t322">322</a></span><span class="t">        <span class="nam">u</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">u_init</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t323" href="#t323">323</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t324" href="#t324">324</a></span><span class="t">        <span class="nam">fx</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">fx</span> <span class="key">if</span> <span class="nam">self</span><span class="op">.</span><span class="nam">base_x</span> <span class="op">==</span> <span class="str">'fft'</span> <span class="key">else</span> <span class="nam">np</span><span class="op">.</span><span class="nam">pi</span> <span class="op">*</span> <span class="nam">self</span><span class="op">.</span><span class="nam">fx</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t325" href="#t325">325</a></span><span class="t">        <span class="nam">fy</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">fy</span> <span class="key">if</span> <span class="nam">self</span><span class="op">.</span><span class="nam">base_y</span> <span class="op">==</span> <span class="str">'fft'</span> <span class="key">else</span> <span class="nam">np</span><span class="op">.</span><span class="nam">pi</span> <span class="op">*</span> <span class="nam">self</span><span class="op">.</span><span class="nam">fy</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t326" href="#t326">326</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t327" href="#t327">327</a></span><span class="t">        <span class="nam">time_dep</span> <span class="op">=</span> <span class="nam">xp</span><span class="op">.</span><span class="nam">exp</span><span class="op">(</span><span class="op">-</span><span class="nam">self</span><span class="op">.</span><span class="nam">nu</span> <span class="op">*</span> <span class="op">(</span><span class="nam">fx</span><span class="op">**</span><span class="num">2</span> <span class="op">+</span> <span class="nam">fy</span><span class="op">**</span><span class="num">2</span><span class="op">)</span> <span class="op">*</span> <span class="nam">t</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t328" href="#t328">328</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t329" href="#t329">329</a></span><span class="t">        <span class="nam">alpha</span> <span class="op">=</span> <span class="op">(</span><span class="nam">self</span><span class="op">.</span><span class="nam">b</span> <span class="op">-</span> <span class="nam">self</span><span class="op">.</span><span class="nam">a</span><span class="op">)</span> <span class="op">/</span> <span class="num">2.0</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t330" href="#t330">330</a></span><span class="t">        <span class="nam">beta</span> <span class="op">=</span> <span class="op">(</span><span class="nam">self</span><span class="op">.</span><span class="nam">c</span> <span class="op">-</span> <span class="nam">self</span><span class="op">.</span><span class="nam">b</span><span class="op">)</span> <span class="op">/</span> <span class="num">2.0</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t331" href="#t331">331</a></span><span class="t">        <span class="nam">gamma</span> <span class="op">=</span> <span class="op">(</span><span class="nam">self</span><span class="op">.</span><span class="nam">c</span> <span class="op">+</span> <span class="nam">self</span><span class="op">.</span><span class="nam">a</span><span class="op">)</span> <span class="op">/</span> <span class="num">2.0</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t332" href="#t332">332</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t333" href="#t333">333</a></span><span class="t">        <span class="nam">u</span><span class="op">[</span><span class="nam">iu</span><span class="op">]</span> <span class="op">=</span> <span class="nam">xp</span><span class="op">.</span><span class="nam">sin</span><span class="op">(</span><span class="nam">fx</span> <span class="op">*</span> <span class="nam">self</span><span class="op">.</span><span class="nam">X</span><span class="op">)</span> <span class="op">*</span> <span class="nam">xp</span><span class="op">.</span><span class="nam">sin</span><span class="op">(</span><span class="nam">fy</span> <span class="op">*</span> <span class="nam">self</span><span class="op">.</span><span class="nam">Y</span><span class="op">)</span> <span class="op">*</span> <span class="nam">time_dep</span> <span class="op">+</span> <span class="nam">alpha</span> <span class="op">*</span> <span class="nam">self</span><span class="op">.</span><span class="nam">X</span> <span class="op">+</span> <span class="nam">beta</span> <span class="op">*</span> <span class="nam">self</span><span class="op">.</span><span class="nam">Y</span> <span class="op">+</span> <span class="nam">gamma</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t334" href="#t334">334</a></span><span class="t">        <span class="nam">u</span><span class="op">[</span><span class="nam">iux</span><span class="op">]</span> <span class="op">=</span> <span class="nam">fx</span> <span class="op">*</span> <span class="nam">xp</span><span class="op">.</span><span class="nam">cos</span><span class="op">(</span><span class="nam">fx</span> <span class="op">*</span> <span class="nam">self</span><span class="op">.</span><span class="nam">X</span><span class="op">)</span> <span class="op">*</span> <span class="nam">xp</span><span class="op">.</span><span class="nam">sin</span><span class="op">(</span><span class="nam">fy</span> <span class="op">*</span> <span class="nam">self</span><span class="op">.</span><span class="nam">Y</span><span class="op">)</span> <span class="op">*</span> <span class="nam">time_dep</span> <span class="op">+</span> <span class="nam">alpha</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t335" href="#t335">335</a></span><span class="t">        <span class="nam">u</span><span class="op">[</span><span class="nam">iuy</span><span class="op">]</span> <span class="op">=</span> <span class="nam">fy</span> <span class="op">*</span> <span class="nam">xp</span><span class="op">.</span><span class="nam">sin</span><span class="op">(</span><span class="nam">fx</span> <span class="op">*</span> <span class="nam">self</span><span class="op">.</span><span class="nam">X</span><span class="op">)</span> <span class="op">*</span> <span class="nam">xp</span><span class="op">.</span><span class="nam">cos</span><span class="op">(</span><span class="nam">fy</span> <span class="op">*</span> <span class="nam">self</span><span class="op">.</span><span class="nam">Y</span><span class="op">)</span> <span class="op">*</span> <span class="nam">time_dep</span> <span class="op">+</span> <span class="nam">beta</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t336" href="#t336">336</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t337" href="#t337">337</a></span><span class="t">        <span class="key">return</span> <span class="nam">u</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t338" href="#t338">338</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t339" href="#t339">339</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t340" href="#t340">340</a></span><span class="t"><span class="key">class</span> <span class="nam">Heat2DUltraspherical</span><span class="op">(</span><span class="nam">GenericSpectralLinear</span><span class="op">)</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t341" href="#t341">341</a></span><span class="t">    <span class="str">"""</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t342" href="#t342">342</a></span><span class="t"><span class="str">    2D Heat equation with Dirichlet Boundary conditions discretized on (-1, 1)x(-1,1) using spectral methods based on FFT and Gegenbauer.</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t343" href="#t343">343</a></span><span class="t"><span class="str">    """</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t344" href="#t344">344</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t345" href="#t345">345</a></span><span class="t">    <span class="nam">dtype_u</span> <span class="op">=</span> <span class="nam">mesh</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t346" href="#t346">346</a></span><span class="t">    <span class="nam">dtype_f</span> <span class="op">=</span> <span class="nam">mesh</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t347" href="#t347">347</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t348" href="#t348">348</a></span><span class="t">    <span class="key">def</span> <span class="nam">__init__</span><span class="op">(</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t349" href="#t349">349</a></span><span class="t">        <span class="nam">self</span><span class="op">,</span> <span class="nam">nx</span><span class="op">=</span><span class="num">128</span><span class="op">,</span> <span class="nam">ny</span><span class="op">=</span><span class="num">128</span><span class="op">,</span> <span class="nam">base_x</span><span class="op">=</span><span class="str">'fft'</span><span class="op">,</span> <span class="nam">base_y</span><span class="op">=</span><span class="str">'ultraspherical'</span><span class="op">,</span> <span class="nam">a</span><span class="op">=</span><span class="num">0</span><span class="op">,</span> <span class="nam">b</span><span class="op">=</span><span class="num">0</span><span class="op">,</span> <span class="nam">c</span><span class="op">=</span><span class="num">0</span><span class="op">,</span> <span class="nam">fx</span><span class="op">=</span><span class="num">1</span><span class="op">,</span> <span class="nam">fy</span><span class="op">=</span><span class="num">1</span><span class="op">,</span> <span class="nam">nu</span><span class="op">=</span><span class="num">1.0</span><span class="op">,</span> <span class="op">**</span><span class="nam">kwargs</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t350" href="#t350">350</a></span><span class="t">    <span class="op">)</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t351" href="#t351">351</a></span><span class="t">        <span class="str">"""</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t352" href="#t352">352</a></span><span class="t"><span class="str">        Constructor. `kwargs` are forwarded to parent class constructor.</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t353" href="#t353">353</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t354" href="#t354">354</a></span><span class="t"><span class="str">        Args:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t355" href="#t355">355</a></span><span class="t"><span class="str">            nx (int): Resolution in x-direction</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t356" href="#t356">356</a></span><span class="t"><span class="str">            ny (int): Resolution in y-direction</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t357" href="#t357">357</a></span><span class="t"><span class="str">            base_x (str): Spectral base in x-direction</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t358" href="#t358">358</a></span><span class="t"><span class="str">            base_y (str): Spectral base in y-direction</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t359" href="#t359">359</a></span><span class="t"><span class="str">            a (float): BC at y=0 and x=-1</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t360" href="#t360">360</a></span><span class="t"><span class="str">            b (float): BC at y=0 and x=+1</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t361" href="#t361">361</a></span><span class="t"><span class="str">            c (float): BC at y=1 and x = -1</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t362" href="#t362">362</a></span><span class="t"><span class="str">            fx (int): Horizontal frequency of initial conditions</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t363" href="#t363">363</a></span><span class="t"><span class="str">            fy (int): Vertical frequency of initial conditions</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t364" href="#t364">364</a></span><span class="t"><span class="str">            nu (float): Diffusion parameter</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t365" href="#t365">365</a></span><span class="t"><span class="str">        """</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t366" href="#t366">366</a></span><span class="t">        <span class="nam">self</span><span class="op">.</span><span class="nam">_makeAttributeAndRegister</span><span class="op">(</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t367" href="#t367">367</a></span><span class="t">            <span class="str">'nx'</span><span class="op">,</span> <span class="str">'ny'</span><span class="op">,</span> <span class="str">'base_x'</span><span class="op">,</span> <span class="str">'base_y'</span><span class="op">,</span> <span class="str">'a'</span><span class="op">,</span> <span class="str">'b'</span><span class="op">,</span> <span class="str">'c'</span><span class="op">,</span> <span class="str">'fx'</span><span class="op">,</span> <span class="str">'fy'</span><span class="op">,</span> <span class="str">'nu'</span><span class="op">,</span> <span class="nam">localVars</span><span class="op">=</span><span class="nam">locals</span><span class="op">(</span><span class="op">)</span><span class="op">,</span> <span class="nam">readOnly</span><span class="op">=</span><span class="key">True</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t368" href="#t368">368</a></span><span class="t">        <span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t369" href="#t369">369</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t370" href="#t370">370</a></span><span class="t">        <span class="nam">bases</span> <span class="op">=</span> <span class="op">[</span><span class="op">{</span><span class="str">'base'</span><span class="op">:</span> <span class="nam">base_x</span><span class="op">,</span> <span class="str">'N'</span><span class="op">:</span> <span class="nam">nx</span><span class="op">}</span><span class="op">,</span> <span class="op">{</span><span class="str">'base'</span><span class="op">:</span> <span class="nam">base_y</span><span class="op">,</span> <span class="str">'N'</span><span class="op">:</span> <span class="nam">ny</span><span class="op">}</span><span class="op">]</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t371" href="#t371">371</a></span><span class="t">        <span class="nam">components</span> <span class="op">=</span> <span class="op">[</span><span class="str">'u'</span><span class="op">]</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t372" href="#t372">372</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t373" href="#t373">373</a></span><span class="t">        <span class="nam">super</span><span class="op">(</span><span class="op">)</span><span class="op">.</span><span class="nam">__init__</span><span class="op">(</span><span class="nam">bases</span><span class="op">,</span> <span class="nam">components</span><span class="op">,</span> <span class="nam">Dirichlet_recombination</span><span class="op">=</span><span class="key">False</span><span class="op">,</span> <span class="nam">spectral_space</span><span class="op">=</span><span class="key">False</span><span class="op">,</span> <span class="op">**</span><span class="nam">kwargs</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t374" href="#t374">374</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t375" href="#t375">375</a></span><span class="t">        <span class="nam">self</span><span class="op">.</span><span class="nam">Y</span><span class="op">,</span> <span class="nam">self</span><span class="op">.</span><span class="nam">X</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">get_grid</span><span class="op">(</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t376" href="#t376">376</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t377" href="#t377">377</a></span><span class="t">        <span class="nam">self</span><span class="op">.</span><span class="nam">Dxx</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">get_differentiation_matrix</span><span class="op">(</span><span class="nam">axes</span><span class="op">=</span><span class="op">(</span><span class="num">0</span><span class="op">,</span><span class="op">)</span><span class="op">,</span> <span class="nam">p</span><span class="op">=</span><span class="num">2</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t378" href="#t378">378</a></span><span class="t">        <span class="nam">self</span><span class="op">.</span><span class="nam">Dyy</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">get_differentiation_matrix</span><span class="op">(</span><span class="nam">axes</span><span class="op">=</span><span class="op">(</span><span class="num">1</span><span class="op">,</span><span class="op">)</span><span class="op">,</span> <span class="nam">p</span><span class="op">=</span><span class="num">2</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t379" href="#t379">379</a></span><span class="t">        <span class="nam">self</span><span class="op">.</span><span class="nam">S2</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">get_basis_change_matrix</span><span class="op">(</span><span class="nam">p</span><span class="op">=</span><span class="num">2</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t380" href="#t380">380</a></span><span class="t">        <span class="nam">I</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">get_Id</span><span class="op">(</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t381" href="#t381">381</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t382" href="#t382">382</a></span><span class="t">        <span class="nam">L_lhs</span> <span class="op">=</span> <span class="op">{</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t383" href="#t383">383</a></span><span class="t">            <span class="str">'u'</span><span class="op">:</span> <span class="op">{</span><span class="str">'u'</span><span class="op">:</span> <span class="op">-</span><span class="nam">nu</span> <span class="op">*</span> <span class="nam">self</span><span class="op">.</span><span class="nam">Dxx</span> <span class="op">-</span> <span class="nam">nu</span> <span class="op">*</span> <span class="nam">self</span><span class="op">.</span><span class="nam">Dyy</span><span class="op">}</span><span class="op">,</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t384" href="#t384">384</a></span><span class="t">        <span class="op">}</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t385" href="#t385">385</a></span><span class="t">        <span class="nam">self</span><span class="op">.</span><span class="nam">setup_L</span><span class="op">(</span><span class="nam">L_lhs</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t386" href="#t386">386</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t387" href="#t387">387</a></span><span class="t">        <span class="nam">M_lhs</span> <span class="op">=</span> <span class="op">{</span><span class="str">'u'</span><span class="op">:</span> <span class="op">{</span><span class="str">'u'</span><span class="op">:</span> <span class="nam">I</span><span class="op">}</span><span class="op">}</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t388" href="#t388">388</a></span><span class="t">        <span class="nam">self</span><span class="op">.</span><span class="nam">setup_M</span><span class="op">(</span><span class="nam">M_lhs</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t389" href="#t389">389</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t390" href="#t390">390</a></span><span class="t">        <span class="key">for</span> <span class="nam">base</span> <span class="key">in</span> <span class="op">[</span><span class="nam">base_x</span><span class="op">,</span> <span class="nam">base_y</span><span class="op">]</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t391" href="#t391">391</a></span><span class="t">            <span class="key">assert</span> <span class="nam">base</span> <span class="key">in</span> <span class="op">[</span><span class="str">'ultraspherical'</span><span class="op">,</span> <span class="str">'fft'</span><span class="op">]</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t392" href="#t392">392</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t393" href="#t393">393</a></span><span class="t">        <span class="nam">alpha</span> <span class="op">=</span> <span class="op">(</span><span class="nam">self</span><span class="op">.</span><span class="nam">b</span> <span class="op">-</span> <span class="nam">self</span><span class="op">.</span><span class="nam">a</span><span class="op">)</span> <span class="op">/</span> <span class="num">2.0</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t394" href="#t394">394</a></span><span class="t">        <span class="nam">beta</span> <span class="op">=</span> <span class="op">(</span><span class="nam">self</span><span class="op">.</span><span class="nam">c</span> <span class="op">-</span> <span class="nam">self</span><span class="op">.</span><span class="nam">b</span><span class="op">)</span> <span class="op">/</span> <span class="num">2.0</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t395" href="#t395">395</a></span><span class="t">        <span class="nam">gamma</span> <span class="op">=</span> <span class="op">(</span><span class="nam">self</span><span class="op">.</span><span class="nam">c</span> <span class="op">+</span> <span class="nam">self</span><span class="op">.</span><span class="nam">a</span><span class="op">)</span> <span class="op">/</span> <span class="num">2.0</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t396" href="#t396">396</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t397" href="#t397">397</a></span><span class="t">        <span class="key">if</span> <span class="nam">base_x</span> <span class="op">==</span> <span class="str">'ultraspherical'</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t398" href="#t398">398</a></span><span class="t">            <span class="nam">y</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">Y</span><span class="op">[</span><span class="num">0</span><span class="op">,</span> <span class="op">:</span><span class="op">]</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t399" href="#t399">399</a></span><span class="t">            <span class="key">if</span> <span class="nam">self</span><span class="op">.</span><span class="nam">base_y</span> <span class="op">==</span> <span class="str">'fft'</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t400" href="#t400">400</a></span><span class="t">                <span class="nam">self</span><span class="op">.</span><span class="nam">add_BC</span><span class="op">(</span><span class="nam">component</span><span class="op">=</span><span class="str">'u'</span><span class="op">,</span> <span class="nam">equation</span><span class="op">=</span><span class="str">'u'</span><span class="op">,</span> <span class="nam">axis</span><span class="op">=</span><span class="num">0</span><span class="op">,</span> <span class="nam">x</span><span class="op">=</span><span class="op">-</span><span class="num">1</span><span class="op">,</span> <span class="nam">v</span><span class="op">=</span><span class="nam">beta</span> <span class="op">*</span> <span class="nam">y</span> <span class="op">-</span> <span class="nam">alpha</span> <span class="op">+</span> <span class="nam">gamma</span><span class="op">,</span> <span class="nam">kind</span><span class="op">=</span><span class="str">'Dirichlet'</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t401" href="#t401">401</a></span><span class="t">            <span class="nam">self</span><span class="op">.</span><span class="nam">add_BC</span><span class="op">(</span><span class="nam">component</span><span class="op">=</span><span class="str">'u'</span><span class="op">,</span> <span class="nam">equation</span><span class="op">=</span><span class="str">'u'</span><span class="op">,</span> <span class="nam">axis</span><span class="op">=</span><span class="num">0</span><span class="op">,</span> <span class="nam">v</span><span class="op">=</span><span class="nam">beta</span> <span class="op">*</span> <span class="nam">y</span> <span class="op">+</span> <span class="nam">alpha</span> <span class="op">+</span> <span class="nam">gamma</span><span class="op">,</span> <span class="nam">x</span><span class="op">=</span><span class="num">1</span><span class="op">,</span> <span class="nam">line</span><span class="op">=</span><span class="op">-</span><span class="num">2</span><span class="op">,</span> <span class="nam">kind</span><span class="op">=</span><span class="str">'Dirichlet'</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t402" href="#t402">402</a></span><span class="t">        <span class="key">else</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t403" href="#t403">403</a></span><span class="t">            <span class="key">assert</span> <span class="nam">a</span> <span class="op">==</span> <span class="nam">b</span><span class="op">,</span> <span class="fst">f'</span><span class="fst">Need periodic boundary conditions in x for </span><span class="op">{</span><span class="nam">base_x</span><span class="op">}</span><span class="fst"> method!</span><span class="fst">'</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t404" href="#t404">404</a></span><span class="t">        <span class="key">if</span> <span class="nam">base_y</span> <span class="op">==</span> <span class="str">'ultraspherical'</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t405" href="#t405">405</a></span><span class="t">            <span class="nam">x</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">X</span><span class="op">[</span><span class="op">:</span><span class="op">,</span> <span class="num">0</span><span class="op">]</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t406" href="#t406">406</a></span><span class="t">            <span class="nam">self</span><span class="op">.</span><span class="nam">add_BC</span><span class="op">(</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t407" href="#t407">407</a></span><span class="t">                <span class="nam">component</span><span class="op">=</span><span class="str">'u'</span><span class="op">,</span> <span class="nam">equation</span><span class="op">=</span><span class="str">'u'</span><span class="op">,</span> <span class="nam">axis</span><span class="op">=</span><span class="num">1</span><span class="op">,</span> <span class="nam">x</span><span class="op">=</span><span class="op">-</span><span class="num">1</span><span class="op">,</span> <span class="nam">v</span><span class="op">=</span><span class="nam">alpha</span> <span class="op">*</span> <span class="nam">x</span> <span class="op">-</span> <span class="nam">beta</span> <span class="op">+</span> <span class="nam">gamma</span><span class="op">,</span> <span class="nam">kind</span><span class="op">=</span><span class="str">'Dirichlet'</span><span class="op">,</span> <span class="nam">line</span><span class="op">=</span><span class="op">-</span><span class="num">1</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t408" href="#t408">408</a></span><span class="t">            <span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t409" href="#t409">409</a></span><span class="t">            <span class="nam">self</span><span class="op">.</span><span class="nam">add_BC</span><span class="op">(</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t410" href="#t410">410</a></span><span class="t">                <span class="nam">component</span><span class="op">=</span><span class="str">'u'</span><span class="op">,</span> <span class="nam">equation</span><span class="op">=</span><span class="str">'u'</span><span class="op">,</span> <span class="nam">axis</span><span class="op">=</span><span class="num">1</span><span class="op">,</span> <span class="nam">x</span><span class="op">=</span><span class="op">+</span><span class="num">1</span><span class="op">,</span> <span class="nam">v</span><span class="op">=</span><span class="nam">alpha</span> <span class="op">*</span> <span class="nam">x</span> <span class="op">+</span> <span class="nam">beta</span> <span class="op">+</span> <span class="nam">gamma</span><span class="op">,</span> <span class="nam">kind</span><span class="op">=</span><span class="str">'Dirichlet'</span><span class="op">,</span> <span class="nam">line</span><span class="op">=</span><span class="op">-</span><span class="num">2</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t411" href="#t411">411</a></span><span class="t">            <span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t412" href="#t412">412</a></span><span class="t">        <span class="key">else</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t413" href="#t413">413</a></span><span class="t">            <span class="key">assert</span> <span class="nam">c</span> <span class="op">==</span> <span class="nam">b</span><span class="op">,</span> <span class="fst">f'</span><span class="fst">Need periodic boundary conditions in y for </span><span class="op">{</span><span class="nam">base_y</span><span class="op">}</span><span class="fst"> method!</span><span class="fst">'</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t414" href="#t414">414</a></span><span class="t">        <span class="nam">self</span><span class="op">.</span><span class="nam">setup_BCs</span><span class="op">(</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t415" href="#t415">415</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t416" href="#t416">416</a></span><span class="t">    <span class="key">def</span> <span class="nam">eval_f</span><span class="op">(</span><span class="nam">self</span><span class="op">,</span> <span class="nam">u</span><span class="op">,</span> <span class="op">*</span><span class="nam">args</span><span class="op">,</span> <span class="op">**</span><span class="nam">kwargs</span><span class="op">)</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t417" href="#t417">417</a></span><span class="t">        <span class="nam">f</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">f_init</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t418" href="#t418">418</a></span><span class="t">        <span class="nam">iu</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">index</span><span class="op">(</span><span class="str">'u'</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t419" href="#t419">419</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t420" href="#t420">420</a></span><span class="t">        <span class="nam">me_hat</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">u_init_forward</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t421" href="#t421">421</a></span><span class="t">        <span class="nam">me_hat</span><span class="op">[</span><span class="op">:</span><span class="op">]</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">transform</span><span class="op">(</span><span class="nam">u</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t422" href="#t422">422</a></span><span class="t">        <span class="nam">me_hat</span><span class="op">[</span><span class="nam">iu</span><span class="op">]</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">nu</span> <span class="op">*</span> <span class="op">(</span><span class="nam">self</span><span class="op">.</span><span class="nam">S2</span> <span class="op">@</span> <span class="op">(</span><span class="nam">self</span><span class="op">.</span><span class="nam">Dxx</span> <span class="op">+</span> <span class="nam">self</span><span class="op">.</span><span class="nam">Dyy</span><span class="op">)</span> <span class="op">@</span> <span class="nam">me_hat</span><span class="op">[</span><span class="nam">iu</span><span class="op">]</span><span class="op">.</span><span class="nam">flatten</span><span class="op">(</span><span class="op">)</span><span class="op">)</span><span class="op">.</span><span class="nam">reshape</span><span class="op">(</span><span class="nam">me_hat</span><span class="op">[</span><span class="nam">iu</span><span class="op">]</span><span class="op">.</span><span class="nam">shape</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t423" href="#t423">423</a></span><span class="t">        <span class="nam">me</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">itransform</span><span class="op">(</span><span class="nam">me_hat</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t424" href="#t424">424</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t425" href="#t425">425</a></span><span class="t">        <span class="nam">f</span><span class="op">[</span><span class="nam">iu</span><span class="op">]</span> <span class="op">=</span> <span class="nam">me</span><span class="op">[</span><span class="nam">iu</span><span class="op">]</span><span class="op">.</span><span class="nam">real</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t426" href="#t426">426</a></span><span class="t">        <span class="key">return</span> <span class="nam">f</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t427" href="#t427">427</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t428" href="#t428">428</a></span><span class="t">    <span class="key">def</span> <span class="nam">u_exact</span><span class="op">(</span><span class="nam">self</span><span class="op">,</span> <span class="nam">t</span><span class="op">)</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t429" href="#t429">429</a></span><span class="t">        <span class="nam">xp</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">xp</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t430" href="#t430">430</a></span><span class="t">        <span class="nam">iu</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">index</span><span class="op">(</span><span class="str">'u'</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t431" href="#t431">431</a></span><span class="t">        <span class="nam">u</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">u_init</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t432" href="#t432">432</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t433" href="#t433">433</a></span><span class="t">        <span class="nam">fx</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">fx</span> <span class="key">if</span> <span class="nam">self</span><span class="op">.</span><span class="nam">base_x</span> <span class="op">==</span> <span class="str">'fft'</span> <span class="key">else</span> <span class="nam">np</span><span class="op">.</span><span class="nam">pi</span> <span class="op">*</span> <span class="nam">self</span><span class="op">.</span><span class="nam">fx</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t434" href="#t434">434</a></span><span class="t">        <span class="nam">fy</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">fy</span> <span class="key">if</span> <span class="nam">self</span><span class="op">.</span><span class="nam">base_y</span> <span class="op">==</span> <span class="str">'fft'</span> <span class="key">else</span> <span class="nam">np</span><span class="op">.</span><span class="nam">pi</span> <span class="op">*</span> <span class="nam">self</span><span class="op">.</span><span class="nam">fy</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t435" href="#t435">435</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t436" href="#t436">436</a></span><span class="t">        <span class="nam">time_dep</span> <span class="op">=</span> <span class="nam">xp</span><span class="op">.</span><span class="nam">exp</span><span class="op">(</span><span class="op">-</span><span class="nam">self</span><span class="op">.</span><span class="nam">nu</span> <span class="op">*</span> <span class="op">(</span><span class="nam">fx</span><span class="op">**</span><span class="num">2</span> <span class="op">+</span> <span class="nam">fy</span><span class="op">**</span><span class="num">2</span><span class="op">)</span> <span class="op">*</span> <span class="nam">t</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t437" href="#t437">437</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t438" href="#t438">438</a></span><span class="t">        <span class="nam">alpha</span> <span class="op">=</span> <span class="op">(</span><span class="nam">self</span><span class="op">.</span><span class="nam">b</span> <span class="op">-</span> <span class="nam">self</span><span class="op">.</span><span class="nam">a</span><span class="op">)</span> <span class="op">/</span> <span class="num">2.0</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t439" href="#t439">439</a></span><span class="t">        <span class="nam">beta</span> <span class="op">=</span> <span class="op">(</span><span class="nam">self</span><span class="op">.</span><span class="nam">c</span> <span class="op">-</span> <span class="nam">self</span><span class="op">.</span><span class="nam">b</span><span class="op">)</span> <span class="op">/</span> <span class="num">2.0</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t440" href="#t440">440</a></span><span class="t">        <span class="nam">gamma</span> <span class="op">=</span> <span class="op">(</span><span class="nam">self</span><span class="op">.</span><span class="nam">c</span> <span class="op">+</span> <span class="nam">self</span><span class="op">.</span><span class="nam">a</span><span class="op">)</span> <span class="op">/</span> <span class="num">2.0</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t441" href="#t441">441</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t442" href="#t442">442</a></span><span class="t">        <span class="nam">u</span><span class="op">[</span><span class="nam">iu</span><span class="op">]</span> <span class="op">=</span> <span class="nam">xp</span><span class="op">.</span><span class="nam">sin</span><span class="op">(</span><span class="nam">fx</span> <span class="op">*</span> <span class="nam">self</span><span class="op">.</span><span class="nam">X</span><span class="op">)</span> <span class="op">*</span> <span class="nam">xp</span><span class="op">.</span><span class="nam">sin</span><span class="op">(</span><span class="nam">fy</span> <span class="op">*</span> <span class="nam">self</span><span class="op">.</span><span class="nam">Y</span><span class="op">)</span> <span class="op">*</span> <span class="nam">time_dep</span> <span class="op">+</span> <span class="nam">alpha</span> <span class="op">*</span> <span class="nam">self</span><span class="op">.</span><span class="nam">X</span> <span class="op">+</span> <span class="nam">beta</span> <span class="op">*</span> <span class="nam">self</span><span class="op">.</span><span class="nam">Y</span> <span class="op">+</span> <span class="nam">gamma</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t443" href="#t443">443</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t444" href="#t444">444</a></span><span class="t">        <span class="key">return</span> <span class="nam">u</span>&nbsp;</span><span class="r"></span></p>
+</main>
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diff --git a/coverage/z_91faa57f8583c837_RayleighBenard_py.html b/coverage/z_91faa57f8583c837_RayleighBenard_py.html
new file mode 100644
index 0000000000..041a734b4b
--- /dev/null
+++ b/coverage/z_91faa57f8583c837_RayleighBenard_py.html
@@ -0,0 +1,657 @@
+<!DOCTYPE html>
+<html lang="en">
+<head>
+    <meta http-equiv="Content-Type" content="text/html; charset=utf-8">
+    <title>Coverage for pySDC/implementations/problem_classes/RayleighBenard.py: 78%</title>
+    <link rel="icon" sizes="32x32" href="favicon_32_cb_58284776.png">
+    <link rel="stylesheet" href="style_cb_8e611ae1.css" type="text/css">
+    <script src="coverage_html_cb_6fb7b396.js" defer></script>
+</head>
+<body class="pyfile">
+<header>
+    <div class="content">
+        <h1>
+            <span class="text">Coverage for </span><b>pySDC/implementations/problem_classes/RayleighBenard.py</b>:
+            <span class="pc_cov">78%</span>
+        </h1>
+        <aside id="help_panel_wrapper">
+            <input id="help_panel_state" type="checkbox">
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+                <img id="keyboard_icon" src="keybd_closed_cb_ce680311.png" alt="Show/hide keyboard shortcuts">
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+            <div id="help_panel">
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+                        <kbd>0</kbd> &nbsp; (zero) top of page
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+                </div>
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+        <h2>
+            <span class="text">209 statements &nbsp;</span>
+            <button type="button" class="run button_toggle_run" value="run" data-shortcut="r" title="Toggle lines run">162<span class="text"> run</span></button>
+            <button type="button" class="mis show_mis button_toggle_mis" value="mis" data-shortcut="m" title="Toggle lines missing">47<span class="text"> missing</span></button>
+            <button type="button" class="exc show_exc button_toggle_exc" value="exc" data-shortcut="x" title="Toggle lines excluded">60<span class="text"> excluded</span></button>
+        </h2>
+        <p class="text">
+            <a id="prevFileLink" class="nav" href="z_91faa57f8583c837_Quench_py.html">&#xab; prev</a> &nbsp; &nbsp;
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+            &nbsp; &nbsp; &nbsp;
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+            <button type="button" class="button_show_hide_help" data-shortcut="?"></button>
+        </aside>
+    </div>
+</header>
+<main id="source">
+    <p class="run"><span class="n"><a id="t1" href="#t1">1</a></span><span class="t"><span class="key">import</span> <span class="nam">numpy</span> <span class="key">as</span> <span class="nam">np</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t2" href="#t2">2</a></span><span class="t"><span class="key">from</span> <span class="nam">mpi4py</span> <span class="key">import</span> <span class="nam">MPI</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t3" href="#t3">3</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t4" href="#t4">4</a></span><span class="t"><span class="key">from</span> <span class="nam">pySDC</span><span class="op">.</span><span class="nam">implementations</span><span class="op">.</span><span class="nam">problem_classes</span><span class="op">.</span><span class="nam">generic_spectral</span> <span class="key">import</span> <span class="nam">GenericSpectralLinear</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t5" href="#t5">5</a></span><span class="t"><span class="key">from</span> <span class="nam">pySDC</span><span class="op">.</span><span class="nam">implementations</span><span class="op">.</span><span class="nam">datatype_classes</span><span class="op">.</span><span class="nam">mesh</span> <span class="key">import</span> <span class="nam">mesh</span><span class="op">,</span> <span class="nam">imex_mesh</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t6" href="#t6">6</a></span><span class="t"><span class="key">from</span> <span class="nam">pySDC</span><span class="op">.</span><span class="nam">core</span><span class="op">.</span><span class="nam">convergence_controller</span> <span class="key">import</span> <span class="nam">ConvergenceController</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t7" href="#t7">7</a></span><span class="t"><span class="key">from</span> <span class="nam">pySDC</span><span class="op">.</span><span class="nam">core</span><span class="op">.</span><span class="nam">hooks</span> <span class="key">import</span> <span class="nam">Hooks</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t8" href="#t8">8</a></span><span class="t"><span class="key">from</span> <span class="nam">pySDC</span><span class="op">.</span><span class="nam">implementations</span><span class="op">.</span><span class="nam">convergence_controller_classes</span><span class="op">.</span><span class="nam">check_convergence</span> <span class="key">import</span> <span class="nam">CheckConvergence</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t9" href="#t9">9</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t10" href="#t10">10</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t11" href="#t11">11</a></span><span class="t"><span class="key">class</span> <span class="nam">RayleighBenard</span><span class="op">(</span><span class="nam">GenericSpectralLinear</span><span class="op">)</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t12" href="#t12">12</a></span><span class="t">    <span class="str">"""</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t13" href="#t13">13</a></span><span class="t"><span class="str">    Rayleigh-Benard Convection is a variation of incompressible Navier-Stokes.</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t14" href="#t14">14</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t15" href="#t15">15</a></span><span class="t"><span class="str">    The equations we solve are</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t16" href="#t16">16</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t17" href="#t17">17</a></span><span class="t"><span class="str">        u_x + v_z = 0</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t18" href="#t18">18</a></span><span class="t"><span class="str">        T_t - kappa (T_xx + T_zz) = -uT_x - vT_z</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t19" href="#t19">19</a></span><span class="t"><span class="str">        u_t - nu (u_xx + u_zz) + p_x = -uu_x - vu_z</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t20" href="#t20">20</a></span><span class="t"><span class="str">        v_t - nu (v_xx + v_zz) + p_z - T = -uv_x - vv_z</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t21" href="#t21">21</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t22" href="#t22">22</a></span><span class="t"><span class="str">    with u the horizontal velocity, v the vertical velocity (in z-direction), T the temperature, p the pressure, indices</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t23" href="#t23">23</a></span><span class="t"><span class="str">    denoting derivatives, kappa=(Rayleigh * Prandl)**(-1/2) and nu = (Rayleigh / Prandl)**(-1/2). Everything on the left</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t24" href="#t24">24</a></span><span class="t"><span class="str">    hand side, that is the viscous part, the pressure gradient and the buoyancy due to temperature are treated</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t25" href="#t25">25</a></span><span class="t"><span class="str">    implicitly, while the non-linear convection part on the right hand side is integrated explicitly.</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t26" href="#t26">26</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t27" href="#t27">27</a></span><span class="t"><span class="str">    The domain, vertical boundary conditions and pressure gauge are</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t28" href="#t28">28</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t29" href="#t29">29</a></span><span class="t"><span class="str">        Omega = [0, 8) x (-1, 1)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t30" href="#t30">30</a></span><span class="t"><span class="str">        T(z=+1) = 0</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t31" href="#t31">31</a></span><span class="t"><span class="str">        T(z=-1) = 2</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t32" href="#t32">32</a></span><span class="t"><span class="str">        u(z=+-1) = v(z=+-1) = 0</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t33" href="#t33">33</a></span><span class="t"><span class="str">        integral over p = 0</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t34" href="#t34">34</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t35" href="#t35">35</a></span><span class="t"><span class="str">    The spectral discretization uses FFT horizontally, implying periodic BCs, and an ultraspherical method vertically to</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t36" href="#t36">36</a></span><span class="t"><span class="str">    facilitate the Dirichlet BCs.</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t37" href="#t37">37</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t38" href="#t38">38</a></span><span class="t"><span class="str">    Parameters:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t39" href="#t39">39</a></span><span class="t"><span class="str">        Prandl (float): Prandl number</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t40" href="#t40">40</a></span><span class="t"><span class="str">        Rayleigh (float): Rayleigh number</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t41" href="#t41">41</a></span><span class="t"><span class="str">        nx (int): Horizontal resolution</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t42" href="#t42">42</a></span><span class="t"><span class="str">        nz (int): Vertical resolution</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t43" href="#t43">43</a></span><span class="t"><span class="str">        BCs (dict): Can specify boundary conditions here</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t44" href="#t44">44</a></span><span class="t"><span class="str">        dealiasing (float): Dealiasing factor for evaluating the non-linear part</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t45" href="#t45">45</a></span><span class="t"><span class="str">        comm (mpi4py.Intracomm): Space communicator</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t46" href="#t46">46</a></span><span class="t"><span class="str">    """</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t47" href="#t47">47</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t48" href="#t48">48</a></span><span class="t">    <span class="nam">dtype_u</span> <span class="op">=</span> <span class="nam">mesh</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t49" href="#t49">49</a></span><span class="t">    <span class="nam">dtype_f</span> <span class="op">=</span> <span class="nam">imex_mesh</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t50" href="#t50">50</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t51" href="#t51">51</a></span><span class="t">    <span class="key">def</span> <span class="nam">__init__</span><span class="op">(</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t52" href="#t52">52</a></span><span class="t">        <span class="nam">self</span><span class="op">,</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t53" href="#t53">53</a></span><span class="t">        <span class="nam">Prandl</span><span class="op">=</span><span class="num">1</span><span class="op">,</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t54" href="#t54">54</a></span><span class="t">        <span class="nam">Rayleigh</span><span class="op">=</span><span class="num">2e6</span><span class="op">,</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t55" href="#t55">55</a></span><span class="t">        <span class="nam">nx</span><span class="op">=</span><span class="num">256</span><span class="op">,</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t56" href="#t56">56</a></span><span class="t">        <span class="nam">nz</span><span class="op">=</span><span class="num">64</span><span class="op">,</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t57" href="#t57">57</a></span><span class="t">        <span class="nam">BCs</span><span class="op">=</span><span class="key">None</span><span class="op">,</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t58" href="#t58">58</a></span><span class="t">        <span class="nam">dealiasing</span><span class="op">=</span><span class="num">3</span> <span class="op">/</span> <span class="num">2</span><span class="op">,</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t59" href="#t59">59</a></span><span class="t">        <span class="nam">comm</span><span class="op">=</span><span class="key">None</span><span class="op">,</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t60" href="#t60">60</a></span><span class="t">        <span class="op">**</span><span class="nam">kwargs</span><span class="op">,</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t61" href="#t61">61</a></span><span class="t">    <span class="op">)</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t62" href="#t62">62</a></span><span class="t">        <span class="str">"""</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t63" href="#t63">63</a></span><span class="t"><span class="str">        Constructor. `kwargs` are forwarded to parent class constructor.</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t64" href="#t64">64</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t65" href="#t65">65</a></span><span class="t"><span class="str">        Args:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t66" href="#t66">66</a></span><span class="t"><span class="str">            Prandl (float): Prandtl number</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t67" href="#t67">67</a></span><span class="t"><span class="str">            Rayleigh (float): Rayleigh number</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t68" href="#t68">68</a></span><span class="t"><span class="str">            nx (int): Resolution in x-direction</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t69" href="#t69">69</a></span><span class="t"><span class="str">            nz (int): Resolution in z direction</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t70" href="#t70">70</a></span><span class="t"><span class="str">            BCs (dict): Vertical boundary conditions</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t71" href="#t71">71</a></span><span class="t"><span class="str">            dealiasing (float): Dealiasing for evaluating the non-linear part in real space</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t72" href="#t72">72</a></span><span class="t"><span class="str">            comm (mpi4py.Intracomm): Space communicator</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t73" href="#t73">73</a></span><span class="t"><span class="str">        """</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t74" href="#t74">74</a></span><span class="t">        <span class="nam">BCs</span> <span class="op">=</span> <span class="op">{</span><span class="op">}</span> <span class="key">if</span> <span class="nam">BCs</span> <span class="key">is</span> <span class="key">None</span> <span class="key">else</span> <span class="nam">BCs</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t75" href="#t75">75</a></span><span class="t">        <span class="nam">BCs</span> <span class="op">=</span> <span class="op">{</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t76" href="#t76">76</a></span><span class="t">            <span class="str">'T_top'</span><span class="op">:</span> <span class="num">0</span><span class="op">,</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t77" href="#t77">77</a></span><span class="t">            <span class="str">'T_bottom'</span><span class="op">:</span> <span class="num">2</span><span class="op">,</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t78" href="#t78">78</a></span><span class="t">            <span class="str">'v_top'</span><span class="op">:</span> <span class="num">0</span><span class="op">,</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t79" href="#t79">79</a></span><span class="t">            <span class="str">'v_bottom'</span><span class="op">:</span> <span class="num">0</span><span class="op">,</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t80" href="#t80">80</a></span><span class="t">            <span class="str">'u_top'</span><span class="op">:</span> <span class="num">0</span><span class="op">,</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t81" href="#t81">81</a></span><span class="t">            <span class="str">'u_bottom'</span><span class="op">:</span> <span class="num">0</span><span class="op">,</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t82" href="#t82">82</a></span><span class="t">            <span class="str">'p_integral'</span><span class="op">:</span> <span class="num">0</span><span class="op">,</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t83" href="#t83">83</a></span><span class="t">            <span class="op">**</span><span class="nam">BCs</span><span class="op">,</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t84" href="#t84">84</a></span><span class="t">        <span class="op">}</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t85" href="#t85">85</a></span><span class="t">        <span class="key">if</span> <span class="nam">comm</span> <span class="key">is</span> <span class="key">None</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t86" href="#t86">86</a></span><span class="t">            <span class="key">try</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t87" href="#t87">87</a></span><span class="t">                <span class="key">from</span> <span class="nam">mpi4py</span> <span class="key">import</span> <span class="nam">MPI</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t88" href="#t88">88</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t89" href="#t89">89</a></span><span class="t">                <span class="nam">comm</span> <span class="op">=</span> <span class="nam">MPI</span><span class="op">.</span><span class="nam">COMM_WORLD</span>&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t90" href="#t90">90</a></span><span class="t">            <span class="key">except</span> <span class="nam">ModuleNotFoundError</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t91" href="#t91">91</a></span><span class="t">                <span class="key">pass</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t92" href="#t92">92</a></span><span class="t">        <span class="nam">self</span><span class="op">.</span><span class="nam">_makeAttributeAndRegister</span><span class="op">(</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t93" href="#t93">93</a></span><span class="t">            <span class="str">'Prandl'</span><span class="op">,</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t94" href="#t94">94</a></span><span class="t">            <span class="str">'Rayleigh'</span><span class="op">,</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t95" href="#t95">95</a></span><span class="t">            <span class="str">'nx'</span><span class="op">,</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t96" href="#t96">96</a></span><span class="t">            <span class="str">'nz'</span><span class="op">,</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t97" href="#t97">97</a></span><span class="t">            <span class="str">'BCs'</span><span class="op">,</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t98" href="#t98">98</a></span><span class="t">            <span class="str">'dealiasing'</span><span class="op">,</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t99" href="#t99">99</a></span><span class="t">            <span class="str">'comm'</span><span class="op">,</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t100" href="#t100">100</a></span><span class="t">            <span class="nam">localVars</span><span class="op">=</span><span class="nam">locals</span><span class="op">(</span><span class="op">)</span><span class="op">,</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t101" href="#t101">101</a></span><span class="t">            <span class="nam">readOnly</span><span class="op">=</span><span class="key">True</span><span class="op">,</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t102" href="#t102">102</a></span><span class="t">        <span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t103" href="#t103">103</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t104" href="#t104">104</a></span><span class="t">        <span class="nam">bases</span> <span class="op">=</span> <span class="op">[</span><span class="op">{</span><span class="str">'base'</span><span class="op">:</span> <span class="str">'fft'</span><span class="op">,</span> <span class="str">'N'</span><span class="op">:</span> <span class="nam">nx</span><span class="op">,</span> <span class="str">'x0'</span><span class="op">:</span> <span class="num">0</span><span class="op">,</span> <span class="str">'x1'</span><span class="op">:</span> <span class="num">8</span><span class="op">}</span><span class="op">,</span> <span class="op">{</span><span class="str">'base'</span><span class="op">:</span> <span class="str">'ultraspherical'</span><span class="op">,</span> <span class="str">'N'</span><span class="op">:</span> <span class="nam">nz</span><span class="op">}</span><span class="op">]</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t105" href="#t105">105</a></span><span class="t">        <span class="nam">components</span> <span class="op">=</span> <span class="op">[</span><span class="str">'u'</span><span class="op">,</span> <span class="str">'v'</span><span class="op">,</span> <span class="str">'T'</span><span class="op">,</span> <span class="str">'p'</span><span class="op">]</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t106" href="#t106">106</a></span><span class="t">        <span class="nam">super</span><span class="op">(</span><span class="op">)</span><span class="op">.</span><span class="nam">__init__</span><span class="op">(</span><span class="nam">bases</span><span class="op">,</span> <span class="nam">components</span><span class="op">,</span> <span class="nam">comm</span><span class="op">=</span><span class="nam">comm</span><span class="op">,</span> <span class="op">**</span><span class="nam">kwargs</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t107" href="#t107">107</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t108" href="#t108">108</a></span><span class="t">        <span class="nam">self</span><span class="op">.</span><span class="nam">Z</span><span class="op">,</span> <span class="nam">self</span><span class="op">.</span><span class="nam">X</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">get_grid</span><span class="op">(</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t109" href="#t109">109</a></span><span class="t">        <span class="nam">self</span><span class="op">.</span><span class="nam">Kz</span><span class="op">,</span> <span class="nam">self</span><span class="op">.</span><span class="nam">Kx</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">get_wavenumbers</span><span class="op">(</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t110" href="#t110">110</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t111" href="#t111">111</a></span><span class="t">        <span class="com"># construct 2D matrices</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t112" href="#t112">112</a></span><span class="t">        <span class="nam">Dzz</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">get_differentiation_matrix</span><span class="op">(</span><span class="nam">axes</span><span class="op">=</span><span class="op">(</span><span class="num">1</span><span class="op">,</span><span class="op">)</span><span class="op">,</span> <span class="nam">p</span><span class="op">=</span><span class="num">2</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t113" href="#t113">113</a></span><span class="t">        <span class="nam">Dz</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">get_differentiation_matrix</span><span class="op">(</span><span class="nam">axes</span><span class="op">=</span><span class="op">(</span><span class="num">1</span><span class="op">,</span><span class="op">)</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t114" href="#t114">114</a></span><span class="t">        <span class="nam">Dx</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">get_differentiation_matrix</span><span class="op">(</span><span class="nam">axes</span><span class="op">=</span><span class="op">(</span><span class="num">0</span><span class="op">,</span><span class="op">)</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t115" href="#t115">115</a></span><span class="t">        <span class="nam">Dxx</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">get_differentiation_matrix</span><span class="op">(</span><span class="nam">axes</span><span class="op">=</span><span class="op">(</span><span class="num">0</span><span class="op">,</span><span class="op">)</span><span class="op">,</span> <span class="nam">p</span><span class="op">=</span><span class="num">2</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t116" href="#t116">116</a></span><span class="t">        <span class="nam">Id</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">get_Id</span><span class="op">(</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t117" href="#t117">117</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t118" href="#t118">118</a></span><span class="t">        <span class="nam">S1</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">get_basis_change_matrix</span><span class="op">(</span><span class="nam">p_out</span><span class="op">=</span><span class="num">0</span><span class="op">,</span> <span class="nam">p_in</span><span class="op">=</span><span class="num">1</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t119" href="#t119">119</a></span><span class="t">        <span class="nam">S2</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">get_basis_change_matrix</span><span class="op">(</span><span class="nam">p_out</span><span class="op">=</span><span class="num">0</span><span class="op">,</span> <span class="nam">p_in</span><span class="op">=</span><span class="num">2</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t120" href="#t120">120</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t121" href="#t121">121</a></span><span class="t">        <span class="nam">U01</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">get_basis_change_matrix</span><span class="op">(</span><span class="nam">p_in</span><span class="op">=</span><span class="num">0</span><span class="op">,</span> <span class="nam">p_out</span><span class="op">=</span><span class="num">1</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t122" href="#t122">122</a></span><span class="t">        <span class="nam">U12</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">get_basis_change_matrix</span><span class="op">(</span><span class="nam">p_in</span><span class="op">=</span><span class="num">1</span><span class="op">,</span> <span class="nam">p_out</span><span class="op">=</span><span class="num">2</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t123" href="#t123">123</a></span><span class="t">        <span class="nam">U02</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">get_basis_change_matrix</span><span class="op">(</span><span class="nam">p_in</span><span class="op">=</span><span class="num">0</span><span class="op">,</span> <span class="nam">p_out</span><span class="op">=</span><span class="num">2</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t124" href="#t124">124</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t125" href="#t125">125</a></span><span class="t">        <span class="nam">self</span><span class="op">.</span><span class="nam">Dx</span> <span class="op">=</span> <span class="nam">Dx</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t126" href="#t126">126</a></span><span class="t">        <span class="nam">self</span><span class="op">.</span><span class="nam">Dxx</span> <span class="op">=</span> <span class="nam">Dxx</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t127" href="#t127">127</a></span><span class="t">        <span class="nam">self</span><span class="op">.</span><span class="nam">Dz</span> <span class="op">=</span> <span class="nam">S1</span> <span class="op">@</span> <span class="nam">Dz</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t128" href="#t128">128</a></span><span class="t">        <span class="nam">self</span><span class="op">.</span><span class="nam">Dzz</span> <span class="op">=</span> <span class="nam">S2</span> <span class="op">@</span> <span class="nam">Dzz</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t129" href="#t129">129</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t130" href="#t130">130</a></span><span class="t">        <span class="nam">kappa</span> <span class="op">=</span> <span class="op">(</span><span class="nam">Rayleigh</span> <span class="op">*</span> <span class="nam">Prandl</span><span class="op">)</span> <span class="op">**</span> <span class="op">(</span><span class="op">-</span><span class="num">1</span> <span class="op">/</span> <span class="num">2.0</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t131" href="#t131">131</a></span><span class="t">        <span class="nam">nu</span> <span class="op">=</span> <span class="op">(</span><span class="nam">Rayleigh</span> <span class="op">/</span> <span class="nam">Prandl</span><span class="op">)</span> <span class="op">**</span> <span class="op">(</span><span class="op">-</span><span class="num">1</span> <span class="op">/</span> <span class="num">2.0</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t132" href="#t132">132</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t133" href="#t133">133</a></span><span class="t">        <span class="com"># construct operators</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t134" href="#t134">134</a></span><span class="t">        <span class="nam">L_lhs</span> <span class="op">=</span> <span class="op">{</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t135" href="#t135">135</a></span><span class="t">            <span class="str">'p'</span><span class="op">:</span> <span class="op">{</span><span class="str">'u'</span><span class="op">:</span> <span class="nam">U01</span> <span class="op">@</span> <span class="nam">Dx</span><span class="op">,</span> <span class="str">'v'</span><span class="op">:</span> <span class="nam">Dz</span><span class="op">}</span><span class="op">,</span>  <span class="com"># divergence free constraint</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t136" href="#t136">136</a></span><span class="t">            <span class="str">'u'</span><span class="op">:</span> <span class="op">{</span><span class="str">'p'</span><span class="op">:</span> <span class="nam">U02</span> <span class="op">@</span> <span class="nam">Dx</span><span class="op">,</span> <span class="str">'u'</span><span class="op">:</span> <span class="op">-</span><span class="nam">nu</span> <span class="op">*</span> <span class="op">(</span><span class="nam">U02</span> <span class="op">@</span> <span class="nam">Dxx</span> <span class="op">+</span> <span class="nam">Dzz</span><span class="op">)</span><span class="op">}</span><span class="op">,</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t137" href="#t137">137</a></span><span class="t">            <span class="str">'v'</span><span class="op">:</span> <span class="op">{</span><span class="str">'p'</span><span class="op">:</span> <span class="nam">U12</span> <span class="op">@</span> <span class="nam">Dz</span><span class="op">,</span> <span class="str">'v'</span><span class="op">:</span> <span class="op">-</span><span class="nam">nu</span> <span class="op">*</span> <span class="op">(</span><span class="nam">U02</span> <span class="op">@</span> <span class="nam">Dxx</span> <span class="op">+</span> <span class="nam">Dzz</span><span class="op">)</span><span class="op">,</span> <span class="str">'T'</span><span class="op">:</span> <span class="op">-</span><span class="nam">U02</span> <span class="op">@</span> <span class="nam">Id</span><span class="op">}</span><span class="op">,</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t138" href="#t138">138</a></span><span class="t">            <span class="str">'T'</span><span class="op">:</span> <span class="op">{</span><span class="str">'T'</span><span class="op">:</span> <span class="op">-</span><span class="nam">kappa</span> <span class="op">*</span> <span class="op">(</span><span class="nam">U02</span> <span class="op">@</span> <span class="nam">Dxx</span> <span class="op">+</span> <span class="nam">Dzz</span><span class="op">)</span><span class="op">}</span><span class="op">,</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t139" href="#t139">139</a></span><span class="t">        <span class="op">}</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t140" href="#t140">140</a></span><span class="t">        <span class="nam">self</span><span class="op">.</span><span class="nam">setup_L</span><span class="op">(</span><span class="nam">L_lhs</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t141" href="#t141">141</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t142" href="#t142">142</a></span><span class="t">        <span class="com"># mass matrix</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t143" href="#t143">143</a></span><span class="t">        <span class="nam">M_lhs</span> <span class="op">=</span> <span class="op">{</span><span class="nam">i</span><span class="op">:</span> <span class="op">{</span><span class="nam">i</span><span class="op">:</span> <span class="nam">U02</span> <span class="op">@</span> <span class="nam">Id</span><span class="op">}</span> <span class="key">for</span> <span class="nam">i</span> <span class="key">in</span> <span class="op">[</span><span class="str">'u'</span><span class="op">,</span> <span class="str">'v'</span><span class="op">,</span> <span class="str">'T'</span><span class="op">]</span><span class="op">}</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t144" href="#t144">144</a></span><span class="t">        <span class="nam">self</span><span class="op">.</span><span class="nam">setup_M</span><span class="op">(</span><span class="nam">M_lhs</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t145" href="#t145">145</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t146" href="#t146">146</a></span><span class="t">        <span class="com"># Prepare going from second (first for divergence free equation) derivative basis back to Chebychov-T</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t147" href="#t147">147</a></span><span class="t">        <span class="nam">self</span><span class="op">.</span><span class="nam">base_change</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">_setup_operator</span><span class="op">(</span><span class="op">{</span><span class="op">**</span><span class="op">{</span><span class="nam">comp</span><span class="op">:</span> <span class="op">{</span><span class="nam">comp</span><span class="op">:</span> <span class="nam">S2</span><span class="op">}</span> <span class="key">for</span> <span class="nam">comp</span> <span class="key">in</span> <span class="op">[</span><span class="str">'u'</span><span class="op">,</span> <span class="str">'v'</span><span class="op">,</span> <span class="str">'T'</span><span class="op">]</span><span class="op">}</span><span class="op">,</span> <span class="str">'p'</span><span class="op">:</span> <span class="op">{</span><span class="str">'p'</span><span class="op">:</span> <span class="nam">S1</span><span class="op">}</span><span class="op">}</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t148" href="#t148">148</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t149" href="#t149">149</a></span><span class="t">        <span class="com"># BCs</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t150" href="#t150">150</a></span><span class="t">        <span class="nam">self</span><span class="op">.</span><span class="nam">add_BC</span><span class="op">(</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t151" href="#t151">151</a></span><span class="t">            <span class="nam">component</span><span class="op">=</span><span class="str">'p'</span><span class="op">,</span> <span class="nam">equation</span><span class="op">=</span><span class="str">'p'</span><span class="op">,</span> <span class="nam">axis</span><span class="op">=</span><span class="num">1</span><span class="op">,</span> <span class="nam">v</span><span class="op">=</span><span class="nam">self</span><span class="op">.</span><span class="nam">BCs</span><span class="op">[</span><span class="str">'p_integral'</span><span class="op">]</span><span class="op">,</span> <span class="nam">kind</span><span class="op">=</span><span class="str">'integral'</span><span class="op">,</span> <span class="nam">line</span><span class="op">=</span><span class="op">-</span><span class="num">1</span><span class="op">,</span> <span class="nam">scalar</span><span class="op">=</span><span class="key">True</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t152" href="#t152">152</a></span><span class="t">        <span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t153" href="#t153">153</a></span><span class="t">        <span class="nam">self</span><span class="op">.</span><span class="nam">add_BC</span><span class="op">(</span><span class="nam">component</span><span class="op">=</span><span class="str">'T'</span><span class="op">,</span> <span class="nam">equation</span><span class="op">=</span><span class="str">'T'</span><span class="op">,</span> <span class="nam">axis</span><span class="op">=</span><span class="num">1</span><span class="op">,</span> <span class="nam">x</span><span class="op">=</span><span class="op">-</span><span class="num">1</span><span class="op">,</span> <span class="nam">v</span><span class="op">=</span><span class="nam">self</span><span class="op">.</span><span class="nam">BCs</span><span class="op">[</span><span class="str">'T_bottom'</span><span class="op">]</span><span class="op">,</span> <span class="nam">kind</span><span class="op">=</span><span class="str">'Dirichlet'</span><span class="op">,</span> <span class="nam">line</span><span class="op">=</span><span class="op">-</span><span class="num">1</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t154" href="#t154">154</a></span><span class="t">        <span class="nam">self</span><span class="op">.</span><span class="nam">add_BC</span><span class="op">(</span><span class="nam">component</span><span class="op">=</span><span class="str">'T'</span><span class="op">,</span> <span class="nam">equation</span><span class="op">=</span><span class="str">'T'</span><span class="op">,</span> <span class="nam">axis</span><span class="op">=</span><span class="num">1</span><span class="op">,</span> <span class="nam">x</span><span class="op">=</span><span class="num">1</span><span class="op">,</span> <span class="nam">v</span><span class="op">=</span><span class="nam">self</span><span class="op">.</span><span class="nam">BCs</span><span class="op">[</span><span class="str">'T_top'</span><span class="op">]</span><span class="op">,</span> <span class="nam">kind</span><span class="op">=</span><span class="str">'Dirichlet'</span><span class="op">,</span> <span class="nam">line</span><span class="op">=</span><span class="op">-</span><span class="num">2</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t155" href="#t155">155</a></span><span class="t">        <span class="nam">self</span><span class="op">.</span><span class="nam">add_BC</span><span class="op">(</span><span class="nam">component</span><span class="op">=</span><span class="str">'v'</span><span class="op">,</span> <span class="nam">equation</span><span class="op">=</span><span class="str">'v'</span><span class="op">,</span> <span class="nam">axis</span><span class="op">=</span><span class="num">1</span><span class="op">,</span> <span class="nam">x</span><span class="op">=</span><span class="num">1</span><span class="op">,</span> <span class="nam">v</span><span class="op">=</span><span class="nam">self</span><span class="op">.</span><span class="nam">BCs</span><span class="op">[</span><span class="str">'v_bottom'</span><span class="op">]</span><span class="op">,</span> <span class="nam">kind</span><span class="op">=</span><span class="str">'Dirichlet'</span><span class="op">,</span> <span class="nam">line</span><span class="op">=</span><span class="op">-</span><span class="num">1</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t156" href="#t156">156</a></span><span class="t">        <span class="nam">self</span><span class="op">.</span><span class="nam">add_BC</span><span class="op">(</span><span class="nam">component</span><span class="op">=</span><span class="str">'v'</span><span class="op">,</span> <span class="nam">equation</span><span class="op">=</span><span class="str">'v'</span><span class="op">,</span> <span class="nam">axis</span><span class="op">=</span><span class="num">1</span><span class="op">,</span> <span class="nam">x</span><span class="op">=</span><span class="op">-</span><span class="num">1</span><span class="op">,</span> <span class="nam">v</span><span class="op">=</span><span class="nam">self</span><span class="op">.</span><span class="nam">BCs</span><span class="op">[</span><span class="str">'v_bottom'</span><span class="op">]</span><span class="op">,</span> <span class="nam">kind</span><span class="op">=</span><span class="str">'Dirichlet'</span><span class="op">,</span> <span class="nam">line</span><span class="op">=</span><span class="op">-</span><span class="num">2</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t157" href="#t157">157</a></span><span class="t">        <span class="nam">self</span><span class="op">.</span><span class="nam">remove_BC</span><span class="op">(</span><span class="nam">component</span><span class="op">=</span><span class="str">'v'</span><span class="op">,</span> <span class="nam">equation</span><span class="op">=</span><span class="str">'v'</span><span class="op">,</span> <span class="nam">axis</span><span class="op">=</span><span class="num">1</span><span class="op">,</span> <span class="nam">x</span><span class="op">=</span><span class="op">-</span><span class="num">1</span><span class="op">,</span> <span class="nam">kind</span><span class="op">=</span><span class="str">'Dirichlet'</span><span class="op">,</span> <span class="nam">line</span><span class="op">=</span><span class="op">-</span><span class="num">2</span><span class="op">,</span> <span class="nam">scalar</span><span class="op">=</span><span class="key">True</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t158" href="#t158">158</a></span><span class="t">        <span class="nam">self</span><span class="op">.</span><span class="nam">add_BC</span><span class="op">(</span><span class="nam">component</span><span class="op">=</span><span class="str">'u'</span><span class="op">,</span> <span class="nam">equation</span><span class="op">=</span><span class="str">'u'</span><span class="op">,</span> <span class="nam">axis</span><span class="op">=</span><span class="num">1</span><span class="op">,</span> <span class="nam">v</span><span class="op">=</span><span class="nam">self</span><span class="op">.</span><span class="nam">BCs</span><span class="op">[</span><span class="str">'u_top'</span><span class="op">]</span><span class="op">,</span> <span class="nam">x</span><span class="op">=</span><span class="num">1</span><span class="op">,</span> <span class="nam">kind</span><span class="op">=</span><span class="str">'Dirichlet'</span><span class="op">,</span> <span class="nam">line</span><span class="op">=</span><span class="op">-</span><span class="num">2</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t159" href="#t159">159</a></span><span class="t">        <span class="nam">self</span><span class="op">.</span><span class="nam">add_BC</span><span class="op">(</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t160" href="#t160">160</a></span><span class="t">            <span class="nam">component</span><span class="op">=</span><span class="str">'u'</span><span class="op">,</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t161" href="#t161">161</a></span><span class="t">            <span class="nam">equation</span><span class="op">=</span><span class="str">'u'</span><span class="op">,</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t162" href="#t162">162</a></span><span class="t">            <span class="nam">axis</span><span class="op">=</span><span class="num">1</span><span class="op">,</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t163" href="#t163">163</a></span><span class="t">            <span class="nam">v</span><span class="op">=</span><span class="nam">self</span><span class="op">.</span><span class="nam">BCs</span><span class="op">[</span><span class="str">'u_bottom'</span><span class="op">]</span><span class="op">,</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t164" href="#t164">164</a></span><span class="t">            <span class="nam">x</span><span class="op">=</span><span class="op">-</span><span class="num">1</span><span class="op">,</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t165" href="#t165">165</a></span><span class="t">            <span class="nam">kind</span><span class="op">=</span><span class="str">'Dirichlet'</span><span class="op">,</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t166" href="#t166">166</a></span><span class="t">            <span class="nam">line</span><span class="op">=</span><span class="op">-</span><span class="num">1</span><span class="op">,</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t167" href="#t167">167</a></span><span class="t">        <span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t168" href="#t168">168</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t169" href="#t169">169</a></span><span class="t">        <span class="com"># eliminate Nyquist mode if needed</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t170" href="#t170">170</a></span><span class="t">        <span class="key">if</span> <span class="nam">nx</span> <span class="op">%</span> <span class="num">2</span> <span class="op">==</span> <span class="num">0</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t171" href="#t171">171</a></span><span class="t">            <span class="nam">Nyquist_mode_index</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">axes</span><span class="op">[</span><span class="num">0</span><span class="op">]</span><span class="op">.</span><span class="nam">get_Nyquist_mode_index</span><span class="op">(</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t172" href="#t172">172</a></span><span class="t">            <span class="key">for</span> <span class="nam">component</span> <span class="key">in</span> <span class="nam">self</span><span class="op">.</span><span class="nam">components</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t173" href="#t173">173</a></span><span class="t">                <span class="nam">self</span><span class="op">.</span><span class="nam">add_BC</span><span class="op">(</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t174" href="#t174">174</a></span><span class="t">                    <span class="nam">component</span><span class="op">=</span><span class="nam">component</span><span class="op">,</span> <span class="nam">equation</span><span class="op">=</span><span class="nam">component</span><span class="op">,</span> <span class="nam">axis</span><span class="op">=</span><span class="num">0</span><span class="op">,</span> <span class="nam">kind</span><span class="op">=</span><span class="str">'Nyquist'</span><span class="op">,</span> <span class="nam">line</span><span class="op">=</span><span class="nam">int</span><span class="op">(</span><span class="nam">Nyquist_mode_index</span><span class="op">)</span><span class="op">,</span> <span class="nam">v</span><span class="op">=</span><span class="num">0</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t175" href="#t175">175</a></span><span class="t">                <span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t176" href="#t176">176</a></span><span class="t">        <span class="nam">self</span><span class="op">.</span><span class="nam">setup_BCs</span><span class="op">(</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t177" href="#t177">177</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t178" href="#t178">178</a></span><span class="t">    <span class="key">def</span> <span class="nam">eval_f</span><span class="op">(</span><span class="nam">self</span><span class="op">,</span> <span class="nam">u</span><span class="op">,</span> <span class="op">*</span><span class="nam">args</span><span class="op">,</span> <span class="op">**</span><span class="nam">kwargs</span><span class="op">)</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t179" href="#t179">179</a></span><span class="t">        <span class="nam">f</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">f_init</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t180" href="#t180">180</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t181" href="#t181">181</a></span><span class="t">        <span class="key">if</span> <span class="nam">self</span><span class="op">.</span><span class="nam">spectral_space</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t182" href="#t182">182</a></span><span class="t">            <span class="nam">u_hat</span> <span class="op">=</span> <span class="nam">u</span><span class="op">.</span><span class="nam">copy</span><span class="op">(</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t183" href="#t183">183</a></span><span class="t">        <span class="key">else</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t184" href="#t184">184</a></span><span class="t">            <span class="nam">u_hat</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">transform</span><span class="op">(</span><span class="nam">u</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t185" href="#t185">185</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t186" href="#t186">186</a></span><span class="t">        <span class="nam">f_impl_hat</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">u_init_forward</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t187" href="#t187">187</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t188" href="#t188">188</a></span><span class="t">        <span class="nam">Dz</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">Dz</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t189" href="#t189">189</a></span><span class="t">        <span class="nam">Dx</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">Dx</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t190" href="#t190">190</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t191" href="#t191">191</a></span><span class="t">        <span class="nam">iu</span><span class="op">,</span> <span class="nam">iv</span><span class="op">,</span> <span class="nam">iT</span><span class="op">,</span> <span class="nam">ip</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">index</span><span class="op">(</span><span class="op">[</span><span class="str">'u'</span><span class="op">,</span> <span class="str">'v'</span><span class="op">,</span> <span class="str">'T'</span><span class="op">,</span> <span class="str">'p'</span><span class="op">]</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t192" href="#t192">192</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t193" href="#t193">193</a></span><span class="t">        <span class="com"># evaluate implicit terms</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t194" href="#t194">194</a></span><span class="t">        <span class="nam">f_impl_hat</span> <span class="op">=</span> <span class="op">-</span><span class="op">(</span><span class="nam">self</span><span class="op">.</span><span class="nam">base_change</span> <span class="op">@</span> <span class="nam">self</span><span class="op">.</span><span class="nam">L</span> <span class="op">@</span> <span class="nam">u_hat</span><span class="op">.</span><span class="nam">flatten</span><span class="op">(</span><span class="op">)</span><span class="op">)</span><span class="op">.</span><span class="nam">reshape</span><span class="op">(</span><span class="nam">u_hat</span><span class="op">.</span><span class="nam">shape</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t195" href="#t195">195</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t196" href="#t196">196</a></span><span class="t">        <span class="key">if</span> <span class="nam">self</span><span class="op">.</span><span class="nam">spectral_space</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t197" href="#t197">197</a></span><span class="t">            <span class="nam">f</span><span class="op">.</span><span class="nam">impl</span><span class="op">[</span><span class="op">:</span><span class="op">]</span> <span class="op">=</span> <span class="nam">f_impl_hat</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t198" href="#t198">198</a></span><span class="t">        <span class="key">else</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t199" href="#t199">199</a></span><span class="t">            <span class="nam">f</span><span class="op">.</span><span class="nam">impl</span><span class="op">[</span><span class="op">:</span><span class="op">]</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">itransform</span><span class="op">(</span><span class="nam">f_impl_hat</span><span class="op">)</span><span class="op">.</span><span class="nam">real</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t200" href="#t200">200</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t201" href="#t201">201</a></span><span class="t">        <span class="com"># treat convection explicitly with dealiasing</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t202" href="#t202">202</a></span><span class="t">        <span class="nam">Dx_u_hat</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">u_init_forward</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t203" href="#t203">203</a></span><span class="t">        <span class="key">for</span> <span class="nam">i</span> <span class="key">in</span> <span class="op">[</span><span class="nam">iu</span><span class="op">,</span> <span class="nam">iv</span><span class="op">,</span> <span class="nam">iT</span><span class="op">]</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t204" href="#t204">204</a></span><span class="t">            <span class="nam">Dx_u_hat</span><span class="op">[</span><span class="nam">i</span><span class="op">]</span><span class="op">[</span><span class="op">:</span><span class="op">]</span> <span class="op">=</span> <span class="op">(</span><span class="nam">Dx</span> <span class="op">@</span> <span class="nam">u_hat</span><span class="op">[</span><span class="nam">i</span><span class="op">]</span><span class="op">.</span><span class="nam">flatten</span><span class="op">(</span><span class="op">)</span><span class="op">)</span><span class="op">.</span><span class="nam">reshape</span><span class="op">(</span><span class="nam">Dx_u_hat</span><span class="op">[</span><span class="nam">i</span><span class="op">]</span><span class="op">.</span><span class="nam">shape</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t205" href="#t205">205</a></span><span class="t">        <span class="nam">Dz_u_hat</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">u_init_forward</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t206" href="#t206">206</a></span><span class="t">        <span class="key">for</span> <span class="nam">i</span> <span class="key">in</span> <span class="op">[</span><span class="nam">iu</span><span class="op">,</span> <span class="nam">iv</span><span class="op">,</span> <span class="nam">iT</span><span class="op">]</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t207" href="#t207">207</a></span><span class="t">            <span class="nam">Dz_u_hat</span><span class="op">[</span><span class="nam">i</span><span class="op">]</span><span class="op">[</span><span class="op">:</span><span class="op">]</span> <span class="op">=</span> <span class="op">(</span><span class="nam">Dz</span> <span class="op">@</span> <span class="nam">u_hat</span><span class="op">[</span><span class="nam">i</span><span class="op">]</span><span class="op">.</span><span class="nam">flatten</span><span class="op">(</span><span class="op">)</span><span class="op">)</span><span class="op">.</span><span class="nam">reshape</span><span class="op">(</span><span class="nam">Dz_u_hat</span><span class="op">[</span><span class="nam">i</span><span class="op">]</span><span class="op">.</span><span class="nam">shape</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t208" href="#t208">208</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t209" href="#t209">209</a></span><span class="t">        <span class="nam">padding</span> <span class="op">=</span> <span class="op">[</span><span class="nam">self</span><span class="op">.</span><span class="nam">dealiasing</span><span class="op">,</span> <span class="nam">self</span><span class="op">.</span><span class="nam">dealiasing</span><span class="op">]</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t210" href="#t210">210</a></span><span class="t">        <span class="nam">Dx_u_pad</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">itransform</span><span class="op">(</span><span class="nam">Dx_u_hat</span><span class="op">,</span> <span class="nam">padding</span><span class="op">=</span><span class="nam">padding</span><span class="op">)</span><span class="op">.</span><span class="nam">real</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t211" href="#t211">211</a></span><span class="t">        <span class="nam">Dz_u_pad</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">itransform</span><span class="op">(</span><span class="nam">Dz_u_hat</span><span class="op">,</span> <span class="nam">padding</span><span class="op">=</span><span class="nam">padding</span><span class="op">)</span><span class="op">.</span><span class="nam">real</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t212" href="#t212">212</a></span><span class="t">        <span class="nam">u_pad</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">itransform</span><span class="op">(</span><span class="nam">u_hat</span><span class="op">,</span> <span class="nam">padding</span><span class="op">=</span><span class="nam">padding</span><span class="op">)</span><span class="op">.</span><span class="nam">real</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t213" href="#t213">213</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t214" href="#t214">214</a></span><span class="t">        <span class="nam">fexpl_pad</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">xp</span><span class="op">.</span><span class="nam">zeros_like</span><span class="op">(</span><span class="nam">u_pad</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t215" href="#t215">215</a></span><span class="t">        <span class="nam">fexpl_pad</span><span class="op">[</span><span class="nam">iu</span><span class="op">]</span><span class="op">[</span><span class="op">:</span><span class="op">]</span> <span class="op">=</span> <span class="op">-</span><span class="op">(</span><span class="nam">u_pad</span><span class="op">[</span><span class="nam">iu</span><span class="op">]</span> <span class="op">*</span> <span class="nam">Dx_u_pad</span><span class="op">[</span><span class="nam">iu</span><span class="op">]</span> <span class="op">+</span> <span class="nam">u_pad</span><span class="op">[</span><span class="nam">iv</span><span class="op">]</span> <span class="op">*</span> <span class="nam">Dz_u_pad</span><span class="op">[</span><span class="nam">iu</span><span class="op">]</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t216" href="#t216">216</a></span><span class="t">        <span class="nam">fexpl_pad</span><span class="op">[</span><span class="nam">iv</span><span class="op">]</span><span class="op">[</span><span class="op">:</span><span class="op">]</span> <span class="op">=</span> <span class="op">-</span><span class="op">(</span><span class="nam">u_pad</span><span class="op">[</span><span class="nam">iu</span><span class="op">]</span> <span class="op">*</span> <span class="nam">Dx_u_pad</span><span class="op">[</span><span class="nam">iv</span><span class="op">]</span> <span class="op">+</span> <span class="nam">u_pad</span><span class="op">[</span><span class="nam">iv</span><span class="op">]</span> <span class="op">*</span> <span class="nam">Dz_u_pad</span><span class="op">[</span><span class="nam">iv</span><span class="op">]</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t217" href="#t217">217</a></span><span class="t">        <span class="nam">fexpl_pad</span><span class="op">[</span><span class="nam">iT</span><span class="op">]</span><span class="op">[</span><span class="op">:</span><span class="op">]</span> <span class="op">=</span> <span class="op">-</span><span class="op">(</span><span class="nam">u_pad</span><span class="op">[</span><span class="nam">iu</span><span class="op">]</span> <span class="op">*</span> <span class="nam">Dx_u_pad</span><span class="op">[</span><span class="nam">iT</span><span class="op">]</span> <span class="op">+</span> <span class="nam">u_pad</span><span class="op">[</span><span class="nam">iv</span><span class="op">]</span> <span class="op">*</span> <span class="nam">Dz_u_pad</span><span class="op">[</span><span class="nam">iT</span><span class="op">]</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t218" href="#t218">218</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t219" href="#t219">219</a></span><span class="t">        <span class="key">if</span> <span class="nam">self</span><span class="op">.</span><span class="nam">spectral_space</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t220" href="#t220">220</a></span><span class="t">            <span class="nam">f</span><span class="op">.</span><span class="nam">expl</span><span class="op">[</span><span class="op">:</span><span class="op">]</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">transform</span><span class="op">(</span><span class="nam">fexpl_pad</span><span class="op">,</span> <span class="nam">padding</span><span class="op">=</span><span class="nam">padding</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t221" href="#t221">221</a></span><span class="t">        <span class="key">else</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t222" href="#t222">222</a></span><span class="t">            <span class="nam">f</span><span class="op">.</span><span class="nam">expl</span><span class="op">[</span><span class="op">:</span><span class="op">]</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">itransform</span><span class="op">(</span><span class="nam">self</span><span class="op">.</span><span class="nam">transform</span><span class="op">(</span><span class="nam">fexpl_pad</span><span class="op">,</span> <span class="nam">padding</span><span class="op">=</span><span class="nam">padding</span><span class="op">)</span><span class="op">)</span><span class="op">.</span><span class="nam">real</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t223" href="#t223">223</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t224" href="#t224">224</a></span><span class="t">        <span class="key">return</span> <span class="nam">f</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t225" href="#t225">225</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t226" href="#t226">226</a></span><span class="t">    <span class="key">def</span> <span class="nam">u_exact</span><span class="op">(</span><span class="nam">self</span><span class="op">,</span> <span class="nam">t</span><span class="op">=</span><span class="num">0</span><span class="op">,</span> <span class="nam">noise_level</span><span class="op">=</span><span class="num">1e-3</span><span class="op">,</span> <span class="nam">seed</span><span class="op">=</span><span class="num">99</span><span class="op">)</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t227" href="#t227">227</a></span><span class="t">        <span class="key">assert</span> <span class="nam">t</span> <span class="op">==</span> <span class="num">0</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t228" href="#t228">228</a></span><span class="t">        <span class="key">assert</span> <span class="op">(</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t229" href="#t229">229</a></span><span class="t">            <span class="nam">self</span><span class="op">.</span><span class="nam">BCs</span><span class="op">[</span><span class="str">'v_top'</span><span class="op">]</span> <span class="op">==</span> <span class="nam">self</span><span class="op">.</span><span class="nam">BCs</span><span class="op">[</span><span class="str">'v_bottom'</span><span class="op">]</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t230" href="#t230">230</a></span><span class="t">        <span class="op">)</span><span class="op">,</span> <span class="str">'Initial conditions are only implemented for zero velocity gradient'</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t231" href="#t231">231</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t232" href="#t232">232</a></span><span class="t">        <span class="nam">me</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">spectral</span><span class="op">.</span><span class="nam">u_init</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t233" href="#t233">233</a></span><span class="t">        <span class="nam">iu</span><span class="op">,</span> <span class="nam">iv</span><span class="op">,</span> <span class="nam">iT</span><span class="op">,</span> <span class="nam">ip</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">index</span><span class="op">(</span><span class="op">[</span><span class="str">'u'</span><span class="op">,</span> <span class="str">'v'</span><span class="op">,</span> <span class="str">'T'</span><span class="op">,</span> <span class="str">'p'</span><span class="op">]</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t234" href="#t234">234</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t235" href="#t235">235</a></span><span class="t">        <span class="com"># linear temperature gradient</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t236" href="#t236">236</a></span><span class="t">        <span class="key">for</span> <span class="nam">comp</span> <span class="key">in</span> <span class="op">[</span><span class="str">'T'</span><span class="op">,</span> <span class="str">'v'</span><span class="op">,</span> <span class="str">'u'</span><span class="op">]</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t237" href="#t237">237</a></span><span class="t">            <span class="nam">a</span> <span class="op">=</span> <span class="op">(</span><span class="nam">self</span><span class="op">.</span><span class="nam">BCs</span><span class="op">[</span><span class="fst">f'</span><span class="op">{</span><span class="nam">comp</span><span class="op">}</span><span class="fst">_top</span><span class="fst">'</span><span class="op">]</span> <span class="op">-</span> <span class="nam">self</span><span class="op">.</span><span class="nam">BCs</span><span class="op">[</span><span class="fst">f'</span><span class="op">{</span><span class="nam">comp</span><span class="op">}</span><span class="fst">_bottom</span><span class="fst">'</span><span class="op">]</span><span class="op">)</span> <span class="op">/</span> <span class="num">2</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t238" href="#t238">238</a></span><span class="t">            <span class="nam">b</span> <span class="op">=</span> <span class="op">(</span><span class="nam">self</span><span class="op">.</span><span class="nam">BCs</span><span class="op">[</span><span class="fst">f'</span><span class="op">{</span><span class="nam">comp</span><span class="op">}</span><span class="fst">_top</span><span class="fst">'</span><span class="op">]</span> <span class="op">+</span> <span class="nam">self</span><span class="op">.</span><span class="nam">BCs</span><span class="op">[</span><span class="fst">f'</span><span class="op">{</span><span class="nam">comp</span><span class="op">}</span><span class="fst">_bottom</span><span class="fst">'</span><span class="op">]</span><span class="op">)</span> <span class="op">/</span> <span class="num">2</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t239" href="#t239">239</a></span><span class="t">            <span class="nam">me</span><span class="op">[</span><span class="nam">self</span><span class="op">.</span><span class="nam">index</span><span class="op">(</span><span class="nam">comp</span><span class="op">)</span><span class="op">]</span> <span class="op">=</span> <span class="nam">a</span> <span class="op">*</span> <span class="nam">self</span><span class="op">.</span><span class="nam">Z</span> <span class="op">+</span> <span class="nam">b</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t240" href="#t240">240</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t241" href="#t241">241</a></span><span class="t">        <span class="com"># perturb slightly</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t242" href="#t242">242</a></span><span class="t">        <span class="nam">rng</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">xp</span><span class="op">.</span><span class="nam">random</span><span class="op">.</span><span class="nam">default_rng</span><span class="op">(</span><span class="nam">seed</span><span class="op">=</span><span class="nam">seed</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t243" href="#t243">243</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t244" href="#t244">244</a></span><span class="t">        <span class="nam">noise</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">spectral</span><span class="op">.</span><span class="nam">u_init</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t245" href="#t245">245</a></span><span class="t">        <span class="nam">noise</span><span class="op">[</span><span class="nam">iT</span><span class="op">]</span> <span class="op">=</span> <span class="nam">rng</span><span class="op">.</span><span class="nam">random</span><span class="op">(</span><span class="nam">size</span><span class="op">=</span><span class="nam">me</span><span class="op">[</span><span class="nam">iT</span><span class="op">]</span><span class="op">.</span><span class="nam">shape</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t246" href="#t246">246</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t247" href="#t247">247</a></span><span class="t">        <span class="nam">me</span><span class="op">[</span><span class="nam">iT</span><span class="op">]</span> <span class="op">+=</span> <span class="nam">noise</span><span class="op">[</span><span class="nam">iT</span><span class="op">]</span><span class="op">.</span><span class="nam">real</span> <span class="op">*</span> <span class="nam">noise_level</span> <span class="op">*</span> <span class="op">(</span><span class="nam">self</span><span class="op">.</span><span class="nam">Z</span> <span class="op">-</span> <span class="num">1</span><span class="op">)</span> <span class="op">*</span> <span class="op">(</span><span class="nam">self</span><span class="op">.</span><span class="nam">Z</span> <span class="op">+</span> <span class="num">1</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t248" href="#t248">248</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t249" href="#t249">249</a></span><span class="t">        <span class="key">if</span> <span class="nam">self</span><span class="op">.</span><span class="nam">spectral_space</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t250" href="#t250">250</a></span><span class="t">            <span class="nam">me_hat</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">spectral</span><span class="op">.</span><span class="nam">u_init_forward</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t251" href="#t251">251</a></span><span class="t">            <span class="nam">me_hat</span><span class="op">[</span><span class="op">:</span><span class="op">]</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">transform</span><span class="op">(</span><span class="nam">me</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t252" href="#t252">252</a></span><span class="t">            <span class="key">return</span> <span class="nam">me_hat</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t253" href="#t253">253</a></span><span class="t">        <span class="key">else</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t254" href="#t254">254</a></span><span class="t">            <span class="key">return</span> <span class="nam">me</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t255" href="#t255">255</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t256" href="#t256">256</a></span><span class="t">    <span class="key">def</span> <span class="nam">get_fig</span><span class="op">(</span><span class="nam">self</span><span class="op">)</span><span class="op">:</span>  <span class="com"># pragma: no cover</span>&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t257" href="#t257">257</a></span><span class="t">        <span class="str">"""</span>&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t258" href="#t258">258</a></span><span class="t"><span class="str">        Get a figure suitable to plot the solution of this problem</span>&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t259" href="#t259">259</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t260" href="#t260">260</a></span><span class="t"><span class="str">        Returns</span>&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t261" href="#t261">261</a></span><span class="t"><span class="str">        -------</span>&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t262" href="#t262">262</a></span><span class="t"><span class="str">        self.fig : matplotlib.pyplot.figure.Figure</span>&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t263" href="#t263">263</a></span><span class="t"><span class="str">        """</span>&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t264" href="#t264">264</a></span><span class="t">        <span class="key">import</span> <span class="nam">matplotlib</span><span class="op">.</span><span class="nam">pyplot</span> <span class="key">as</span> <span class="nam">plt</span>&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t265" href="#t265">265</a></span><span class="t">        <span class="key">from</span> <span class="nam">mpl_toolkits</span><span class="op">.</span><span class="nam">axes_grid1</span> <span class="key">import</span> <span class="nam">make_axes_locatable</span>&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t266" href="#t266">266</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t267" href="#t267">267</a></span><span class="t">        <span class="nam">plt</span><span class="op">.</span><span class="nam">rcParams</span><span class="op">[</span><span class="str">'figure.constrained_layout.use'</span><span class="op">]</span> <span class="op">=</span> <span class="key">True</span>&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t268" href="#t268">268</a></span><span class="t">        <span class="nam">self</span><span class="op">.</span><span class="nam">fig</span><span class="op">,</span> <span class="nam">axs</span> <span class="op">=</span> <span class="nam">plt</span><span class="op">.</span><span class="nam">subplots</span><span class="op">(</span><span class="num">2</span><span class="op">,</span> <span class="num">1</span><span class="op">,</span> <span class="nam">sharex</span><span class="op">=</span><span class="key">True</span><span class="op">,</span> <span class="nam">sharey</span><span class="op">=</span><span class="key">True</span><span class="op">,</span> <span class="nam">figsize</span><span class="op">=</span><span class="op">(</span><span class="op">(</span><span class="num">10</span><span class="op">,</span> <span class="num">5</span><span class="op">)</span><span class="op">)</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t269" href="#t269">269</a></span><span class="t">        <span class="nam">self</span><span class="op">.</span><span class="nam">cax</span> <span class="op">=</span> <span class="op">[</span><span class="op">]</span>&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t270" href="#t270">270</a></span><span class="t">        <span class="nam">divider</span> <span class="op">=</span> <span class="nam">make_axes_locatable</span><span class="op">(</span><span class="nam">axs</span><span class="op">[</span><span class="num">0</span><span class="op">]</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t271" href="#t271">271</a></span><span class="t">        <span class="nam">self</span><span class="op">.</span><span class="nam">cax</span> <span class="op">+=</span> <span class="op">[</span><span class="nam">divider</span><span class="op">.</span><span class="nam">append_axes</span><span class="op">(</span><span class="str">'right'</span><span class="op">,</span> <span class="nam">size</span><span class="op">=</span><span class="str">'3%'</span><span class="op">,</span> <span class="nam">pad</span><span class="op">=</span><span class="num">0.03</span><span class="op">)</span><span class="op">]</span>&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t272" href="#t272">272</a></span><span class="t">        <span class="nam">divider2</span> <span class="op">=</span> <span class="nam">make_axes_locatable</span><span class="op">(</span><span class="nam">axs</span><span class="op">[</span><span class="num">1</span><span class="op">]</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t273" href="#t273">273</a></span><span class="t">        <span class="nam">self</span><span class="op">.</span><span class="nam">cax</span> <span class="op">+=</span> <span class="op">[</span><span class="nam">divider2</span><span class="op">.</span><span class="nam">append_axes</span><span class="op">(</span><span class="str">'right'</span><span class="op">,</span> <span class="nam">size</span><span class="op">=</span><span class="str">'3%'</span><span class="op">,</span> <span class="nam">pad</span><span class="op">=</span><span class="num">0.03</span><span class="op">)</span><span class="op">]</span>&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t274" href="#t274">274</a></span><span class="t">        <span class="key">return</span> <span class="nam">self</span><span class="op">.</span><span class="nam">fig</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t275" href="#t275">275</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t276" href="#t276">276</a></span><span class="t">    <span class="key">def</span> <span class="nam">plot</span><span class="op">(</span><span class="nam">self</span><span class="op">,</span> <span class="nam">u</span><span class="op">,</span> <span class="nam">t</span><span class="op">=</span><span class="key">None</span><span class="op">,</span> <span class="nam">fig</span><span class="op">=</span><span class="key">None</span><span class="op">,</span> <span class="nam">quantity</span><span class="op">=</span><span class="str">'T'</span><span class="op">)</span><span class="op">:</span>  <span class="com"># pragma: no cover</span>&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t277" href="#t277">277</a></span><span class="t">        <span class="str">r"""</span>&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t278" href="#t278">278</a></span><span class="t"><span class="str">        Plot the solution.</span>&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t279" href="#t279">279</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t280" href="#t280">280</a></span><span class="t"><span class="str">        Parameters</span>&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t281" href="#t281">281</a></span><span class="t"><span class="str">        ----------</span>&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t282" href="#t282">282</a></span><span class="t"><span class="str">        u : dtype_u</span>&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t283" href="#t283">283</a></span><span class="t"><span class="str">            Solution to be plotted</span>&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t284" href="#t284">284</a></span><span class="t"><span class="str">        t : float</span>&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t285" href="#t285">285</a></span><span class="t"><span class="str">            Time to display at the top of the figure</span>&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t286" href="#t286">286</a></span><span class="t"><span class="str">        fig : matplotlib.pyplot.figure.Figure</span>&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t287" href="#t287">287</a></span><span class="t"><span class="str">            Figure with the same structure as a figure generated by `self.get_fig`. If none is supplied, a new figure will be generated.</span>&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t288" href="#t288">288</a></span><span class="t"><span class="str">        quantity : (str)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t289" href="#t289">289</a></span><span class="t"><span class="str">            quantity you want to plot</span>&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t290" href="#t290">290</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t291" href="#t291">291</a></span><span class="t"><span class="str">        Returns</span>&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t292" href="#t292">292</a></span><span class="t"><span class="str">        -------</span>&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t293" href="#t293">293</a></span><span class="t"><span class="str">        None</span>&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t294" href="#t294">294</a></span><span class="t"><span class="str">        """</span>&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t295" href="#t295">295</a></span><span class="t">        <span class="nam">fig</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">get_fig</span><span class="op">(</span><span class="op">)</span> <span class="key">if</span> <span class="nam">fig</span> <span class="key">is</span> <span class="key">None</span> <span class="key">else</span> <span class="nam">fig</span>&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t296" href="#t296">296</a></span><span class="t">        <span class="nam">axs</span> <span class="op">=</span> <span class="nam">fig</span><span class="op">.</span><span class="nam">axes</span>&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t297" href="#t297">297</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t298" href="#t298">298</a></span><span class="t">        <span class="nam">imV</span> <span class="op">=</span> <span class="nam">axs</span><span class="op">[</span><span class="num">1</span><span class="op">]</span><span class="op">.</span><span class="nam">pcolormesh</span><span class="op">(</span><span class="nam">self</span><span class="op">.</span><span class="nam">X</span><span class="op">,</span> <span class="nam">self</span><span class="op">.</span><span class="nam">Z</span><span class="op">,</span> <span class="nam">self</span><span class="op">.</span><span class="nam">compute_vorticity</span><span class="op">(</span><span class="nam">u</span><span class="op">)</span><span class="op">.</span><span class="nam">real</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t299" href="#t299">299</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t300" href="#t300">300</a></span><span class="t">        <span class="key">if</span> <span class="nam">self</span><span class="op">.</span><span class="nam">spectral_space</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t301" href="#t301">301</a></span><span class="t">            <span class="nam">u</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">itransform</span><span class="op">(</span><span class="nam">u</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t302" href="#t302">302</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t303" href="#t303">303</a></span><span class="t">        <span class="nam">imT</span> <span class="op">=</span> <span class="nam">axs</span><span class="op">[</span><span class="num">0</span><span class="op">]</span><span class="op">.</span><span class="nam">pcolormesh</span><span class="op">(</span><span class="nam">self</span><span class="op">.</span><span class="nam">X</span><span class="op">,</span> <span class="nam">self</span><span class="op">.</span><span class="nam">Z</span><span class="op">,</span> <span class="nam">u</span><span class="op">[</span><span class="nam">self</span><span class="op">.</span><span class="nam">index</span><span class="op">(</span><span class="nam">quantity</span><span class="op">)</span><span class="op">]</span><span class="op">.</span><span class="nam">real</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t304" href="#t304">304</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t305" href="#t305">305</a></span><span class="t">        <span class="key">for</span> <span class="nam">i</span><span class="op">,</span> <span class="nam">label</span> <span class="key">in</span> <span class="nam">zip</span><span class="op">(</span><span class="op">[</span><span class="num">0</span><span class="op">,</span> <span class="num">1</span><span class="op">]</span><span class="op">,</span> <span class="op">[</span><span class="fst">rf'</span><span class="fst">$</span><span class="op">{</span><span class="nam">quantity</span><span class="op">}</span><span class="fst">$</span><span class="fst">'</span><span class="op">,</span> <span class="str">'vorticity'</span><span class="op">]</span><span class="op">)</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t306" href="#t306">306</a></span><span class="t">            <span class="nam">axs</span><span class="op">[</span><span class="nam">i</span><span class="op">]</span><span class="op">.</span><span class="nam">set_aspect</span><span class="op">(</span><span class="num">1</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t307" href="#t307">307</a></span><span class="t">            <span class="nam">axs</span><span class="op">[</span><span class="nam">i</span><span class="op">]</span><span class="op">.</span><span class="nam">set_title</span><span class="op">(</span><span class="nam">label</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t308" href="#t308">308</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t309" href="#t309">309</a></span><span class="t">        <span class="key">if</span> <span class="nam">t</span> <span class="key">is</span> <span class="key">not</span> <span class="key">None</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t310" href="#t310">310</a></span><span class="t">            <span class="nam">fig</span><span class="op">.</span><span class="nam">suptitle</span><span class="op">(</span><span class="fst">f'</span><span class="fst">t = </span><span class="op">{</span><span class="nam">t</span><span class="op">:</span><span class="fst">.2f</span><span class="op">}</span><span class="fst">'</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t311" href="#t311">311</a></span><span class="t">        <span class="nam">axs</span><span class="op">[</span><span class="num">1</span><span class="op">]</span><span class="op">.</span><span class="nam">set_xlabel</span><span class="op">(</span><span class="str">r'$x$'</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t312" href="#t312">312</a></span><span class="t">        <span class="nam">axs</span><span class="op">[</span><span class="num">1</span><span class="op">]</span><span class="op">.</span><span class="nam">set_ylabel</span><span class="op">(</span><span class="str">r'$z$'</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t313" href="#t313">313</a></span><span class="t">        <span class="nam">fig</span><span class="op">.</span><span class="nam">colorbar</span><span class="op">(</span><span class="nam">imT</span><span class="op">,</span> <span class="nam">self</span><span class="op">.</span><span class="nam">cax</span><span class="op">[</span><span class="num">0</span><span class="op">]</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t314" href="#t314">314</a></span><span class="t">        <span class="nam">fig</span><span class="op">.</span><span class="nam">colorbar</span><span class="op">(</span><span class="nam">imV</span><span class="op">,</span> <span class="nam">self</span><span class="op">.</span><span class="nam">cax</span><span class="op">[</span><span class="num">1</span><span class="op">]</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t315" href="#t315">315</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t316" href="#t316">316</a></span><span class="t">    <span class="key">def</span> <span class="nam">compute_vorticity</span><span class="op">(</span><span class="nam">self</span><span class="op">,</span> <span class="nam">u</span><span class="op">)</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t317" href="#t317">317</a></span><span class="t">        <span class="key">if</span> <span class="nam">self</span><span class="op">.</span><span class="nam">spectral_space</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t318" href="#t318">318</a></span><span class="t">            <span class="nam">u_hat</span> <span class="op">=</span> <span class="nam">u</span><span class="op">.</span><span class="nam">copy</span><span class="op">(</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t319" href="#t319">319</a></span><span class="t">        <span class="key">else</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t320" href="#t320">320</a></span><span class="t">            <span class="nam">u_hat</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">transform</span><span class="op">(</span><span class="nam">u</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t321" href="#t321">321</a></span><span class="t">        <span class="nam">Dz</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">Dz</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t322" href="#t322">322</a></span><span class="t">        <span class="nam">Dx</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">Dx</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t323" href="#t323">323</a></span><span class="t">        <span class="nam">iu</span><span class="op">,</span> <span class="nam">iv</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">index</span><span class="op">(</span><span class="op">[</span><span class="str">'u'</span><span class="op">,</span> <span class="str">'v'</span><span class="op">]</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t324" href="#t324">324</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t325" href="#t325">325</a></span><span class="t">        <span class="nam">vorticity_hat</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">spectral</span><span class="op">.</span><span class="nam">u_init_forward</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t326" href="#t326">326</a></span><span class="t">        <span class="nam">vorticity_hat</span><span class="op">[</span><span class="num">0</span><span class="op">]</span> <span class="op">=</span> <span class="op">(</span><span class="nam">Dx</span> <span class="op">*</span> <span class="nam">u_hat</span><span class="op">[</span><span class="nam">iv</span><span class="op">]</span><span class="op">.</span><span class="nam">flatten</span><span class="op">(</span><span class="op">)</span> <span class="op">+</span> <span class="nam">Dz</span> <span class="op">@</span> <span class="nam">u_hat</span><span class="op">[</span><span class="nam">iu</span><span class="op">]</span><span class="op">.</span><span class="nam">flatten</span><span class="op">(</span><span class="op">)</span><span class="op">)</span><span class="op">.</span><span class="nam">reshape</span><span class="op">(</span><span class="nam">u</span><span class="op">[</span><span class="nam">iu</span><span class="op">]</span><span class="op">.</span><span class="nam">shape</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t327" href="#t327">327</a></span><span class="t">        <span class="key">return</span> <span class="nam">self</span><span class="op">.</span><span class="nam">itransform</span><span class="op">(</span><span class="nam">vorticity_hat</span><span class="op">)</span><span class="op">[</span><span class="num">0</span><span class="op">]</span><span class="op">.</span><span class="nam">real</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t328" href="#t328">328</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t329" href="#t329">329</a></span><span class="t">    <span class="key">def</span> <span class="nam">compute_Nusselt_numbers</span><span class="op">(</span><span class="nam">self</span><span class="op">,</span> <span class="nam">u</span><span class="op">)</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t330" href="#t330">330</a></span><span class="t">        <span class="str">"""</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t331" href="#t331">331</a></span><span class="t"><span class="str">        Compute the various versions of the Nusselt number. This reflects the type of heat transport.</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t332" href="#t332">332</a></span><span class="t"><span class="str">        If the Nusselt number is equal to one, it indicates heat transport due to conduction. If it is larger,</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t333" href="#t333">333</a></span><span class="t"><span class="str">        advection is present.</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t334" href="#t334">334</a></span><span class="t"><span class="str">        Computing the Nusselt number at various places can be used to check the code.</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t335" href="#t335">335</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t336" href="#t336">336</a></span><span class="t"><span class="str">        Args:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t337" href="#t337">337</a></span><span class="t"><span class="str">            u: The solution you want to compute the Nusselt numbers of</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t338" href="#t338">338</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t339" href="#t339">339</a></span><span class="t"><span class="str">        Returns:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t340" href="#t340">340</a></span><span class="t"><span class="str">            dict: Nusselt number averaged over the entire volume and horizontally averaged at the top and bottom.</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t341" href="#t341">341</a></span><span class="t"><span class="str">        """</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t342" href="#t342">342</a></span><span class="t">        <span class="nam">iv</span><span class="op">,</span> <span class="nam">iT</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">index</span><span class="op">(</span><span class="op">[</span><span class="str">'v'</span><span class="op">,</span> <span class="str">'T'</span><span class="op">]</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t343" href="#t343">343</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t344" href="#t344">344</a></span><span class="t">        <span class="nam">DzT_hat</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">spectral</span><span class="op">.</span><span class="nam">u_init_forward</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t345" href="#t345">345</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t346" href="#t346">346</a></span><span class="t">        <span class="key">if</span> <span class="nam">self</span><span class="op">.</span><span class="nam">spectral_space</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t347" href="#t347">347</a></span><span class="t">            <span class="nam">u_hat</span> <span class="op">=</span> <span class="nam">u</span><span class="op">.</span><span class="nam">copy</span><span class="op">(</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t348" href="#t348">348</a></span><span class="t">        <span class="key">else</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t349" href="#t349">349</a></span><span class="t">            <span class="nam">u_hat</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">transform</span><span class="op">(</span><span class="nam">u</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t350" href="#t350">350</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t351" href="#t351">351</a></span><span class="t">        <span class="nam">DzT_hat</span><span class="op">[</span><span class="nam">iT</span><span class="op">]</span> <span class="op">=</span> <span class="op">(</span><span class="nam">self</span><span class="op">.</span><span class="nam">Dz</span> <span class="op">@</span> <span class="nam">u_hat</span><span class="op">[</span><span class="nam">iT</span><span class="op">]</span><span class="op">.</span><span class="nam">flatten</span><span class="op">(</span><span class="op">)</span><span class="op">)</span><span class="op">.</span><span class="nam">reshape</span><span class="op">(</span><span class="nam">DzT_hat</span><span class="op">[</span><span class="nam">iT</span><span class="op">]</span><span class="op">.</span><span class="nam">shape</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t352" href="#t352">352</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t353" href="#t353">353</a></span><span class="t">        <span class="com"># compute vT with dealiasing</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t354" href="#t354">354</a></span><span class="t">        <span class="nam">padding</span> <span class="op">=</span> <span class="op">[</span><span class="nam">self</span><span class="op">.</span><span class="nam">dealiasing</span><span class="op">,</span> <span class="nam">self</span><span class="op">.</span><span class="nam">dealiasing</span><span class="op">]</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t355" href="#t355">355</a></span><span class="t">        <span class="nam">u_pad</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">itransform</span><span class="op">(</span><span class="nam">u_hat</span><span class="op">,</span> <span class="nam">padding</span><span class="op">=</span><span class="nam">padding</span><span class="op">)</span><span class="op">.</span><span class="nam">real</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t356" href="#t356">356</a></span><span class="t">        <span class="nam">_me</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">xp</span><span class="op">.</span><span class="nam">zeros_like</span><span class="op">(</span><span class="nam">u_pad</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t357" href="#t357">357</a></span><span class="t">        <span class="nam">_me</span><span class="op">[</span><span class="num">0</span><span class="op">]</span> <span class="op">=</span> <span class="nam">u_pad</span><span class="op">[</span><span class="nam">iv</span><span class="op">]</span> <span class="op">*</span> <span class="nam">u_pad</span><span class="op">[</span><span class="nam">iT</span><span class="op">]</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t358" href="#t358">358</a></span><span class="t">        <span class="nam">vT_hat</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">transform</span><span class="op">(</span><span class="nam">_me</span><span class="op">,</span> <span class="nam">padding</span><span class="op">=</span><span class="nam">padding</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t359" href="#t359">359</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t360" href="#t360">360</a></span><span class="t">        <span class="nam">nusselt_hat</span> <span class="op">=</span> <span class="op">(</span><span class="nam">vT_hat</span><span class="op">[</span><span class="num">0</span><span class="op">]</span> <span class="op">-</span> <span class="nam">DzT_hat</span><span class="op">[</span><span class="nam">iT</span><span class="op">]</span><span class="op">)</span> <span class="op">/</span> <span class="nam">self</span><span class="op">.</span><span class="nam">nx</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t361" href="#t361">361</a></span><span class="t">        <span class="nam">nusselt_no_v_hat</span> <span class="op">=</span> <span class="op">(</span><span class="op">-</span><span class="nam">DzT_hat</span><span class="op">[</span><span class="nam">iT</span><span class="op">]</span><span class="op">)</span> <span class="op">/</span> <span class="nam">self</span><span class="op">.</span><span class="nam">nx</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t362" href="#t362">362</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t363" href="#t363">363</a></span><span class="t">        <span class="nam">integral_z</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">xp</span><span class="op">.</span><span class="nam">sum</span><span class="op">(</span><span class="nam">nusselt_hat</span> <span class="op">*</span> <span class="nam">self</span><span class="op">.</span><span class="nam">spectral</span><span class="op">.</span><span class="nam">axes</span><span class="op">[</span><span class="num">1</span><span class="op">]</span><span class="op">.</span><span class="nam">get_BC</span><span class="op">(</span><span class="nam">kind</span><span class="op">=</span><span class="str">'integral'</span><span class="op">)</span><span class="op">,</span> <span class="nam">axis</span><span class="op">=</span><span class="op">-</span><span class="num">1</span><span class="op">)</span><span class="op">.</span><span class="nam">real</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t364" href="#t364">364</a></span><span class="t">        <span class="nam">integral_V</span> <span class="op">=</span> <span class="op">(</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t365" href="#t365">365</a></span><span class="t">            <span class="nam">integral_z</span><span class="op">[</span><span class="num">0</span><span class="op">]</span> <span class="op">*</span> <span class="nam">self</span><span class="op">.</span><span class="nam">axes</span><span class="op">[</span><span class="num">0</span><span class="op">]</span><span class="op">.</span><span class="nam">L</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t366" href="#t366">366</a></span><span class="t">        <span class="op">)</span>  <span class="com"># only the first Fourier mode has non-zero integral with periodic BCs</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t367" href="#t367">367</a></span><span class="t">        <span class="nam">Nusselt_V</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">comm</span><span class="op">.</span><span class="nam">bcast</span><span class="op">(</span><span class="nam">integral_V</span> <span class="op">/</span> <span class="nam">self</span><span class="op">.</span><span class="nam">spectral</span><span class="op">.</span><span class="nam">V</span><span class="op">,</span> <span class="nam">root</span><span class="op">=</span><span class="num">0</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t368" href="#t368">368</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t369" href="#t369">369</a></span><span class="t">        <span class="nam">Nusselt_t</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">comm</span><span class="op">.</span><span class="nam">bcast</span><span class="op">(</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t370" href="#t370">370</a></span><span class="t">            <span class="nam">self</span><span class="op">.</span><span class="nam">xp</span><span class="op">.</span><span class="nam">sum</span><span class="op">(</span><span class="nam">nusselt_hat</span> <span class="op">*</span> <span class="nam">self</span><span class="op">.</span><span class="nam">spectral</span><span class="op">.</span><span class="nam">axes</span><span class="op">[</span><span class="num">1</span><span class="op">]</span><span class="op">.</span><span class="nam">get_BC</span><span class="op">(</span><span class="nam">kind</span><span class="op">=</span><span class="str">'Dirichlet'</span><span class="op">,</span> <span class="nam">x</span><span class="op">=</span><span class="num">1</span><span class="op">)</span><span class="op">,</span> <span class="nam">axis</span><span class="op">=</span><span class="op">-</span><span class="num">1</span><span class="op">)</span><span class="op">.</span><span class="nam">real</span><span class="op">[</span><span class="num">0</span><span class="op">]</span><span class="op">,</span> <span class="nam">root</span><span class="op">=</span><span class="num">0</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t371" href="#t371">371</a></span><span class="t">        <span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t372" href="#t372">372</a></span><span class="t">        <span class="nam">Nusselt_b</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">comm</span><span class="op">.</span><span class="nam">bcast</span><span class="op">(</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t373" href="#t373">373</a></span><span class="t">            <span class="nam">self</span><span class="op">.</span><span class="nam">xp</span><span class="op">.</span><span class="nam">sum</span><span class="op">(</span><span class="nam">nusselt_hat</span> <span class="op">*</span> <span class="nam">self</span><span class="op">.</span><span class="nam">spectral</span><span class="op">.</span><span class="nam">axes</span><span class="op">[</span><span class="num">1</span><span class="op">]</span><span class="op">.</span><span class="nam">get_BC</span><span class="op">(</span><span class="nam">kind</span><span class="op">=</span><span class="str">'Dirichlet'</span><span class="op">,</span> <span class="nam">x</span><span class="op">=</span><span class="op">-</span><span class="num">1</span><span class="op">)</span><span class="op">,</span> <span class="nam">axis</span><span class="op">=</span><span class="op">-</span><span class="num">1</span><span class="op">)</span><span class="op">.</span><span class="nam">real</span><span class="op">[</span><span class="num">0</span><span class="op">]</span><span class="op">,</span> <span class="nam">root</span><span class="op">=</span><span class="num">0</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t374" href="#t374">374</a></span><span class="t">        <span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t375" href="#t375">375</a></span><span class="t">        <span class="nam">Nusselt_no_v_t</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">comm</span><span class="op">.</span><span class="nam">bcast</span><span class="op">(</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t376" href="#t376">376</a></span><span class="t">            <span class="nam">self</span><span class="op">.</span><span class="nam">xp</span><span class="op">.</span><span class="nam">sum</span><span class="op">(</span><span class="nam">nusselt_no_v_hat</span> <span class="op">*</span> <span class="nam">self</span><span class="op">.</span><span class="nam">spectral</span><span class="op">.</span><span class="nam">axes</span><span class="op">[</span><span class="num">1</span><span class="op">]</span><span class="op">.</span><span class="nam">get_BC</span><span class="op">(</span><span class="nam">kind</span><span class="op">=</span><span class="str">'Dirichlet'</span><span class="op">,</span> <span class="nam">x</span><span class="op">=</span><span class="num">1</span><span class="op">)</span><span class="op">,</span> <span class="nam">axis</span><span class="op">=</span><span class="op">-</span><span class="num">1</span><span class="op">)</span><span class="op">.</span><span class="nam">real</span><span class="op">[</span><span class="num">0</span><span class="op">]</span><span class="op">,</span> <span class="nam">root</span><span class="op">=</span><span class="num">0</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t377" href="#t377">377</a></span><span class="t">        <span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t378" href="#t378">378</a></span><span class="t">        <span class="nam">Nusselt_no_v_b</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">comm</span><span class="op">.</span><span class="nam">bcast</span><span class="op">(</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t379" href="#t379">379</a></span><span class="t">            <span class="nam">self</span><span class="op">.</span><span class="nam">xp</span><span class="op">.</span><span class="nam">sum</span><span class="op">(</span><span class="nam">nusselt_no_v_hat</span> <span class="op">*</span> <span class="nam">self</span><span class="op">.</span><span class="nam">spectral</span><span class="op">.</span><span class="nam">axes</span><span class="op">[</span><span class="num">1</span><span class="op">]</span><span class="op">.</span><span class="nam">get_BC</span><span class="op">(</span><span class="nam">kind</span><span class="op">=</span><span class="str">'Dirichlet'</span><span class="op">,</span> <span class="nam">x</span><span class="op">=</span><span class="op">-</span><span class="num">1</span><span class="op">)</span><span class="op">,</span> <span class="nam">axis</span><span class="op">=</span><span class="op">-</span><span class="num">1</span><span class="op">)</span><span class="op">.</span><span class="nam">real</span><span class="op">[</span><span class="num">0</span><span class="op">]</span><span class="op">,</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t380" href="#t380">380</a></span><span class="t">            <span class="nam">root</span><span class="op">=</span><span class="num">0</span><span class="op">,</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t381" href="#t381">381</a></span><span class="t">        <span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t382" href="#t382">382</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t383" href="#t383">383</a></span><span class="t">        <span class="key">return</span> <span class="op">{</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t384" href="#t384">384</a></span><span class="t">            <span class="str">'V'</span><span class="op">:</span> <span class="nam">Nusselt_V</span><span class="op">,</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t385" href="#t385">385</a></span><span class="t">            <span class="str">'t'</span><span class="op">:</span> <span class="nam">Nusselt_t</span><span class="op">,</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t386" href="#t386">386</a></span><span class="t">            <span class="str">'b'</span><span class="op">:</span> <span class="nam">Nusselt_b</span><span class="op">,</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t387" href="#t387">387</a></span><span class="t">            <span class="str">'t_no_v'</span><span class="op">:</span> <span class="nam">Nusselt_no_v_t</span><span class="op">,</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t388" href="#t388">388</a></span><span class="t">            <span class="str">'b_no_v'</span><span class="op">:</span> <span class="nam">Nusselt_no_v_b</span><span class="op">,</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t389" href="#t389">389</a></span><span class="t">        <span class="op">}</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t390" href="#t390">390</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t391" href="#t391">391</a></span><span class="t">    <span class="key">def</span> <span class="nam">compute_viscous_dissipation</span><span class="op">(</span><span class="nam">self</span><span class="op">,</span> <span class="nam">u</span><span class="op">)</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t392" href="#t392">392</a></span><span class="t">        <span class="nam">iu</span><span class="op">,</span> <span class="nam">iv</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">index</span><span class="op">(</span><span class="op">[</span><span class="str">'u'</span><span class="op">,</span> <span class="str">'v'</span><span class="op">]</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t393" href="#t393">393</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t394" href="#t394">394</a></span><span class="t">        <span class="nam">Lap_u_hat</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">spectral</span><span class="op">.</span><span class="nam">u_init_forward</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t395" href="#t395">395</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t396" href="#t396">396</a></span><span class="t">        <span class="key">if</span> <span class="nam">self</span><span class="op">.</span><span class="nam">spectral_space</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t397" href="#t397">397</a></span><span class="t">            <span class="nam">u_hat</span> <span class="op">=</span> <span class="nam">u</span><span class="op">.</span><span class="nam">copy</span><span class="op">(</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t398" href="#t398">398</a></span><span class="t">        <span class="key">else</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t399" href="#t399">399</a></span><span class="t">            <span class="nam">u_hat</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">transform</span><span class="op">(</span><span class="nam">u</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t400" href="#t400">400</a></span><span class="t">        <span class="nam">Lap_u_hat</span><span class="op">[</span><span class="nam">iu</span><span class="op">]</span> <span class="op">=</span> <span class="op">(</span><span class="op">(</span><span class="nam">self</span><span class="op">.</span><span class="nam">Dzz</span> <span class="op">+</span> <span class="nam">self</span><span class="op">.</span><span class="nam">Dxx</span><span class="op">)</span> <span class="op">@</span> <span class="nam">u_hat</span><span class="op">[</span><span class="nam">iu</span><span class="op">]</span><span class="op">.</span><span class="nam">flatten</span><span class="op">(</span><span class="op">)</span><span class="op">)</span><span class="op">.</span><span class="nam">reshape</span><span class="op">(</span><span class="nam">u_hat</span><span class="op">[</span><span class="nam">iu</span><span class="op">]</span><span class="op">.</span><span class="nam">shape</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t401" href="#t401">401</a></span><span class="t">        <span class="nam">Lap_u_hat</span><span class="op">[</span><span class="nam">iv</span><span class="op">]</span> <span class="op">=</span> <span class="op">(</span><span class="op">(</span><span class="nam">self</span><span class="op">.</span><span class="nam">Dzz</span> <span class="op">+</span> <span class="nam">self</span><span class="op">.</span><span class="nam">Dxx</span><span class="op">)</span> <span class="op">@</span> <span class="nam">u_hat</span><span class="op">[</span><span class="nam">iv</span><span class="op">]</span><span class="op">.</span><span class="nam">flatten</span><span class="op">(</span><span class="op">)</span><span class="op">)</span><span class="op">.</span><span class="nam">reshape</span><span class="op">(</span><span class="nam">u_hat</span><span class="op">[</span><span class="nam">iu</span><span class="op">]</span><span class="op">.</span><span class="nam">shape</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t402" href="#t402">402</a></span><span class="t">        <span class="nam">Lap_u</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">itransform</span><span class="op">(</span><span class="nam">Lap_u_hat</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t403" href="#t403">403</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t404" href="#t404">404</a></span><span class="t">        <span class="key">return</span> <span class="nam">abs</span><span class="op">(</span><span class="nam">u</span><span class="op">[</span><span class="nam">iu</span><span class="op">]</span> <span class="op">*</span> <span class="nam">Lap_u</span><span class="op">[</span><span class="nam">iu</span><span class="op">]</span> <span class="op">+</span> <span class="nam">u</span><span class="op">[</span><span class="nam">iv</span><span class="op">]</span> <span class="op">*</span> <span class="nam">Lap_u</span><span class="op">[</span><span class="nam">iv</span><span class="op">]</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t405" href="#t405">405</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t406" href="#t406">406</a></span><span class="t">    <span class="key">def</span> <span class="nam">compute_buoyancy_generation</span><span class="op">(</span><span class="nam">self</span><span class="op">,</span> <span class="nam">u</span><span class="op">)</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t407" href="#t407">407</a></span><span class="t">        <span class="key">if</span> <span class="nam">self</span><span class="op">.</span><span class="nam">spectral_space</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t408" href="#t408">408</a></span><span class="t">            <span class="nam">u</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">itransform</span><span class="op">(</span><span class="nam">u</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t409" href="#t409">409</a></span><span class="t">        <span class="nam">iv</span><span class="op">,</span> <span class="nam">iT</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">index</span><span class="op">(</span><span class="op">[</span><span class="str">'v'</span><span class="op">,</span> <span class="str">'T'</span><span class="op">]</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t410" href="#t410">410</a></span><span class="t">        <span class="key">return</span> <span class="nam">abs</span><span class="op">(</span><span class="nam">u</span><span class="op">[</span><span class="nam">iv</span><span class="op">]</span> <span class="op">*</span> <span class="nam">self</span><span class="op">.</span><span class="nam">Rayleigh</span> <span class="op">*</span> <span class="nam">u</span><span class="op">[</span><span class="nam">iT</span><span class="op">]</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t411" href="#t411">411</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t412" href="#t412">412</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t413" href="#t413">413</a></span><span class="t"><span class="key">class</span> <span class="nam">CFLLimit</span><span class="op">(</span><span class="nam">ConvergenceController</span><span class="op">)</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t414" href="#t414">414</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t415" href="#t415">415</a></span><span class="t">    <span class="key">def</span> <span class="nam">dependencies</span><span class="op">(</span><span class="nam">self</span><span class="op">,</span> <span class="nam">controller</span><span class="op">,</span> <span class="op">*</span><span class="nam">args</span><span class="op">,</span> <span class="op">**</span><span class="nam">kwargs</span><span class="op">)</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t416" href="#t416">416</a></span><span class="t">        <span class="key">from</span> <span class="nam">pySDC</span><span class="op">.</span><span class="nam">implementations</span><span class="op">.</span><span class="nam">hooks</span><span class="op">.</span><span class="nam">log_step_size</span> <span class="key">import</span> <span class="nam">LogStepSize</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t417" href="#t417">417</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t418" href="#t418">418</a></span><span class="t">        <span class="nam">controller</span><span class="op">.</span><span class="nam">add_hook</span><span class="op">(</span><span class="nam">LogCFL</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t419" href="#t419">419</a></span><span class="t">        <span class="nam">controller</span><span class="op">.</span><span class="nam">add_hook</span><span class="op">(</span><span class="nam">LogStepSize</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t420" href="#t420">420</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t421" href="#t421">421</a></span><span class="t">    <span class="key">def</span> <span class="nam">setup_status_variables</span><span class="op">(</span><span class="nam">self</span><span class="op">,</span> <span class="nam">controller</span><span class="op">,</span> <span class="op">**</span><span class="nam">kwargs</span><span class="op">)</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t422" href="#t422">422</a></span><span class="t">        <span class="str">"""</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t423" href="#t423">423</a></span><span class="t"><span class="str">        Add the embedded error variable to the error function.</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t424" href="#t424">424</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t425" href="#t425">425</a></span><span class="t"><span class="str">        Args:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t426" href="#t426">426</a></span><span class="t"><span class="str">            controller (pySDC.Controller): The controller</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t427" href="#t427">427</a></span><span class="t"><span class="str">        """</span>&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t428" href="#t428">428</a></span><span class="t">        <span class="nam">self</span><span class="op">.</span><span class="nam">add_status_variable_to_level</span><span class="op">(</span><span class="str">'CFL_limit'</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t429" href="#t429">429</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t430" href="#t430">430</a></span><span class="t">    <span class="key">def</span> <span class="nam">setup</span><span class="op">(</span><span class="nam">self</span><span class="op">,</span> <span class="nam">controller</span><span class="op">,</span> <span class="nam">params</span><span class="op">,</span> <span class="nam">description</span><span class="op">,</span> <span class="op">**</span><span class="nam">kwargs</span><span class="op">)</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t431" href="#t431">431</a></span><span class="t">        <span class="str">"""</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t432" href="#t432">432</a></span><span class="t"><span class="str">        Define default parameters here.</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t433" href="#t433">433</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t434" href="#t434">434</a></span><span class="t"><span class="str">        Default parameters are:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t435" href="#t435">435</a></span><span class="t"><span class="str">         - control_order (int): The order relative to other convergence controllers</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t436" href="#t436">436</a></span><span class="t"><span class="str">         - dt_max (float): maximal step size</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t437" href="#t437">437</a></span><span class="t"><span class="str">         - dt_min (float): minimal step size</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t438" href="#t438">438</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t439" href="#t439">439</a></span><span class="t"><span class="str">        Args:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t440" href="#t440">440</a></span><span class="t"><span class="str">            controller (pySDC.Controller): The controller</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t441" href="#t441">441</a></span><span class="t"><span class="str">            params (dict): The params passed for this specific convergence controller</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t442" href="#t442">442</a></span><span class="t"><span class="str">            description (dict): The description object used to instantiate the controller</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t443" href="#t443">443</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t444" href="#t444">444</a></span><span class="t"><span class="str">        Returns:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t445" href="#t445">445</a></span><span class="t"><span class="str">            (dict): The updated params dictionary</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t446" href="#t446">446</a></span><span class="t"><span class="str">        """</span>&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t447" href="#t447">447</a></span><span class="t">        <span class="nam">defaults</span> <span class="op">=</span> <span class="op">{</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t448" href="#t448">448</a></span><span class="t">            <span class="str">"control_order"</span><span class="op">:</span> <span class="op">-</span><span class="num">50</span><span class="op">,</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t449" href="#t449">449</a></span><span class="t">            <span class="str">"dt_max"</span><span class="op">:</span> <span class="nam">np</span><span class="op">.</span><span class="nam">inf</span><span class="op">,</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t450" href="#t450">450</a></span><span class="t">            <span class="str">"dt_min"</span><span class="op">:</span> <span class="num">0</span><span class="op">,</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t451" href="#t451">451</a></span><span class="t">            <span class="str">"cfl"</span><span class="op">:</span> <span class="num">0.4</span><span class="op">,</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t452" href="#t452">452</a></span><span class="t">        <span class="op">}</span>&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t453" href="#t453">453</a></span><span class="t">        <span class="key">return</span> <span class="op">{</span><span class="op">**</span><span class="nam">defaults</span><span class="op">,</span> <span class="op">**</span><span class="nam">super</span><span class="op">(</span><span class="op">)</span><span class="op">.</span><span class="nam">setup</span><span class="op">(</span><span class="nam">controller</span><span class="op">,</span> <span class="nam">params</span><span class="op">,</span> <span class="nam">description</span><span class="op">,</span> <span class="op">**</span><span class="nam">kwargs</span><span class="op">)</span><span class="op">}</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t454" href="#t454">454</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t455" href="#t455">455</a></span><span class="t">    <span class="op">@</span><span class="nam">staticmethod</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t456" href="#t456">456</a></span><span class="t">    <span class="key">def</span> <span class="nam">compute_max_step_size</span><span class="op">(</span><span class="nam">P</span><span class="op">,</span> <span class="nam">u</span><span class="op">)</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t457" href="#t457">457</a></span><span class="t">        <span class="nam">grid_spacing_x</span> <span class="op">=</span> <span class="nam">P</span><span class="op">.</span><span class="nam">X</span><span class="op">[</span><span class="num">1</span><span class="op">,</span> <span class="num">0</span><span class="op">]</span> <span class="op">-</span> <span class="nam">P</span><span class="op">.</span><span class="nam">X</span><span class="op">[</span><span class="num">0</span><span class="op">,</span> <span class="num">0</span><span class="op">]</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t458" href="#t458">458</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t459" href="#t459">459</a></span><span class="t">        <span class="nam">cell_wallz</span> <span class="op">=</span> <span class="nam">P</span><span class="op">.</span><span class="nam">xp</span><span class="op">.</span><span class="nam">zeros</span><span class="op">(</span><span class="nam">P</span><span class="op">.</span><span class="nam">nz</span> <span class="op">+</span> <span class="num">1</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t460" href="#t460">460</a></span><span class="t">        <span class="nam">cell_wallz</span><span class="op">[</span><span class="num">0</span><span class="op">]</span> <span class="op">=</span> <span class="num">1</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t461" href="#t461">461</a></span><span class="t">        <span class="nam">cell_wallz</span><span class="op">[</span><span class="op">-</span><span class="num">1</span><span class="op">]</span> <span class="op">=</span> <span class="op">-</span><span class="num">1</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t462" href="#t462">462</a></span><span class="t">        <span class="nam">cell_wallz</span><span class="op">[</span><span class="num">1</span><span class="op">:</span><span class="op">-</span><span class="num">1</span><span class="op">]</span> <span class="op">=</span> <span class="op">(</span><span class="nam">P</span><span class="op">.</span><span class="nam">Z</span><span class="op">[</span><span class="num">0</span><span class="op">,</span> <span class="op">:</span><span class="op">-</span><span class="num">1</span><span class="op">]</span> <span class="op">+</span> <span class="nam">P</span><span class="op">.</span><span class="nam">Z</span><span class="op">[</span><span class="num">0</span><span class="op">,</span> <span class="num">1</span><span class="op">:</span><span class="op">]</span><span class="op">)</span> <span class="op">/</span> <span class="num">2</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t463" href="#t463">463</a></span><span class="t">        <span class="nam">grid_spacing_z</span> <span class="op">=</span> <span class="nam">cell_wallz</span><span class="op">[</span><span class="op">:</span><span class="op">-</span><span class="num">1</span><span class="op">]</span> <span class="op">-</span> <span class="nam">cell_wallz</span><span class="op">[</span><span class="num">1</span><span class="op">:</span><span class="op">]</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t464" href="#t464">464</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t465" href="#t465">465</a></span><span class="t">        <span class="nam">iu</span><span class="op">,</span> <span class="nam">iv</span> <span class="op">=</span> <span class="nam">P</span><span class="op">.</span><span class="nam">index</span><span class="op">(</span><span class="op">[</span><span class="str">'u'</span><span class="op">,</span> <span class="str">'v'</span><span class="op">]</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t466" href="#t466">466</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t467" href="#t467">467</a></span><span class="t">        <span class="key">if</span> <span class="nam">P</span><span class="op">.</span><span class="nam">spectral_space</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t468" href="#t468">468</a></span><span class="t">            <span class="nam">u</span> <span class="op">=</span> <span class="nam">P</span><span class="op">.</span><span class="nam">itransform</span><span class="op">(</span><span class="nam">u</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t469" href="#t469">469</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t470" href="#t470">470</a></span><span class="t">        <span class="nam">max_step_size_x</span> <span class="op">=</span> <span class="nam">P</span><span class="op">.</span><span class="nam">xp</span><span class="op">.</span><span class="nam">min</span><span class="op">(</span><span class="nam">grid_spacing_x</span> <span class="op">/</span> <span class="nam">P</span><span class="op">.</span><span class="nam">xp</span><span class="op">.</span><span class="nam">abs</span><span class="op">(</span><span class="nam">u</span><span class="op">[</span><span class="nam">iu</span><span class="op">]</span><span class="op">)</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t471" href="#t471">471</a></span><span class="t">        <span class="nam">max_step_size_z</span> <span class="op">=</span> <span class="nam">P</span><span class="op">.</span><span class="nam">xp</span><span class="op">.</span><span class="nam">min</span><span class="op">(</span><span class="nam">grid_spacing_z</span> <span class="op">/</span> <span class="nam">P</span><span class="op">.</span><span class="nam">xp</span><span class="op">.</span><span class="nam">abs</span><span class="op">(</span><span class="nam">u</span><span class="op">[</span><span class="nam">iv</span><span class="op">]</span><span class="op">)</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t472" href="#t472">472</a></span><span class="t">        <span class="nam">max_step_size</span> <span class="op">=</span> <span class="nam">min</span><span class="op">(</span><span class="op">[</span><span class="nam">max_step_size_x</span><span class="op">,</span> <span class="nam">max_step_size_z</span><span class="op">]</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t473" href="#t473">473</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t474" href="#t474">474</a></span><span class="t">        <span class="key">if</span> <span class="nam">hasattr</span><span class="op">(</span><span class="nam">P</span><span class="op">,</span> <span class="str">'comm'</span><span class="op">)</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t475" href="#t475">475</a></span><span class="t">            <span class="nam">max_step_size</span> <span class="op">=</span> <span class="nam">P</span><span class="op">.</span><span class="nam">comm</span><span class="op">.</span><span class="nam">allreduce</span><span class="op">(</span><span class="nam">max_step_size</span><span class="op">,</span> <span class="nam">op</span><span class="op">=</span><span class="nam">MPI</span><span class="op">.</span><span class="nam">MIN</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t476" href="#t476">476</a></span><span class="t">        <span class="key">return</span> <span class="nam">float</span><span class="op">(</span><span class="nam">max_step_size</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t477" href="#t477">477</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t478" href="#t478">478</a></span><span class="t">    <span class="key">def</span> <span class="nam">get_new_step_size</span><span class="op">(</span><span class="nam">self</span><span class="op">,</span> <span class="nam">controller</span><span class="op">,</span> <span class="nam">step</span><span class="op">,</span> <span class="op">**</span><span class="nam">kwargs</span><span class="op">)</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t479" href="#t479">479</a></span><span class="t">        <span class="key">if</span> <span class="key">not</span> <span class="nam">CheckConvergence</span><span class="op">.</span><span class="nam">check_convergence</span><span class="op">(</span><span class="nam">step</span><span class="op">)</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t480" href="#t480">480</a></span><span class="t">            <span class="key">return</span> <span class="key">None</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t481" href="#t481">481</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t482" href="#t482">482</a></span><span class="t">        <span class="nam">L</span> <span class="op">=</span> <span class="nam">step</span><span class="op">.</span><span class="nam">levels</span><span class="op">[</span><span class="num">0</span><span class="op">]</span>&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t483" href="#t483">483</a></span><span class="t">        <span class="nam">P</span> <span class="op">=</span> <span class="nam">step</span><span class="op">.</span><span class="nam">levels</span><span class="op">[</span><span class="num">0</span><span class="op">]</span><span class="op">.</span><span class="nam">prob</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t484" href="#t484">484</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t485" href="#t485">485</a></span><span class="t">        <span class="nam">L</span><span class="op">.</span><span class="nam">sweep</span><span class="op">.</span><span class="nam">compute_end_point</span><span class="op">(</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t486" href="#t486">486</a></span><span class="t">        <span class="nam">max_step_size</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">compute_max_step_size</span><span class="op">(</span><span class="nam">P</span><span class="op">,</span> <span class="nam">L</span><span class="op">.</span><span class="nam">uend</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t487" href="#t487">487</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t488" href="#t488">488</a></span><span class="t">        <span class="nam">L</span><span class="op">.</span><span class="nam">status</span><span class="op">.</span><span class="nam">CFL_limit</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">params</span><span class="op">.</span><span class="nam">cfl</span> <span class="op">*</span> <span class="nam">max_step_size</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t489" href="#t489">489</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t490" href="#t490">490</a></span><span class="t">        <span class="nam">dt_new</span> <span class="op">=</span> <span class="nam">L</span><span class="op">.</span><span class="nam">status</span><span class="op">.</span><span class="nam">dt_new</span> <span class="key">if</span> <span class="nam">L</span><span class="op">.</span><span class="nam">status</span><span class="op">.</span><span class="nam">dt_new</span> <span class="key">else</span> <span class="nam">max</span><span class="op">(</span><span class="op">[</span><span class="nam">self</span><span class="op">.</span><span class="nam">params</span><span class="op">.</span><span class="nam">dt_max</span><span class="op">,</span> <span class="nam">L</span><span class="op">.</span><span class="nam">params</span><span class="op">.</span><span class="nam">dt</span><span class="op">]</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t491" href="#t491">491</a></span><span class="t">        <span class="nam">L</span><span class="op">.</span><span class="nam">status</span><span class="op">.</span><span class="nam">dt_new</span> <span class="op">=</span> <span class="nam">min</span><span class="op">(</span><span class="op">[</span><span class="nam">dt_new</span><span class="op">,</span> <span class="nam">self</span><span class="op">.</span><span class="nam">params</span><span class="op">.</span><span class="nam">cfl</span> <span class="op">*</span> <span class="nam">max_step_size</span><span class="op">]</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t492" href="#t492">492</a></span><span class="t">        <span class="nam">L</span><span class="op">.</span><span class="nam">status</span><span class="op">.</span><span class="nam">dt_new</span> <span class="op">=</span> <span class="nam">max</span><span class="op">(</span><span class="op">[</span><span class="nam">self</span><span class="op">.</span><span class="nam">params</span><span class="op">.</span><span class="nam">dt_min</span><span class="op">,</span> <span class="nam">L</span><span class="op">.</span><span class="nam">status</span><span class="op">.</span><span class="nam">dt_new</span><span class="op">]</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t493" href="#t493">493</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t494" href="#t494">494</a></span><span class="t">        <span class="nam">self</span><span class="op">.</span><span class="nam">log</span><span class="op">(</span><span class="fst">f'</span><span class="fst">dt max: </span><span class="op">{</span><span class="nam">max_step_size</span><span class="op">:</span><span class="fst">.2e</span><span class="op">}</span><span class="fst"> -> New step size: </span><span class="op">{</span><span class="nam">L</span><span class="op">.</span><span class="nam">status</span><span class="op">.</span><span class="nam">dt_new</span><span class="op">:</span><span class="fst">.2e</span><span class="op">}</span><span class="fst">'</span><span class="op">,</span> <span class="nam">step</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t495" href="#t495">495</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t496" href="#t496">496</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t497" href="#t497">497</a></span><span class="t"><span class="key">class</span> <span class="nam">LogCFL</span><span class="op">(</span><span class="nam">Hooks</span><span class="op">)</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t498" href="#t498">498</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t499" href="#t499">499</a></span><span class="t">    <span class="key">def</span> <span class="nam">post_step</span><span class="op">(</span><span class="nam">self</span><span class="op">,</span> <span class="nam">step</span><span class="op">,</span> <span class="nam">level_number</span><span class="op">)</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t500" href="#t500">500</a></span><span class="t">        <span class="str">"""</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t501" href="#t501">501</a></span><span class="t"><span class="str">        Record CFL limit.</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t502" href="#t502">502</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t503" href="#t503">503</a></span><span class="t"><span class="str">        Args:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t504" href="#t504">504</a></span><span class="t"><span class="str">            step (pySDC.Step.step): the current step</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t505" href="#t505">505</a></span><span class="t"><span class="str">            level_number (int): the current level number</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t506" href="#t506">506</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t507" href="#t507">507</a></span><span class="t"><span class="str">        Returns:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t508" href="#t508">508</a></span><span class="t"><span class="str">            None</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t509" href="#t509">509</a></span><span class="t"><span class="str">        """</span>&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t510" href="#t510">510</a></span><span class="t">        <span class="nam">super</span><span class="op">(</span><span class="op">)</span><span class="op">.</span><span class="nam">post_step</span><span class="op">(</span><span class="nam">step</span><span class="op">,</span> <span class="nam">level_number</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t511" href="#t511">511</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t512" href="#t512">512</a></span><span class="t">        <span class="nam">L</span> <span class="op">=</span> <span class="nam">step</span><span class="op">.</span><span class="nam">levels</span><span class="op">[</span><span class="nam">level_number</span><span class="op">]</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t513" href="#t513">513</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t514" href="#t514">514</a></span><span class="t">        <span class="nam">self</span><span class="op">.</span><span class="nam">add_to_stats</span><span class="op">(</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t515" href="#t515">515</a></span><span class="t">            <span class="nam">process</span><span class="op">=</span><span class="nam">step</span><span class="op">.</span><span class="nam">status</span><span class="op">.</span><span class="nam">slot</span><span class="op">,</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t516" href="#t516">516</a></span><span class="t">            <span class="nam">time</span><span class="op">=</span><span class="nam">L</span><span class="op">.</span><span class="nam">time</span> <span class="op">+</span> <span class="nam">L</span><span class="op">.</span><span class="nam">dt</span><span class="op">,</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t517" href="#t517">517</a></span><span class="t">            <span class="nam">level</span><span class="op">=</span><span class="nam">L</span><span class="op">.</span><span class="nam">level_index</span><span class="op">,</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t518" href="#t518">518</a></span><span class="t">            <span class="nam">iter</span><span class="op">=</span><span class="nam">step</span><span class="op">.</span><span class="nam">status</span><span class="op">.</span><span class="nam">iter</span><span class="op">,</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t519" href="#t519">519</a></span><span class="t">            <span class="nam">sweep</span><span class="op">=</span><span class="nam">L</span><span class="op">.</span><span class="nam">status</span><span class="op">.</span><span class="nam">sweep</span><span class="op">,</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t520" href="#t520">520</a></span><span class="t">            <span class="nam">type</span><span class="op">=</span><span class="str">'CFL_limit'</span><span class="op">,</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t521" href="#t521">521</a></span><span class="t">            <span class="nam">value</span><span class="op">=</span><span class="nam">L</span><span class="op">.</span><span class="nam">status</span><span class="op">.</span><span class="nam">CFL_limit</span><span class="op">,</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t522" href="#t522">522</a></span><span class="t">        <span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t523" href="#t523">523</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t524" href="#t524">524</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t525" href="#t525">525</a></span><span class="t"><span class="key">class</span> <span class="nam">LogAnalysisVariables</span><span class="op">(</span><span class="nam">Hooks</span><span class="op">)</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t526" href="#t526">526</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t527" href="#t527">527</a></span><span class="t">    <span class="key">def</span> <span class="nam">post_step</span><span class="op">(</span><span class="nam">self</span><span class="op">,</span> <span class="nam">step</span><span class="op">,</span> <span class="nam">level_number</span><span class="op">)</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t528" href="#t528">528</a></span><span class="t">        <span class="str">"""</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t529" href="#t529">529</a></span><span class="t"><span class="str">        Record Nusselt numbers.</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t530" href="#t530">530</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t531" href="#t531">531</a></span><span class="t"><span class="str">        Args:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t532" href="#t532">532</a></span><span class="t"><span class="str">            step (pySDC.Step.step): the current step</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t533" href="#t533">533</a></span><span class="t"><span class="str">            level_number (int): the current level number</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t534" href="#t534">534</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t535" href="#t535">535</a></span><span class="t"><span class="str">        Returns:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t536" href="#t536">536</a></span><span class="t"><span class="str">            None</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t537" href="#t537">537</a></span><span class="t"><span class="str">        """</span>&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t538" href="#t538">538</a></span><span class="t">        <span class="nam">super</span><span class="op">(</span><span class="op">)</span><span class="op">.</span><span class="nam">post_step</span><span class="op">(</span><span class="nam">step</span><span class="op">,</span> <span class="nam">level_number</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t539" href="#t539">539</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t540" href="#t540">540</a></span><span class="t">        <span class="nam">L</span> <span class="op">=</span> <span class="nam">step</span><span class="op">.</span><span class="nam">levels</span><span class="op">[</span><span class="nam">level_number</span><span class="op">]</span>&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t541" href="#t541">541</a></span><span class="t">        <span class="nam">P</span> <span class="op">=</span> <span class="nam">L</span><span class="op">.</span><span class="nam">prob</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t542" href="#t542">542</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t543" href="#t543">543</a></span><span class="t">        <span class="nam">L</span><span class="op">.</span><span class="nam">sweep</span><span class="op">.</span><span class="nam">compute_end_point</span><span class="op">(</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t544" href="#t544">544</a></span><span class="t">        <span class="nam">Nusselt</span> <span class="op">=</span> <span class="nam">P</span><span class="op">.</span><span class="nam">compute_Nusselt_numbers</span><span class="op">(</span><span class="nam">L</span><span class="op">.</span><span class="nam">uend</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t545" href="#t545">545</a></span><span class="t">        <span class="nam">buoyancy_production</span> <span class="op">=</span> <span class="nam">P</span><span class="op">.</span><span class="nam">compute_buoyancy_generation</span><span class="op">(</span><span class="nam">L</span><span class="op">.</span><span class="nam">uend</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t546" href="#t546">546</a></span><span class="t">        <span class="nam">viscous_dissipation</span> <span class="op">=</span> <span class="nam">P</span><span class="op">.</span><span class="nam">compute_viscous_dissipation</span><span class="op">(</span><span class="nam">L</span><span class="op">.</span><span class="nam">uend</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t547" href="#t547">547</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t548" href="#t548">548</a></span><span class="t">        <span class="key">for</span> <span class="nam">key</span><span class="op">,</span> <span class="nam">value</span> <span class="key">in</span> <span class="nam">zip</span><span class="op">(</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t549" href="#t549">549</a></span><span class="t">            <span class="op">[</span><span class="str">'Nusselt'</span><span class="op">,</span> <span class="str">'buoyancy_production'</span><span class="op">,</span> <span class="str">'viscous_dissipation'</span><span class="op">]</span><span class="op">,</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t550" href="#t550">550</a></span><span class="t">            <span class="op">[</span><span class="nam">Nusselt</span><span class="op">,</span> <span class="nam">buoyancy_production</span><span class="op">,</span> <span class="nam">viscous_dissipation</span><span class="op">]</span><span class="op">,</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t551" href="#t551">551</a></span><span class="t">        <span class="op">)</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t552" href="#t552">552</a></span><span class="t">            <span class="nam">self</span><span class="op">.</span><span class="nam">add_to_stats</span><span class="op">(</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t553" href="#t553">553</a></span><span class="t">                <span class="nam">process</span><span class="op">=</span><span class="nam">step</span><span class="op">.</span><span class="nam">status</span><span class="op">.</span><span class="nam">slot</span><span class="op">,</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t554" href="#t554">554</a></span><span class="t">                <span class="nam">time</span><span class="op">=</span><span class="nam">L</span><span class="op">.</span><span class="nam">time</span> <span class="op">+</span> <span class="nam">L</span><span class="op">.</span><span class="nam">dt</span><span class="op">,</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t555" href="#t555">555</a></span><span class="t">                <span class="nam">level</span><span class="op">=</span><span class="nam">L</span><span class="op">.</span><span class="nam">level_index</span><span class="op">,</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t556" href="#t556">556</a></span><span class="t">                <span class="nam">iter</span><span class="op">=</span><span class="nam">step</span><span class="op">.</span><span class="nam">status</span><span class="op">.</span><span class="nam">iter</span><span class="op">,</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t557" href="#t557">557</a></span><span class="t">                <span class="nam">sweep</span><span class="op">=</span><span class="nam">L</span><span class="op">.</span><span class="nam">status</span><span class="op">.</span><span class="nam">sweep</span><span class="op">,</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t558" href="#t558">558</a></span><span class="t">                <span class="nam">type</span><span class="op">=</span><span class="nam">key</span><span class="op">,</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t559" href="#t559">559</a></span><span class="t">                <span class="nam">value</span><span class="op">=</span><span class="nam">value</span><span class="op">,</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t560" href="#t560">560</a></span><span class="t">            <span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+</main>
+<footer>
+    <div class="content">
+        <p>
+            <a class="nav" href="z_91faa57f8583c837_Quench_py.html">&#xab; prev</a> &nbsp; &nbsp;
+            <a class="nav" href="index.html">&Hat; index</a> &nbsp; &nbsp;
+            <a class="nav" href="z_91faa57f8583c837_TestEquation_0D_py.html">&#xbb; next</a>
+            &nbsp; &nbsp; &nbsp;
+            <a class="nav" href="https://coverage.readthedocs.io/en/7.6.1">coverage.py v7.6.1</a>,
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+        </p>
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+</footer>
+</body>
+</html>
diff --git a/coverage/z_91faa57f8583c837_TestEquation_0D_py.html b/coverage/z_91faa57f8583c837_TestEquation_0D_py.html
index e1690aab08..370a641e68 100644
--- a/coverage/z_91faa57f8583c837_TestEquation_0D_py.html
+++ b/coverage/z_91faa57f8583c837_TestEquation_0D_py.html
@@ -60,12 +60,12 @@ <h2>
             <button type="button" class="exc show_exc button_toggle_exc" value="exc" data-shortcut="x" title="Toggle lines excluded">1<span class="text"> excluded</span></button>
         </h2>
         <p class="text">
-            <a id="prevFileLink" class="nav" href="z_91faa57f8583c837_Quench_py.html">&#xab; prev</a> &nbsp; &nbsp;
+            <a id="prevFileLink" class="nav" href="z_91faa57f8583c837_RayleighBenard_py.html">&#xab; prev</a> &nbsp; &nbsp;
             <a id="indexLink" class="nav" href="index.html">&Hat; index</a> &nbsp; &nbsp;
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             &nbsp; &nbsp; &nbsp;
             <a class="nav" href="https://coverage.readthedocs.io/en/7.6.1">coverage.py v7.6.1</a>,
-            created at 2024-09-19 09:13 +0000
+            created at 2024-09-20 16:55 +0000
         </p>
         <aside class="hidden">
             <button type="button" class="button_next_chunk" data-shortcut="j"></button>
@@ -231,12 +231,12 @@ <h2>
 <footer>
     <div class="content">
         <p>
-            <a class="nav" href="z_91faa57f8583c837_Quench_py.html">&#xab; prev</a> &nbsp; &nbsp;
+            <a class="nav" href="z_91faa57f8583c837_RayleighBenard_py.html">&#xab; prev</a> &nbsp; &nbsp;
             <a class="nav" href="index.html">&Hat; index</a> &nbsp; &nbsp;
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             &nbsp; &nbsp; &nbsp;
             <a class="nav" href="https://coverage.readthedocs.io/en/7.6.1">coverage.py v7.6.1</a>,
-            created at 2024-09-19 09:13 +0000
+            created at 2024-09-20 16:55 +0000
         </p>
     </div>
 </footer>
diff --git a/coverage/z_91faa57f8583c837_Van_der_Pol_implicit_py.html b/coverage/z_91faa57f8583c837_Van_der_Pol_implicit_py.html
index dd7649c4d1..8bcaa77718 100644
--- a/coverage/z_91faa57f8583c837_Van_der_Pol_implicit_py.html
+++ b/coverage/z_91faa57f8583c837_Van_der_Pol_implicit_py.html
@@ -2,7 +2,7 @@
 <html lang="en">
 <head>
     <meta http-equiv="Content-Type" content="text/html; charset=utf-8">
-    <title>Coverage for pySDC/implementations/problem_classes/Van_der_Pol_implicit.py: 96%</title>
+    <title>Coverage for pySDC/implementations/problem_classes/Van_der_Pol_implicit.py: 95%</title>
     <link rel="icon" sizes="32x32" href="favicon_32_cb_58284776.png">
     <link rel="stylesheet" href="style_cb_8e611ae1.css" type="text/css">
     <script src="coverage_html_cb_6fb7b396.js" defer></script>
@@ -12,7 +12,7 @@
     <div class="content">
         <h1>
             <span class="text">Coverage for </span><b>pySDC/implementations/problem_classes/Van_der_Pol_implicit.py</b>:
-            <span class="pc_cov">96%</span>
+            <span class="pc_cov">95%</span>
         </h1>
         <aside id="help_panel_wrapper">
             <input id="help_panel_state" type="checkbox">
@@ -55,8 +55,8 @@ <h1>
         </aside>
         <h2>
             <span class="text">57 statements &nbsp;</span>
-            <button type="button" class="run button_toggle_run" value="run" data-shortcut="r" title="Toggle lines run">55<span class="text"> run</span></button>
-            <button type="button" class="mis show_mis button_toggle_mis" value="mis" data-shortcut="m" title="Toggle lines missing">2<span class="text"> missing</span></button>
+            <button type="button" class="run button_toggle_run" value="run" data-shortcut="r" title="Toggle lines run">54<span class="text"> run</span></button>
+            <button type="button" class="mis show_mis button_toggle_mis" value="mis" data-shortcut="m" title="Toggle lines missing">3<span class="text"> missing</span></button>
             <button type="button" class="exc show_exc button_toggle_exc" value="exc" data-shortcut="x" title="Toggle lines excluded">2<span class="text"> excluded</span></button>
         </h2>
         <p class="text">
@@ -65,7 +65,7 @@ <h2>
             <a id="nextFileLink" class="nav" href="z_91faa57f8583c837_VorticityVelocity_2D_FEniCS_periodic_py.html">&#xbb; next</a>
             &nbsp; &nbsp; &nbsp;
             <a class="nav" href="https://coverage.readthedocs.io/en/7.6.1">coverage.py v7.6.1</a>,
-            created at 2024-09-19 09:13 +0000
+            created at 2024-09-20 16:55 +0000
         </p>
         <aside class="hidden">
             <button type="button" class="button_next_chunk" data-shortcut="j"></button>
@@ -267,7 +267,7 @@ <h2>
     <p class="mis show_mis"><span class="n"><a id="t185" href="#t185">185</a></span><span class="t">            <span class="nam">self</span><span class="op">.</span><span class="nam">logger</span><span class="op">.</span><span class="nam">warning</span><span class="op">(</span><span class="str">'Newton got nan after %i iterations...'</span> <span class="op">%</span> <span class="nam">n</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
     <p class="exc show_exc"><span class="n"><a id="t186" href="#t186">186</a></span><span class="t">            <span class="key">raise</span> <span class="nam">ProblemError</span><span class="op">(</span><span class="str">'Newton got nan after %i iterations, aborting...'</span> <span class="op">%</span> <span class="nam">n</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
     <p class="run"><span class="n"><a id="t187" href="#t187">187</a></span><span class="t">        <span class="key">elif</span> <span class="nam">np</span><span class="op">.</span><span class="nam">isnan</span><span class="op">(</span><span class="nam">res</span><span class="op">)</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
-    <p class="run"><span class="n"><a id="t188" href="#t188">188</a></span><span class="t">            <span class="nam">self</span><span class="op">.</span><span class="nam">logger</span><span class="op">.</span><span class="nam">warning</span><span class="op">(</span><span class="str">'Newton got nan after %i iterations...'</span> <span class="op">%</span> <span class="nam">n</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t188" href="#t188">188</a></span><span class="t">            <span class="nam">self</span><span class="op">.</span><span class="nam">logger</span><span class="op">.</span><span class="nam">warning</span><span class="op">(</span><span class="str">'Newton got nan after %i iterations...'</span> <span class="op">%</span> <span class="nam">n</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
     <p class="pln"><span class="n"><a id="t189" href="#t189">189</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
     <p class="run"><span class="n"><a id="t190" href="#t190">190</a></span><span class="t">        <span class="key">if</span> <span class="nam">n</span> <span class="op">==</span> <span class="nam">self</span><span class="op">.</span><span class="nam">newton_maxiter</span> <span class="key">and</span> <span class="nam">self</span><span class="op">.</span><span class="nam">crash_at_maxiter</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
     <p class="exc show_exc"><span class="n"><a id="t191" href="#t191">191</a></span><span class="t">            <span class="key">raise</span> <span class="nam">ProblemError</span><span class="op">(</span><span class="str">'Newton did not converge after %i iterations, error is %s'</span> <span class="op">%</span> <span class="op">(</span><span class="nam">n</span><span class="op">,</span> <span class="nam">res</span><span class="op">)</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
@@ -282,7 +282,7 @@ <h2>
             <a class="nav" href="z_91faa57f8583c837_VorticityVelocity_2D_FEniCS_periodic_py.html">&#xbb; next</a>
             &nbsp; &nbsp; &nbsp;
             <a class="nav" href="https://coverage.readthedocs.io/en/7.6.1">coverage.py v7.6.1</a>,
-            created at 2024-09-19 09:13 +0000
+            created at 2024-09-20 16:55 +0000
         </p>
     </div>
 </footer>
diff --git a/coverage/z_91faa57f8583c837_VorticityVelocity_2D_FEniCS_periodic_py.html b/coverage/z_91faa57f8583c837_VorticityVelocity_2D_FEniCS_periodic_py.html
index 7e0e3e7167..c27b96d16a 100644
--- a/coverage/z_91faa57f8583c837_VorticityVelocity_2D_FEniCS_periodic_py.html
+++ b/coverage/z_91faa57f8583c837_VorticityVelocity_2D_FEniCS_periodic_py.html
@@ -65,7 +65,7 @@ <h2>
             <a id="nextFileLink" class="nav" href="z_c032a2f084ee6f96_buildFDMatrix_py.html">&#xbb; next</a>
             &nbsp; &nbsp; &nbsp;
             <a class="nav" href="https://coverage.readthedocs.io/en/7.6.1">coverage.py v7.6.1</a>,
-            created at 2024-09-19 09:13 +0000
+            created at 2024-09-20 16:55 +0000
         </p>
         <aside class="hidden">
             <button type="button" class="button_next_chunk" data-shortcut="j"></button>
@@ -572,7 +572,7 @@ <h2>
             <a class="nav" href="z_c032a2f084ee6f96_buildFDMatrix_py.html">&#xbb; next</a>
             &nbsp; &nbsp; &nbsp;
             <a class="nav" href="https://coverage.readthedocs.io/en/7.6.1">coverage.py v7.6.1</a>,
-            created at 2024-09-19 09:13 +0000
+            created at 2024-09-20 16:55 +0000
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     </div>
 </footer>
diff --git a/coverage/z_91faa57f8583c837_generic_MPIFFT_Laplacian_py.html b/coverage/z_91faa57f8583c837_generic_MPIFFT_Laplacian_py.html
index 3290b3610c..77a4665ab6 100644
--- a/coverage/z_91faa57f8583c837_generic_MPIFFT_Laplacian_py.html
+++ b/coverage/z_91faa57f8583c837_generic_MPIFFT_Laplacian_py.html
@@ -65,7 +65,7 @@ <h2>
             <a id="nextFileLink" class="nav" href="z_91faa57f8583c837_generic_ND_FD_py.html">&#xbb; next</a>
             &nbsp; &nbsp; &nbsp;
             <a class="nav" href="https://coverage.readthedocs.io/en/7.6.1">coverage.py v7.6.1</a>,
-            created at 2024-09-19 09:13 +0000
+            created at 2024-09-20 16:55 +0000
         </p>
         <aside class="hidden">
             <button type="button" class="button_next_chunk" data-shortcut="j"></button>
@@ -299,7 +299,7 @@ <h2>
             <a class="nav" href="z_91faa57f8583c837_generic_ND_FD_py.html">&#xbb; next</a>
             &nbsp; &nbsp; &nbsp;
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-            created at 2024-09-19 09:13 +0000
+            created at 2024-09-20 16:55 +0000
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     </div>
 </footer>
diff --git a/coverage/z_91faa57f8583c837_generic_ND_FD_py.html b/coverage/z_91faa57f8583c837_generic_ND_FD_py.html
index 2781fd273e..52a180e788 100644
--- a/coverage/z_91faa57f8583c837_generic_ND_FD_py.html
+++ b/coverage/z_91faa57f8583c837_generic_ND_FD_py.html
@@ -62,10 +62,10 @@ <h2>
         <p class="text">
             <a id="prevFileLink" class="nav" href="z_91faa57f8583c837_generic_MPIFFT_Laplacian_py.html">&#xab; prev</a> &nbsp; &nbsp;
             <a id="indexLink" class="nav" href="index.html">&Hat; index</a> &nbsp; &nbsp;
-            <a id="nextFileLink" class="nav" href="z_91faa57f8583c837_nonlinear_ODE_1_py.html">&#xbb; next</a>
+            <a id="nextFileLink" class="nav" href="z_91faa57f8583c837_generic_spectral_py.html">&#xbb; next</a>
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+            created at 2024-09-20 16:55 +0000
         </p>
         <aside class="hidden">
             <button type="button" class="button_next_chunk" data-shortcut="j"></button>
@@ -349,10 +349,10 @@ <h2>
         <p>
             <a class="nav" href="z_91faa57f8583c837_generic_MPIFFT_Laplacian_py.html">&#xab; prev</a> &nbsp; &nbsp;
             <a class="nav" href="index.html">&Hat; index</a> &nbsp; &nbsp;
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+            <a class="nav" href="z_91faa57f8583c837_generic_spectral_py.html">&#xbb; next</a>
             &nbsp; &nbsp; &nbsp;
             <a class="nav" href="https://coverage.readthedocs.io/en/7.6.1">coverage.py v7.6.1</a>,
-            created at 2024-09-19 09:13 +0000
+            created at 2024-09-20 16:55 +0000
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 </footer>
diff --git a/coverage/z_91faa57f8583c837_generic_spectral_py.html b/coverage/z_91faa57f8583c837_generic_spectral_py.html
new file mode 100644
index 0000000000..4dad9b24a9
--- /dev/null
+++ b/coverage/z_91faa57f8583c837_generic_spectral_py.html
@@ -0,0 +1,486 @@
+<!DOCTYPE html>
+<html lang="en">
+<head>
+    <meta http-equiv="Content-Type" content="text/html; charset=utf-8">
+    <title>Coverage for pySDC/implementations/problem_classes/generic_spectral.py: 57%</title>
+    <link rel="icon" sizes="32x32" href="favicon_32_cb_58284776.png">
+    <link rel="stylesheet" href="style_cb_8e611ae1.css" type="text/css">
+    <script src="coverage_html_cb_6fb7b396.js" defer></script>
+</head>
+<body class="pyfile">
+<header>
+    <div class="content">
+        <h1>
+            <span class="text">Coverage for </span><b>pySDC/implementations/problem_classes/generic_spectral.py</b>:
+            <span class="pc_cov">57%</span>
+        </h1>
+        <aside id="help_panel_wrapper">
+            <input id="help_panel_state" type="checkbox">
+            <label for="help_panel_state">
+                <img id="keyboard_icon" src="keybd_closed_cb_ce680311.png" alt="Show/hide keyboard shortcuts">
+            </label>
+            <div id="help_panel">
+                <p class="legend">Shortcuts on this page</p>
+                <div class="keyhelp">
+                    <p>
+                        <kbd>r</kbd>
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+                        &nbsp; toggle line displays
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+                        <kbd>0</kbd> &nbsp; (zero) top of page
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+                        <kbd>1</kbd> &nbsp; (one) first highlighted chunk
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+        <h2>
+            <span class="text">141 statements &nbsp;</span>
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+            <button type="button" class="mis show_mis button_toggle_mis" value="mis" data-shortcut="m" title="Toggle lines missing">60<span class="text"> missing</span></button>
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+<main id="source">
+    <p class="run"><span class="n"><a id="t1" href="#t1">1</a></span><span class="t"><span class="key">from</span> <span class="nam">pySDC</span><span class="op">.</span><span class="nam">core</span><span class="op">.</span><span class="nam">problem</span> <span class="key">import</span> <span class="nam">Problem</span><span class="op">,</span> <span class="nam">WorkCounter</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t2" href="#t2">2</a></span><span class="t"><span class="key">from</span> <span class="nam">pySDC</span><span class="op">.</span><span class="nam">helpers</span><span class="op">.</span><span class="nam">spectral_helper</span> <span class="key">import</span> <span class="nam">SpectralHelper</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t3" href="#t3">3</a></span><span class="t"><span class="key">import</span> <span class="nam">numpy</span> <span class="key">as</span> <span class="nam">np</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t4" href="#t4">4</a></span><span class="t"><span class="key">from</span> <span class="nam">pySDC</span><span class="op">.</span><span class="nam">core</span><span class="op">.</span><span class="nam">errors</span> <span class="key">import</span> <span class="nam">ParameterError</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t5" href="#t5">5</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t6" href="#t6">6</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t7" href="#t7">7</a></span><span class="t"><span class="key">class</span> <span class="nam">GenericSpectralLinear</span><span class="op">(</span><span class="nam">Problem</span><span class="op">)</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t8" href="#t8">8</a></span><span class="t">    <span class="str">"""</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t9" href="#t9">9</a></span><span class="t"><span class="str">    Generic class to solve problems of the form M u_t + L u = y, with mass matrix M, linear operator L and some right</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t10" href="#t10">10</a></span><span class="t"><span class="str">    hand side y using spectral methods.</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t11" href="#t11">11</a></span><span class="t"><span class="str">    L may contain algebraic conditions, as long as (M + dt L) is invertible.</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t12" href="#t12">12</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t13" href="#t13">13</a></span><span class="t"><span class="str">    Note that the `__getattr__` method is overloaded to pass requests on to the spectral helper if they are not</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t14" href="#t14">14</a></span><span class="t"><span class="str">    attributes of this class itself. For instance, you can add a BC by calling `self.spectral.add_BC` or equivalently</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t15" href="#t15">15</a></span><span class="t"><span class="str">    `self.add_BC`.</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t16" href="#t16">16</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t17" href="#t17">17</a></span><span class="t"><span class="str">    You can port problems derived from this more or less seamlessly to GPU by using the numerical libraries that are</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t18" href="#t18">18</a></span><span class="t"><span class="str">    class attributes of the spectral helper. This class will automatically switch the datatype using the `setup_GPU` class method.</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t19" href="#t19">19</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t20" href="#t20">20</a></span><span class="t"><span class="str">    Attributes:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t21" href="#t21">21</a></span><span class="t"><span class="str">        spectral (pySDC.helpers.spectral_helper.SpectralHelper): Spectral helper</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t22" href="#t22">22</a></span><span class="t"><span class="str">        work_counters (dict): Dictionary for counting work</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t23" href="#t23">23</a></span><span class="t"><span class="str">        cached_factorizations (dict): Dictionary of cached matrix factorizations for solving</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t24" href="#t24">24</a></span><span class="t"><span class="str">        L (sparse matrix): Linear operator</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t25" href="#t25">25</a></span><span class="t"><span class="str">        M (sparse matrix): Mass matrix</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t26" href="#t26">26</a></span><span class="t"><span class="str">        diff_mask (list): Mask for separating differential and algebraic terms</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t27" href="#t27">27</a></span><span class="t"><span class="str">        Pl (sparse matrix): Left preconditioner</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t28" href="#t28">28</a></span><span class="t"><span class="str">        Pr (sparse matrix): Right preconditioner</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t29" href="#t29">29</a></span><span class="t"><span class="str">    """</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t30" href="#t30">30</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t31" href="#t31">31</a></span><span class="t">    <span class="op">@</span><span class="nam">classmethod</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t32" href="#t32">32</a></span><span class="t">    <span class="key">def</span> <span class="nam">setup_GPU</span><span class="op">(</span><span class="nam">cls</span><span class="op">)</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t33" href="#t33">33</a></span><span class="t">        <span class="str">"""switch to GPU modules"""</span>&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t34" href="#t34">34</a></span><span class="t">        <span class="key">import</span> <span class="nam">cupy</span> <span class="key">as</span> <span class="nam">cp</span>&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t35" href="#t35">35</a></span><span class="t">        <span class="key">from</span> <span class="nam">pySDC</span><span class="op">.</span><span class="nam">implementations</span><span class="op">.</span><span class="nam">datatype_classes</span><span class="op">.</span><span class="nam">cupy_mesh</span> <span class="key">import</span> <span class="nam">cupy_mesh</span><span class="op">,</span> <span class="nam">imex_cupy_mesh</span>&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t36" href="#t36">36</a></span><span class="t">        <span class="key">from</span> <span class="nam">pySDC</span><span class="op">.</span><span class="nam">implementations</span><span class="op">.</span><span class="nam">datatype_classes</span><span class="op">.</span><span class="nam">mesh</span> <span class="key">import</span> <span class="nam">mesh</span><span class="op">,</span> <span class="nam">imex_mesh</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t37" href="#t37">37</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t38" href="#t38">38</a></span><span class="t">        <span class="nam">cls</span><span class="op">.</span><span class="nam">dtype_u</span> <span class="op">=</span> <span class="nam">cupy_mesh</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t39" href="#t39">39</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t40" href="#t40">40</a></span><span class="t">        <span class="nam">GPU_versions</span> <span class="op">=</span> <span class="op">{</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t41" href="#t41">41</a></span><span class="t">            <span class="nam">mesh</span><span class="op">:</span> <span class="nam">cupy_mesh</span><span class="op">,</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t42" href="#t42">42</a></span><span class="t">            <span class="nam">imex_mesh</span><span class="op">:</span> <span class="nam">imex_cupy_mesh</span><span class="op">,</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t43" href="#t43">43</a></span><span class="t">        <span class="op">}</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t44" href="#t44">44</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t45" href="#t45">45</a></span><span class="t">        <span class="nam">cls</span><span class="op">.</span><span class="nam">dtype_f</span> <span class="op">=</span> <span class="nam">GPU_versions</span><span class="op">[</span><span class="nam">cls</span><span class="op">.</span><span class="nam">dtype_f</span><span class="op">]</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t46" href="#t46">46</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t47" href="#t47">47</a></span><span class="t">    <span class="key">def</span> <span class="nam">__init__</span><span class="op">(</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t48" href="#t48">48</a></span><span class="t">        <span class="nam">self</span><span class="op">,</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t49" href="#t49">49</a></span><span class="t">        <span class="nam">bases</span><span class="op">,</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t50" href="#t50">50</a></span><span class="t">        <span class="nam">components</span><span class="op">,</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t51" href="#t51">51</a></span><span class="t">        <span class="nam">comm</span><span class="op">=</span><span class="key">None</span><span class="op">,</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t52" href="#t52">52</a></span><span class="t">        <span class="nam">Dirichlet_recombination</span><span class="op">=</span><span class="key">True</span><span class="op">,</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t53" href="#t53">53</a></span><span class="t">        <span class="nam">left_preconditioner</span><span class="op">=</span><span class="key">True</span><span class="op">,</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t54" href="#t54">54</a></span><span class="t">        <span class="nam">solver_type</span><span class="op">=</span><span class="str">'cached_direct'</span><span class="op">,</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t55" href="#t55">55</a></span><span class="t">        <span class="nam">solver_args</span><span class="op">=</span><span class="key">None</span><span class="op">,</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t56" href="#t56">56</a></span><span class="t">        <span class="nam">useGPU</span><span class="op">=</span><span class="key">False</span><span class="op">,</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t57" href="#t57">57</a></span><span class="t">        <span class="nam">max_cached_factorizations</span><span class="op">=</span><span class="num">12</span><span class="op">,</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t58" href="#t58">58</a></span><span class="t">        <span class="nam">spectral_space</span><span class="op">=</span><span class="key">True</span><span class="op">,</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t59" href="#t59">59</a></span><span class="t">        <span class="nam">real_spectral_coefficients</span><span class="op">=</span><span class="key">False</span><span class="op">,</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t60" href="#t60">60</a></span><span class="t">        <span class="nam">debug</span><span class="op">=</span><span class="key">False</span><span class="op">,</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t61" href="#t61">61</a></span><span class="t">    <span class="op">)</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t62" href="#t62">62</a></span><span class="t">        <span class="str">"""</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t63" href="#t63">63</a></span><span class="t"><span class="str">        Base class for problems discretized with spectral methods.</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t64" href="#t64">64</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t65" href="#t65">65</a></span><span class="t"><span class="str">        Args:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t66" href="#t66">66</a></span><span class="t"><span class="str">            bases (list of dictionaries): 1D Bases</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t67" href="#t67">67</a></span><span class="t"><span class="str">            components (list of strings): Components of the equations</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t68" href="#t68">68</a></span><span class="t"><span class="str">            comm (mpi4py.Intracomm or None): MPI communicator</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t69" href="#t69">69</a></span><span class="t"><span class="str">            Dirichlet_recombination (bool): Use Dirichlet recombination in the last axis as right preconditioner</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t70" href="#t70">70</a></span><span class="t"><span class="str">            left_preconditioner (bool): Reverse the Kronecker product if yes</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t71" href="#t71">71</a></span><span class="t"><span class="str">            solver_type (str): Solver for linear systems</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t72" href="#t72">72</a></span><span class="t"><span class="str">            solver_args (dict): Arguments for linear solver</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t73" href="#t73">73</a></span><span class="t"><span class="str">            useGPU (bool): Run on GPU or CPU</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t74" href="#t74">74</a></span><span class="t"><span class="str">            max_cached_factorizations (int): Number of matrix decompositions to cache before starting eviction</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t75" href="#t75">75</a></span><span class="t"><span class="str">            spectral_space (bool): If yes, the solution will not be transformed back after solving and evaluating the RHS, and is expected as input in spectral space to these functions</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t76" href="#t76">76</a></span><span class="t"><span class="str">            real_spectral_coefficients (bool): If yes, allow only real values in spectral space, otherwise, allow complex.</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t77" href="#t77">77</a></span><span class="t"><span class="str">            debug (bool): Make additional tests at extra computational cost</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t78" href="#t78">78</a></span><span class="t"><span class="str">        """</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t79" href="#t79">79</a></span><span class="t">        <span class="nam">solver_args</span> <span class="op">=</span> <span class="op">{</span><span class="op">}</span> <span class="key">if</span> <span class="nam">solver_args</span> <span class="key">is</span> <span class="key">None</span> <span class="key">else</span> <span class="nam">solver_args</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t80" href="#t80">80</a></span><span class="t">        <span class="nam">self</span><span class="op">.</span><span class="nam">_makeAttributeAndRegister</span><span class="op">(</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t81" href="#t81">81</a></span><span class="t">            <span class="str">'max_cached_factorizations'</span><span class="op">,</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t82" href="#t82">82</a></span><span class="t">            <span class="str">'useGPU'</span><span class="op">,</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t83" href="#t83">83</a></span><span class="t">            <span class="str">'solver_type'</span><span class="op">,</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t84" href="#t84">84</a></span><span class="t">            <span class="str">'solver_args'</span><span class="op">,</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t85" href="#t85">85</a></span><span class="t">            <span class="str">'left_preconditioner'</span><span class="op">,</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t86" href="#t86">86</a></span><span class="t">            <span class="str">'Dirichlet_recombination'</span><span class="op">,</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t87" href="#t87">87</a></span><span class="t">            <span class="str">'comm'</span><span class="op">,</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t88" href="#t88">88</a></span><span class="t">            <span class="str">'spectral_space'</span><span class="op">,</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t89" href="#t89">89</a></span><span class="t">            <span class="str">'real_spectral_coefficients'</span><span class="op">,</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t90" href="#t90">90</a></span><span class="t">            <span class="str">'debug'</span><span class="op">,</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t91" href="#t91">91</a></span><span class="t">            <span class="nam">localVars</span><span class="op">=</span><span class="nam">locals</span><span class="op">(</span><span class="op">)</span><span class="op">,</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t92" href="#t92">92</a></span><span class="t">        <span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t93" href="#t93">93</a></span><span class="t">        <span class="nam">self</span><span class="op">.</span><span class="nam">spectral</span> <span class="op">=</span> <span class="nam">SpectralHelper</span><span class="op">(</span><span class="nam">comm</span><span class="op">=</span><span class="nam">comm</span><span class="op">,</span> <span class="nam">useGPU</span><span class="op">=</span><span class="nam">useGPU</span><span class="op">,</span> <span class="nam">debug</span><span class="op">=</span><span class="nam">debug</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t94" href="#t94">94</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t95" href="#t95">95</a></span><span class="t">        <span class="key">if</span> <span class="nam">useGPU</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t96" href="#t96">96</a></span><span class="t">            <span class="nam">self</span><span class="op">.</span><span class="nam">setup_GPU</span><span class="op">(</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t97" href="#t97">97</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t98" href="#t98">98</a></span><span class="t">        <span class="key">for</span> <span class="nam">base</span> <span class="key">in</span> <span class="nam">bases</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t99" href="#t99">99</a></span><span class="t">            <span class="nam">self</span><span class="op">.</span><span class="nam">spectral</span><span class="op">.</span><span class="nam">add_axis</span><span class="op">(</span><span class="op">**</span><span class="nam">base</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t100" href="#t100">100</a></span><span class="t">        <span class="nam">self</span><span class="op">.</span><span class="nam">spectral</span><span class="op">.</span><span class="nam">add_component</span><span class="op">(</span><span class="nam">components</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t101" href="#t101">101</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t102" href="#t102">102</a></span><span class="t">        <span class="nam">self</span><span class="op">.</span><span class="nam">spectral</span><span class="op">.</span><span class="nam">setup_fft</span><span class="op">(</span><span class="nam">real_spectral_coefficients</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t103" href="#t103">103</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t104" href="#t104">104</a></span><span class="t">        <span class="nam">super</span><span class="op">(</span><span class="op">)</span><span class="op">.</span><span class="nam">__init__</span><span class="op">(</span><span class="nam">init</span><span class="op">=</span><span class="nam">self</span><span class="op">.</span><span class="nam">spectral</span><span class="op">.</span><span class="nam">init_forward</span> <span class="key">if</span> <span class="nam">spectral_space</span> <span class="key">else</span> <span class="nam">self</span><span class="op">.</span><span class="nam">spectral</span><span class="op">.</span><span class="nam">init</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t105" href="#t105">105</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t106" href="#t106">106</a></span><span class="t">        <span class="nam">self</span><span class="op">.</span><span class="nam">work_counters</span><span class="op">[</span><span class="nam">solver_type</span><span class="op">]</span> <span class="op">=</span> <span class="nam">WorkCounter</span><span class="op">(</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t107" href="#t107">107</a></span><span class="t">        <span class="nam">self</span><span class="op">.</span><span class="nam">work_counters</span><span class="op">[</span><span class="str">'factorizations'</span><span class="op">]</span> <span class="op">=</span> <span class="nam">WorkCounter</span><span class="op">(</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t108" href="#t108">108</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t109" href="#t109">109</a></span><span class="t">        <span class="nam">self</span><span class="op">.</span><span class="nam">setup_preconditioner</span><span class="op">(</span><span class="nam">Dirichlet_recombination</span><span class="op">,</span> <span class="nam">left_preconditioner</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t110" href="#t110">110</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t111" href="#t111">111</a></span><span class="t">        <span class="nam">self</span><span class="op">.</span><span class="nam">cached_factorizations</span> <span class="op">=</span> <span class="op">{</span><span class="op">}</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t112" href="#t112">112</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t113" href="#t113">113</a></span><span class="t">    <span class="key">def</span> <span class="nam">__getattr__</span><span class="op">(</span><span class="nam">self</span><span class="op">,</span> <span class="nam">name</span><span class="op">)</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t114" href="#t114">114</a></span><span class="t">        <span class="str">"""</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t115" href="#t115">115</a></span><span class="t"><span class="str">        Pass requests on to the helper if they are not directly attributes of this class for convenience.</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t116" href="#t116">116</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t117" href="#t117">117</a></span><span class="t"><span class="str">        Args:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t118" href="#t118">118</a></span><span class="t"><span class="str">            name (str): Name of the attribute you want</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t119" href="#t119">119</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t120" href="#t120">120</a></span><span class="t"><span class="str">        Returns:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t121" href="#t121">121</a></span><span class="t"><span class="str">            request</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t122" href="#t122">122</a></span><span class="t"><span class="str">        """</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t123" href="#t123">123</a></span><span class="t">        <span class="key">return</span> <span class="nam">getattr</span><span class="op">(</span><span class="nam">self</span><span class="op">.</span><span class="nam">spectral</span><span class="op">,</span> <span class="nam">name</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t124" href="#t124">124</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t125" href="#t125">125</a></span><span class="t">    <span class="key">def</span> <span class="nam">_setup_operator</span><span class="op">(</span><span class="nam">self</span><span class="op">,</span> <span class="nam">LHS</span><span class="op">)</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t126" href="#t126">126</a></span><span class="t">        <span class="str">"""</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t127" href="#t127">127</a></span><span class="t"><span class="str">        Setup a sparse linear operator by adding relationships. See documentation for ``GenericSpectralLinear.setup_L`` to learn more.</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t128" href="#t128">128</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t129" href="#t129">129</a></span><span class="t"><span class="str">        Args:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t130" href="#t130">130</a></span><span class="t"><span class="str">            LHS (dict): Equations to be added to the operator</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t131" href="#t131">131</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t132" href="#t132">132</a></span><span class="t"><span class="str">        Returns:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t133" href="#t133">133</a></span><span class="t"><span class="str">            sparse linear operator</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t134" href="#t134">134</a></span><span class="t"><span class="str">        """</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t135" href="#t135">135</a></span><span class="t">        <span class="nam">operator</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">spectral</span><span class="op">.</span><span class="nam">get_empty_operator_matrix</span><span class="op">(</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t136" href="#t136">136</a></span><span class="t">        <span class="key">for</span> <span class="nam">line</span><span class="op">,</span> <span class="nam">equation</span> <span class="key">in</span> <span class="nam">LHS</span><span class="op">.</span><span class="nam">items</span><span class="op">(</span><span class="op">)</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t137" href="#t137">137</a></span><span class="t">            <span class="nam">self</span><span class="op">.</span><span class="nam">spectral</span><span class="op">.</span><span class="nam">add_equation_lhs</span><span class="op">(</span><span class="nam">operator</span><span class="op">,</span> <span class="nam">line</span><span class="op">,</span> <span class="nam">equation</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t138" href="#t138">138</a></span><span class="t">        <span class="key">return</span> <span class="nam">self</span><span class="op">.</span><span class="nam">spectral</span><span class="op">.</span><span class="nam">convert_operator_matrix_to_operator</span><span class="op">(</span><span class="nam">operator</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t139" href="#t139">139</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t140" href="#t140">140</a></span><span class="t">    <span class="key">def</span> <span class="nam">setup_L</span><span class="op">(</span><span class="nam">self</span><span class="op">,</span> <span class="nam">LHS</span><span class="op">)</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t141" href="#t141">141</a></span><span class="t">        <span class="str">"""</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t142" href="#t142">142</a></span><span class="t"><span class="str">        Setup the left hand side of the linear operator L and store it in ``self.L``.</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t143" href="#t143">143</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t144" href="#t144">144</a></span><span class="t"><span class="str">        The argument is meant to be a dictionary with the line you want to write the equation in as the key and the relationship between components as another dictionary. For instance, you can add an algebraic condition capturing a first derivative relationship between u and ux as follows:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t145" href="#t145">145</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t146" href="#t146">146</a></span><span class="t"><span class="str">        ```</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t147" href="#t147">147</a></span><span class="t"><span class="str">        Dx = self.get_differentiation_matrix(axes=(0,))</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t148" href="#t148">148</a></span><span class="t"><span class="str">        I = self.get_Id()</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t149" href="#t149">149</a></span><span class="t"><span class="str">        LHS = {'ux': {'u': Dx, 'ux': -I}}</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t150" href="#t150">150</a></span><span class="t"><span class="str">        self.setup_L(LHS)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t151" href="#t151">151</a></span><span class="t"><span class="str">        ```</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t152" href="#t152">152</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t153" href="#t153">153</a></span><span class="t"><span class="str">        If you put zero as right hand side for the solver in the line for ux, ux will contain the x-derivative of u afterwards.</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t154" href="#t154">154</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t155" href="#t155">155</a></span><span class="t"><span class="str">        Args:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t156" href="#t156">156</a></span><span class="t"><span class="str">            LHS (dict): Dictionary containing the equations.</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t157" href="#t157">157</a></span><span class="t"><span class="str">        """</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t158" href="#t158">158</a></span><span class="t">        <span class="nam">self</span><span class="op">.</span><span class="nam">L</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">_setup_operator</span><span class="op">(</span><span class="nam">LHS</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t159" href="#t159">159</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t160" href="#t160">160</a></span><span class="t">    <span class="key">def</span> <span class="nam">setup_M</span><span class="op">(</span><span class="nam">self</span><span class="op">,</span> <span class="nam">LHS</span><span class="op">)</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t161" href="#t161">161</a></span><span class="t">        <span class="str">'''</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t162" href="#t162">162</a></span><span class="t"><span class="str">        Setup mass matrix, see documentation of ``GenericSpectralLinear.setup_L``.</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t163" href="#t163">163</a></span><span class="t"><span class="str">        '''</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t164" href="#t164">164</a></span><span class="t">        <span class="nam">diff_index</span> <span class="op">=</span> <span class="nam">list</span><span class="op">(</span><span class="nam">LHS</span><span class="op">.</span><span class="nam">keys</span><span class="op">(</span><span class="op">)</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t165" href="#t165">165</a></span><span class="t">        <span class="nam">self</span><span class="op">.</span><span class="nam">diff_mask</span> <span class="op">=</span> <span class="op">[</span><span class="nam">me</span> <span class="key">in</span> <span class="nam">diff_index</span> <span class="key">for</span> <span class="nam">me</span> <span class="key">in</span> <span class="nam">self</span><span class="op">.</span><span class="nam">components</span><span class="op">]</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t166" href="#t166">166</a></span><span class="t">        <span class="nam">self</span><span class="op">.</span><span class="nam">M</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">_setup_operator</span><span class="op">(</span><span class="nam">LHS</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t167" href="#t167">167</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t168" href="#t168">168</a></span><span class="t">    <span class="key">def</span> <span class="nam">setup_preconditioner</span><span class="op">(</span><span class="nam">self</span><span class="op">,</span> <span class="nam">Dirichlet_recombination</span><span class="op">=</span><span class="key">True</span><span class="op">,</span> <span class="nam">left_preconditioner</span><span class="op">=</span><span class="key">True</span><span class="op">)</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t169" href="#t169">169</a></span><span class="t">        <span class="str">"""</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t170" href="#t170">170</a></span><span class="t"><span class="str">        Get left and right preconditioners.</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t171" href="#t171">171</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t172" href="#t172">172</a></span><span class="t"><span class="str">        Args:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t173" href="#t173">173</a></span><span class="t"><span class="str">            Dirichlet_recombination (bool): Basis conversion for right preconditioner. Useful for Chebychev and Ultraspherical methods. 10/10 would recommend.</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t174" href="#t174">174</a></span><span class="t"><span class="str">            left_preconditioner (bool): If True, it will interleave the variables and reverse the Kronecker product</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t175" href="#t175">175</a></span><span class="t"><span class="str">        """</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t176" href="#t176">176</a></span><span class="t">        <span class="nam">sp</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">spectral</span><span class="op">.</span><span class="nam">sparse_lib</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t177" href="#t177">177</a></span><span class="t">        <span class="nam">N</span> <span class="op">=</span> <span class="nam">np</span><span class="op">.</span><span class="nam">prod</span><span class="op">(</span><span class="nam">self</span><span class="op">.</span><span class="nam">init</span><span class="op">[</span><span class="num">0</span><span class="op">]</span><span class="op">[</span><span class="num">1</span><span class="op">:</span><span class="op">]</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t178" href="#t178">178</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t179" href="#t179">179</a></span><span class="t">        <span class="nam">Id</span> <span class="op">=</span> <span class="nam">sp</span><span class="op">.</span><span class="nam">eye</span><span class="op">(</span><span class="nam">N</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t180" href="#t180">180</a></span><span class="t">        <span class="nam">Pl_lhs</span> <span class="op">=</span> <span class="op">{</span><span class="nam">comp</span><span class="op">:</span> <span class="op">{</span><span class="nam">comp</span><span class="op">:</span> <span class="nam">Id</span><span class="op">}</span> <span class="key">for</span> <span class="nam">comp</span> <span class="key">in</span> <span class="nam">self</span><span class="op">.</span><span class="nam">components</span><span class="op">}</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t181" href="#t181">181</a></span><span class="t">        <span class="nam">self</span><span class="op">.</span><span class="nam">Pl</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">_setup_operator</span><span class="op">(</span><span class="nam">Pl_lhs</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t182" href="#t182">182</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t183" href="#t183">183</a></span><span class="t">        <span class="key">if</span> <span class="nam">left_preconditioner</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t184" href="#t184">184</a></span><span class="t">            <span class="com"># reverse Kronecker product</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t185" href="#t185">185</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t186" href="#t186">186</a></span><span class="t">            <span class="key">if</span> <span class="nam">self</span><span class="op">.</span><span class="nam">spectral</span><span class="op">.</span><span class="nam">useGPU</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t187" href="#t187">187</a></span><span class="t">                <span class="nam">R</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">Pl</span><span class="op">.</span><span class="nam">get</span><span class="op">(</span><span class="op">)</span><span class="op">.</span><span class="nam">tolil</span><span class="op">(</span><span class="op">)</span> <span class="op">*</span> <span class="num">0</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t188" href="#t188">188</a></span><span class="t">            <span class="key">else</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t189" href="#t189">189</a></span><span class="t">                <span class="nam">R</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">Pl</span><span class="op">.</span><span class="nam">tolil</span><span class="op">(</span><span class="op">)</span> <span class="op">*</span> <span class="num">0</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t190" href="#t190">190</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t191" href="#t191">191</a></span><span class="t">            <span class="key">for</span> <span class="nam">j</span> <span class="key">in</span> <span class="nam">range</span><span class="op">(</span><span class="nam">self</span><span class="op">.</span><span class="nam">ncomponents</span><span class="op">)</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t192" href="#t192">192</a></span><span class="t">                <span class="key">for</span> <span class="nam">i</span> <span class="key">in</span> <span class="nam">range</span><span class="op">(</span><span class="nam">N</span><span class="op">)</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t193" href="#t193">193</a></span><span class="t">                    <span class="nam">R</span><span class="op">[</span><span class="nam">i</span> <span class="op">*</span> <span class="nam">self</span><span class="op">.</span><span class="nam">ncomponents</span> <span class="op">+</span> <span class="nam">j</span><span class="op">,</span> <span class="nam">j</span> <span class="op">*</span> <span class="nam">N</span> <span class="op">+</span> <span class="nam">i</span><span class="op">]</span> <span class="op">=</span> <span class="num">1.0</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t194" href="#t194">194</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t195" href="#t195">195</a></span><span class="t">            <span class="nam">self</span><span class="op">.</span><span class="nam">Pl</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">spectral</span><span class="op">.</span><span class="nam">sparse_lib</span><span class="op">.</span><span class="nam">csc_matrix</span><span class="op">(</span><span class="nam">R</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t196" href="#t196">196</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t197" href="#t197">197</a></span><span class="t">        <span class="key">if</span> <span class="nam">Dirichlet_recombination</span> <span class="key">and</span> <span class="nam">type</span><span class="op">(</span><span class="nam">self</span><span class="op">.</span><span class="nam">axes</span><span class="op">[</span><span class="op">-</span><span class="num">1</span><span class="op">]</span><span class="op">)</span><span class="op">.</span><span class="nam">__name__</span> <span class="key">in</span> <span class="op">[</span><span class="str">'ChebychevHelper, Ultraspherical'</span><span class="op">]</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t198" href="#t198">198</a></span><span class="t">            <span class="nam">_Pr</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">spectral</span><span class="op">.</span><span class="nam">get_Dirichlet_recombination_matrix</span><span class="op">(</span><span class="nam">axis</span><span class="op">=</span><span class="op">-</span><span class="num">1</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t199" href="#t199">199</a></span><span class="t">        <span class="key">else</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t200" href="#t200">200</a></span><span class="t">            <span class="nam">_Pr</span> <span class="op">=</span> <span class="nam">Id</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t201" href="#t201">201</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t202" href="#t202">202</a></span><span class="t">        <span class="nam">Pr_lhs</span> <span class="op">=</span> <span class="op">{</span><span class="nam">comp</span><span class="op">:</span> <span class="op">{</span><span class="nam">comp</span><span class="op">:</span> <span class="nam">_Pr</span><span class="op">}</span> <span class="key">for</span> <span class="nam">comp</span> <span class="key">in</span> <span class="nam">self</span><span class="op">.</span><span class="nam">components</span><span class="op">}</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t203" href="#t203">203</a></span><span class="t">        <span class="nam">self</span><span class="op">.</span><span class="nam">Pr</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">_setup_operator</span><span class="op">(</span><span class="nam">Pr_lhs</span><span class="op">)</span> <span class="op">@</span> <span class="nam">self</span><span class="op">.</span><span class="nam">Pl</span><span class="op">.</span><span class="nam">T</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t204" href="#t204">204</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t205" href="#t205">205</a></span><span class="t">    <span class="key">def</span> <span class="nam">solve_system</span><span class="op">(</span><span class="nam">self</span><span class="op">,</span> <span class="nam">rhs</span><span class="op">,</span> <span class="nam">dt</span><span class="op">,</span> <span class="nam">u0</span><span class="op">=</span><span class="key">None</span><span class="op">,</span> <span class="op">*</span><span class="nam">args</span><span class="op">,</span> <span class="nam">skip_itransform</span><span class="op">=</span><span class="key">False</span><span class="op">,</span> <span class="op">**</span><span class="nam">kwargs</span><span class="op">)</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t206" href="#t206">206</a></span><span class="t">        <span class="str">"""</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t207" href="#t207">207</a></span><span class="t"><span class="str">        Do an implicit Euler step to solve M u_t + Lu = rhs, with M the mass matrix and L the linear operator as setup by</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t208" href="#t208">208</a></span><span class="t"><span class="str">        ``GenericSpectralLinear.setup_L`` and ``GenericSpectralLinear.setup_M``.</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t209" href="#t209">209</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t210" href="#t210">210</a></span><span class="t"><span class="str">        The implicit Euler step is (M - dt L) u = M rhs. Note that M need not be invertible as long as (M + dt*L) is.</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t211" href="#t211">211</a></span><span class="t"><span class="str">        This means solving with dt=0 to mimic explicit methods does not work for all problems, in particular simple DAEs.</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t212" href="#t212">212</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t213" href="#t213">213</a></span><span class="t"><span class="str">        Note that by putting M rhs on the right hand side, this function can only solve algebraic conditions equal to</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t214" href="#t214">214</a></span><span class="t"><span class="str">        zero. If you want something else, it should be easy to overload this function.</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t215" href="#t215">215</a></span><span class="t"><span class="str">        """</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t216" href="#t216">216</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t217" href="#t217">217</a></span><span class="t">        <span class="nam">sp</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">spectral</span><span class="op">.</span><span class="nam">sparse_lib</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t218" href="#t218">218</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t219" href="#t219">219</a></span><span class="t">        <span class="key">if</span> <span class="nam">self</span><span class="op">.</span><span class="nam">spectral_space</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t220" href="#t220">220</a></span><span class="t">            <span class="nam">rhs_hat</span> <span class="op">=</span> <span class="nam">rhs</span><span class="op">.</span><span class="nam">copy</span><span class="op">(</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t221" href="#t221">221</a></span><span class="t">        <span class="key">else</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t222" href="#t222">222</a></span><span class="t">            <span class="nam">rhs_hat</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">spectral</span><span class="op">.</span><span class="nam">transform</span><span class="op">(</span><span class="nam">rhs</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t223" href="#t223">223</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t224" href="#t224">224</a></span><span class="t">        <span class="nam">rhs_hat</span> <span class="op">=</span> <span class="op">(</span><span class="nam">self</span><span class="op">.</span><span class="nam">M</span> <span class="op">@</span> <span class="nam">rhs_hat</span><span class="op">.</span><span class="nam">flatten</span><span class="op">(</span><span class="op">)</span><span class="op">)</span><span class="op">.</span><span class="nam">reshape</span><span class="op">(</span><span class="nam">rhs_hat</span><span class="op">.</span><span class="nam">shape</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t225" href="#t225">225</a></span><span class="t">        <span class="nam">rhs_hat</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">spectral</span><span class="op">.</span><span class="nam">put_BCs_in_rhs_hat</span><span class="op">(</span><span class="nam">rhs_hat</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t226" href="#t226">226</a></span><span class="t">        <span class="nam">rhs_hat</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">Pl</span> <span class="op">@</span> <span class="nam">rhs_hat</span><span class="op">.</span><span class="nam">flatten</span><span class="op">(</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t227" href="#t227">227</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t228" href="#t228">228</a></span><span class="t">        <span class="key">if</span> <span class="nam">dt</span> <span class="key">not</span> <span class="key">in</span> <span class="nam">self</span><span class="op">.</span><span class="nam">cached_factorizations</span><span class="op">.</span><span class="nam">keys</span><span class="op">(</span><span class="op">)</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t229" href="#t229">229</a></span><span class="t">            <span class="nam">A</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">M</span> <span class="op">+</span> <span class="nam">dt</span> <span class="op">*</span> <span class="nam">self</span><span class="op">.</span><span class="nam">L</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t230" href="#t230">230</a></span><span class="t">            <span class="nam">A</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">Pl</span> <span class="op">@</span> <span class="nam">self</span><span class="op">.</span><span class="nam">spectral</span><span class="op">.</span><span class="nam">put_BCs_in_matrix</span><span class="op">(</span><span class="nam">A</span><span class="op">)</span> <span class="op">@</span> <span class="nam">self</span><span class="op">.</span><span class="nam">Pr</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t231" href="#t231">231</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t232" href="#t232">232</a></span><span class="t">        <span class="com"># import numpy as np</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t233" href="#t233">233</a></span><span class="t">        <span class="com"># if A.shape[0] &lt; 200:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t234" href="#t234">234</a></span><span class="t">        <span class="com">#     import matplotlib.pyplot as plt</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t235" href="#t235">235</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t236" href="#t236">236</a></span><span class="t">        <span class="com">#     # M = self.spectral.put_BCs_in_matrix(self.L.copy())</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t237" href="#t237">237</a></span><span class="t">        <span class="com">#     M = A  # self.L</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t238" href="#t238">238</a></span><span class="t">        <span class="com">#     im = plt.imshow((M / abs(M)).real)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t239" href="#t239">239</a></span><span class="t">        <span class="com">#     # im = plt.imshow(np.log10(abs(A.toarray())).real)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t240" href="#t240">240</a></span><span class="t">        <span class="com">#     # im = plt.imshow(((A.toarray())).real)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t241" href="#t241">241</a></span><span class="t">        <span class="com">#     plt.colorbar(im)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t242" href="#t242">242</a></span><span class="t">        <span class="com">#     plt.show()</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t243" href="#t243">243</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t244" href="#t244">244</a></span><span class="t">        <span class="key">if</span> <span class="nam">self</span><span class="op">.</span><span class="nam">solver_type</span><span class="op">.</span><span class="nam">lower</span><span class="op">(</span><span class="op">)</span> <span class="op">==</span> <span class="str">'cached_direct'</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t245" href="#t245">245</a></span><span class="t">            <span class="key">if</span> <span class="nam">dt</span> <span class="key">not</span> <span class="key">in</span> <span class="nam">self</span><span class="op">.</span><span class="nam">cached_factorizations</span><span class="op">.</span><span class="nam">keys</span><span class="op">(</span><span class="op">)</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t246" href="#t246">246</a></span><span class="t">                <span class="key">if</span> <span class="nam">len</span><span class="op">(</span><span class="nam">self</span><span class="op">.</span><span class="nam">cached_factorizations</span><span class="op">)</span> <span class="op">>=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">max_cached_factorizations</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t247" href="#t247">247</a></span><span class="t">                    <span class="nam">self</span><span class="op">.</span><span class="nam">cached_factorizations</span><span class="op">.</span><span class="nam">pop</span><span class="op">(</span><span class="nam">list</span><span class="op">(</span><span class="nam">self</span><span class="op">.</span><span class="nam">cached_factorizations</span><span class="op">.</span><span class="nam">keys</span><span class="op">(</span><span class="op">)</span><span class="op">)</span><span class="op">[</span><span class="num">0</span><span class="op">]</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t248" href="#t248">248</a></span><span class="t">                    <span class="nam">self</span><span class="op">.</span><span class="nam">logger</span><span class="op">.</span><span class="nam">debug</span><span class="op">(</span><span class="fst">f'</span><span class="fst">Evicted matrix factorization for </span><span class="op">{</span><span class="nam">dt</span><span class="op">=</span><span class="op">:</span><span class="fst">.6f</span><span class="op">}</span><span class="fst"> from cache</span><span class="fst">'</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t249" href="#t249">249</a></span><span class="t">                <span class="nam">self</span><span class="op">.</span><span class="nam">cached_factorizations</span><span class="op">[</span><span class="nam">dt</span><span class="op">]</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">spectral</span><span class="op">.</span><span class="nam">linalg</span><span class="op">.</span><span class="nam">factorized</span><span class="op">(</span><span class="nam">A</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t250" href="#t250">250</a></span><span class="t">                <span class="nam">self</span><span class="op">.</span><span class="nam">logger</span><span class="op">.</span><span class="nam">debug</span><span class="op">(</span><span class="fst">f'</span><span class="fst">Cached matrix factorization for </span><span class="op">{</span><span class="nam">dt</span><span class="op">=</span><span class="op">:</span><span class="fst">.6f</span><span class="op">}</span><span class="fst">'</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t251" href="#t251">251</a></span><span class="t">                <span class="nam">self</span><span class="op">.</span><span class="nam">work_counters</span><span class="op">[</span><span class="str">'factorizations'</span><span class="op">]</span><span class="op">(</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t252" href="#t252">252</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t253" href="#t253">253</a></span><span class="t">            <span class="nam">_sol_hat</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">cached_factorizations</span><span class="op">[</span><span class="nam">dt</span><span class="op">]</span><span class="op">(</span><span class="nam">rhs_hat</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t254" href="#t254">254</a></span><span class="t">            <span class="nam">self</span><span class="op">.</span><span class="nam">logger</span><span class="op">.</span><span class="nam">debug</span><span class="op">(</span><span class="fst">f'</span><span class="fst">Used cached matrix factorization for </span><span class="op">{</span><span class="nam">dt</span><span class="op">=</span><span class="op">:</span><span class="fst">.6f</span><span class="op">}</span><span class="fst">'</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t255" href="#t255">255</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t256" href="#t256">256</a></span><span class="t">        <span class="key">elif</span> <span class="nam">self</span><span class="op">.</span><span class="nam">solver_type</span><span class="op">.</span><span class="nam">lower</span><span class="op">(</span><span class="op">)</span> <span class="op">==</span> <span class="str">'direct'</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t257" href="#t257">257</a></span><span class="t">            <span class="nam">_sol_hat</span> <span class="op">=</span> <span class="nam">sp</span><span class="op">.</span><span class="nam">linalg</span><span class="op">.</span><span class="nam">spsolve</span><span class="op">(</span><span class="nam">A</span><span class="op">,</span> <span class="nam">rhs_hat</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t258" href="#t258">258</a></span><span class="t">        <span class="key">elif</span> <span class="nam">self</span><span class="op">.</span><span class="nam">solver_type</span><span class="op">.</span><span class="nam">lower</span><span class="op">(</span><span class="op">)</span> <span class="op">==</span> <span class="str">'gmres'</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t259" href="#t259">259</a></span><span class="t">            <span class="nam">_sol_hat</span><span class="op">,</span> <span class="nam">_</span> <span class="op">=</span> <span class="nam">sp</span><span class="op">.</span><span class="nam">linalg</span><span class="op">.</span><span class="nam">gmres</span><span class="op">(</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t260" href="#t260">260</a></span><span class="t">                <span class="nam">A</span><span class="op">,</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t261" href="#t261">261</a></span><span class="t">                <span class="nam">rhs_hat</span><span class="op">,</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t262" href="#t262">262</a></span><span class="t">                <span class="nam">x0</span><span class="op">=</span><span class="nam">u0</span><span class="op">.</span><span class="nam">flatten</span><span class="op">(</span><span class="op">)</span><span class="op">,</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t263" href="#t263">263</a></span><span class="t">                <span class="op">**</span><span class="nam">self</span><span class="op">.</span><span class="nam">solver_args</span><span class="op">,</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t264" href="#t264">264</a></span><span class="t">                <span class="nam">callback</span><span class="op">=</span><span class="nam">self</span><span class="op">.</span><span class="nam">work_counters</span><span class="op">[</span><span class="nam">self</span><span class="op">.</span><span class="nam">solver_type</span><span class="op">]</span><span class="op">,</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t265" href="#t265">265</a></span><span class="t">                <span class="nam">callback_type</span><span class="op">=</span><span class="str">'legacy'</span><span class="op">,</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t266" href="#t266">266</a></span><span class="t">            <span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t267" href="#t267">267</a></span><span class="t">        <span class="key">elif</span> <span class="nam">self</span><span class="op">.</span><span class="nam">solver_type</span><span class="op">.</span><span class="nam">lower</span><span class="op">(</span><span class="op">)</span> <span class="op">==</span> <span class="str">'cg'</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t268" href="#t268">268</a></span><span class="t">            <span class="nam">_sol_hat</span><span class="op">,</span> <span class="nam">_</span> <span class="op">=</span> <span class="nam">sp</span><span class="op">.</span><span class="nam">linalg</span><span class="op">.</span><span class="nam">cg</span><span class="op">(</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t269" href="#t269">269</a></span><span class="t">                <span class="nam">A</span><span class="op">,</span> <span class="nam">rhs_hat</span><span class="op">,</span> <span class="nam">x0</span><span class="op">=</span><span class="nam">u0</span><span class="op">.</span><span class="nam">flatten</span><span class="op">(</span><span class="op">)</span><span class="op">,</span> <span class="op">**</span><span class="nam">self</span><span class="op">.</span><span class="nam">solver_args</span><span class="op">,</span> <span class="nam">callback</span><span class="op">=</span><span class="nam">self</span><span class="op">.</span><span class="nam">work_counters</span><span class="op">[</span><span class="nam">self</span><span class="op">.</span><span class="nam">solver_type</span><span class="op">]</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t270" href="#t270">270</a></span><span class="t">            <span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t271" href="#t271">271</a></span><span class="t">        <span class="key">else</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t272" href="#t272">272</a></span><span class="t">            <span class="key">raise</span> <span class="nam">NotImplementedError</span><span class="op">(</span><span class="fst">f'</span><span class="fst">Solver </span><span class="op">{</span><span class="nam">self</span><span class="op">.</span><span class="nam">solver_type</span><span class="op">:</span><span class="fst">!</span><span class="op">}</span><span class="fst"> not implemented in </span><span class="op">{</span><span class="nam">type</span><span class="op">(</span><span class="nam">self</span><span class="op">)</span><span class="op">.</span><span class="nam">__name__</span><span class="op">}</span><span class="fst">!</span><span class="fst">'</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t273" href="#t273">273</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t274" href="#t274">274</a></span><span class="t">        <span class="nam">sol_hat</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">spectral</span><span class="op">.</span><span class="nam">u_init_forward</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t275" href="#t275">275</a></span><span class="t">        <span class="nam">sol_hat</span><span class="op">[</span><span class="op">...</span><span class="op">]</span> <span class="op">=</span> <span class="op">(</span><span class="nam">self</span><span class="op">.</span><span class="nam">Pr</span> <span class="op">@</span> <span class="nam">_sol_hat</span><span class="op">)</span><span class="op">.</span><span class="nam">reshape</span><span class="op">(</span><span class="nam">sol_hat</span><span class="op">.</span><span class="nam">shape</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t276" href="#t276">276</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t277" href="#t277">277</a></span><span class="t">        <span class="key">if</span> <span class="nam">self</span><span class="op">.</span><span class="nam">spectral_space</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t278" href="#t278">278</a></span><span class="t">            <span class="key">return</span> <span class="nam">sol_hat</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t279" href="#t279">279</a></span><span class="t">        <span class="key">else</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t280" href="#t280">280</a></span><span class="t">            <span class="nam">sol</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">spectral</span><span class="op">.</span><span class="nam">u_init</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t281" href="#t281">281</a></span><span class="t">            <span class="nam">sol</span><span class="op">[</span><span class="op">:</span><span class="op">]</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">spectral</span><span class="op">.</span><span class="nam">itransform</span><span class="op">(</span><span class="nam">sol_hat</span><span class="op">)</span><span class="op">.</span><span class="nam">real</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t282" href="#t282">282</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t283" href="#t283">283</a></span><span class="t">            <span class="key">if</span> <span class="nam">self</span><span class="op">.</span><span class="nam">spectral</span><span class="op">.</span><span class="nam">debug</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t284" href="#t284">284</a></span><span class="t">                <span class="nam">self</span><span class="op">.</span><span class="nam">spectral</span><span class="op">.</span><span class="nam">check_BCs</span><span class="op">(</span><span class="nam">sol</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t285" href="#t285">285</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t286" href="#t286">286</a></span><span class="t">            <span class="key">return</span> <span class="nam">sol</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t287" href="#t287">287</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t288" href="#t288">288</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t289" href="#t289">289</a></span><span class="t"><span class="key">def</span> <span class="nam">compute_residual_DAE</span><span class="op">(</span><span class="nam">self</span><span class="op">,</span> <span class="nam">stage</span><span class="op">=</span><span class="str">''</span><span class="op">)</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t290" href="#t290">290</a></span><span class="t">    <span class="str">"""</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t291" href="#t291">291</a></span><span class="t"><span class="str">    Computation of the residual that does not add u_0 - u_m in algebraic equations.</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t292" href="#t292">292</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t293" href="#t293">293</a></span><span class="t"><span class="str">    Args:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t294" href="#t294">294</a></span><span class="t"><span class="str">        stage (str): The current stage of the step the level belongs to</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t295" href="#t295">295</a></span><span class="t"><span class="str">    """</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t296" href="#t296">296</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t297" href="#t297">297</a></span><span class="t">    <span class="com"># get current level and problem description</span>&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t298" href="#t298">298</a></span><span class="t">    <span class="nam">L</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">level</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t299" href="#t299">299</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t300" href="#t300">300</a></span><span class="t">    <span class="com"># Check if we want to skip the residual computation to gain performance</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t301" href="#t301">301</a></span><span class="t">    <span class="com"># Keep in mind that skipping any residual computation is likely to give incorrect outputs of the residual!</span>&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t302" href="#t302">302</a></span><span class="t">    <span class="key">if</span> <span class="nam">stage</span> <span class="key">in</span> <span class="nam">self</span><span class="op">.</span><span class="nam">params</span><span class="op">.</span><span class="nam">skip_residual_computation</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t303" href="#t303">303</a></span><span class="t">        <span class="nam">L</span><span class="op">.</span><span class="nam">status</span><span class="op">.</span><span class="nam">residual</span> <span class="op">=</span> <span class="num">0.0</span> <span class="key">if</span> <span class="nam">L</span><span class="op">.</span><span class="nam">status</span><span class="op">.</span><span class="nam">residual</span> <span class="key">is</span> <span class="key">None</span> <span class="key">else</span> <span class="nam">L</span><span class="op">.</span><span class="nam">status</span><span class="op">.</span><span class="nam">residual</span>&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t304" href="#t304">304</a></span><span class="t">        <span class="key">return</span> <span class="key">None</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t305" href="#t305">305</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t306" href="#t306">306</a></span><span class="t">    <span class="com"># check if there are new values (e.g. from a sweep)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t307" href="#t307">307</a></span><span class="t">    <span class="com"># assert L.status.updated</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t308" href="#t308">308</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t309" href="#t309">309</a></span><span class="t">    <span class="com"># compute the residual for each node</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t310" href="#t310">310</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t311" href="#t311">311</a></span><span class="t">    <span class="com"># build QF(u)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t312" href="#t312">312</a></span><span class="t">    <span class="nam">res_norm</span> <span class="op">=</span> <span class="op">[</span><span class="op">]</span>&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t313" href="#t313">313</a></span><span class="t">    <span class="nam">res</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">integrate</span><span class="op">(</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t314" href="#t314">314</a></span><span class="t">    <span class="nam">mask</span> <span class="op">=</span> <span class="nam">L</span><span class="op">.</span><span class="nam">prob</span><span class="op">.</span><span class="nam">diff_mask</span>&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t315" href="#t315">315</a></span><span class="t">    <span class="key">for</span> <span class="nam">m</span> <span class="key">in</span> <span class="nam">range</span><span class="op">(</span><span class="nam">self</span><span class="op">.</span><span class="nam">coll</span><span class="op">.</span><span class="nam">num_nodes</span><span class="op">)</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t316" href="#t316">316</a></span><span class="t">        <span class="nam">res</span><span class="op">[</span><span class="nam">m</span><span class="op">]</span><span class="op">[</span><span class="nam">mask</span><span class="op">]</span> <span class="op">+=</span> <span class="nam">L</span><span class="op">.</span><span class="nam">u</span><span class="op">[</span><span class="num">0</span><span class="op">]</span><span class="op">[</span><span class="nam">mask</span><span class="op">]</span> <span class="op">-</span> <span class="nam">L</span><span class="op">.</span><span class="nam">u</span><span class="op">[</span><span class="nam">m</span> <span class="op">+</span> <span class="num">1</span><span class="op">]</span><span class="op">[</span><span class="nam">mask</span><span class="op">]</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t317" href="#t317">317</a></span><span class="t">        <span class="com"># add tau if associated</span>&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t318" href="#t318">318</a></span><span class="t">        <span class="key">if</span> <span class="nam">L</span><span class="op">.</span><span class="nam">tau</span><span class="op">[</span><span class="nam">m</span><span class="op">]</span> <span class="key">is</span> <span class="key">not</span> <span class="key">None</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t319" href="#t319">319</a></span><span class="t">            <span class="nam">res</span><span class="op">[</span><span class="nam">m</span><span class="op">]</span> <span class="op">+=</span> <span class="nam">L</span><span class="op">.</span><span class="nam">tau</span><span class="op">[</span><span class="nam">m</span><span class="op">]</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t320" href="#t320">320</a></span><span class="t">        <span class="com"># use abs function from data type here</span>&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t321" href="#t321">321</a></span><span class="t">        <span class="nam">res_norm</span><span class="op">.</span><span class="nam">append</span><span class="op">(</span><span class="nam">abs</span><span class="op">(</span><span class="nam">res</span><span class="op">[</span><span class="nam">m</span><span class="op">]</span><span class="op">)</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t322" href="#t322">322</a></span><span class="t">        <span class="com"># print(m, [abs(me) for me in res[m]], [abs(me) for me in L.u[0] - L.u[m + 1]])</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t323" href="#t323">323</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t324" href="#t324">324</a></span><span class="t">    <span class="com"># find maximal residual over the nodes</span>&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t325" href="#t325">325</a></span><span class="t">    <span class="key">if</span> <span class="nam">L</span><span class="op">.</span><span class="nam">params</span><span class="op">.</span><span class="nam">residual_type</span> <span class="op">==</span> <span class="str">'full_abs'</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t326" href="#t326">326</a></span><span class="t">        <span class="nam">L</span><span class="op">.</span><span class="nam">status</span><span class="op">.</span><span class="nam">residual</span> <span class="op">=</span> <span class="nam">max</span><span class="op">(</span><span class="nam">res_norm</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t327" href="#t327">327</a></span><span class="t">    <span class="key">elif</span> <span class="nam">L</span><span class="op">.</span><span class="nam">params</span><span class="op">.</span><span class="nam">residual_type</span> <span class="op">==</span> <span class="str">'last_abs'</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t328" href="#t328">328</a></span><span class="t">        <span class="nam">L</span><span class="op">.</span><span class="nam">status</span><span class="op">.</span><span class="nam">residual</span> <span class="op">=</span> <span class="nam">res_norm</span><span class="op">[</span><span class="op">-</span><span class="num">1</span><span class="op">]</span>&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t329" href="#t329">329</a></span><span class="t">    <span class="key">elif</span> <span class="nam">L</span><span class="op">.</span><span class="nam">params</span><span class="op">.</span><span class="nam">residual_type</span> <span class="op">==</span> <span class="str">'full_rel'</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t330" href="#t330">330</a></span><span class="t">        <span class="nam">L</span><span class="op">.</span><span class="nam">status</span><span class="op">.</span><span class="nam">residual</span> <span class="op">=</span> <span class="nam">max</span><span class="op">(</span><span class="nam">res_norm</span><span class="op">)</span> <span class="op">/</span> <span class="nam">abs</span><span class="op">(</span><span class="nam">L</span><span class="op">.</span><span class="nam">u</span><span class="op">[</span><span class="num">0</span><span class="op">]</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t331" href="#t331">331</a></span><span class="t">    <span class="key">elif</span> <span class="nam">L</span><span class="op">.</span><span class="nam">params</span><span class="op">.</span><span class="nam">residual_type</span> <span class="op">==</span> <span class="str">'last_rel'</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t332" href="#t332">332</a></span><span class="t">        <span class="nam">L</span><span class="op">.</span><span class="nam">status</span><span class="op">.</span><span class="nam">residual</span> <span class="op">=</span> <span class="nam">res_norm</span><span class="op">[</span><span class="op">-</span><span class="num">1</span><span class="op">]</span> <span class="op">/</span> <span class="nam">abs</span><span class="op">(</span><span class="nam">L</span><span class="op">.</span><span class="nam">u</span><span class="op">[</span><span class="num">0</span><span class="op">]</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t333" href="#t333">333</a></span><span class="t">    <span class="key">else</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t334" href="#t334">334</a></span><span class="t">        <span class="key">raise</span> <span class="nam">ParameterError</span><span class="op">(</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t335" href="#t335">335</a></span><span class="t">            <span class="fst">f'</span><span class="fst">residual_type = </span><span class="op">{</span><span class="nam">L</span><span class="op">.</span><span class="nam">params</span><span class="op">.</span><span class="nam">residual_type</span><span class="op">}</span><span class="fst"> not implemented, choose </span><span class="fst">'</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t336" href="#t336">336</a></span><span class="t">            <span class="fst">f'</span><span class="fst">full_abs, last_abs, full_rel or last_rel instead</span><span class="fst">'</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t337" href="#t337">337</a></span><span class="t">        <span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t338" href="#t338">338</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t339" href="#t339">339</a></span><span class="t">    <span class="com"># indicate that the residual has seen the new values</span>&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t340" href="#t340">340</a></span><span class="t">    <span class="nam">L</span><span class="op">.</span><span class="nam">status</span><span class="op">.</span><span class="nam">updated</span> <span class="op">=</span> <span class="key">False</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t341" href="#t341">341</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t342" href="#t342">342</a></span><span class="t">    <span class="key">return</span> <span class="key">None</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t343" href="#t343">343</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t344" href="#t344">344</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="run"><span class="n"><a id="t345" href="#t345">345</a></span><span class="t"><span class="key">def</span> <span class="nam">compute_residual_DAE_MPI</span><span class="op">(</span><span class="nam">self</span><span class="op">,</span> <span class="nam">stage</span><span class="op">=</span><span class="key">None</span><span class="op">)</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t346" href="#t346">346</a></span><span class="t">    <span class="str">"""</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t347" href="#t347">347</a></span><span class="t"><span class="str">    Computation of the residual using the collocation matrix Q</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t348" href="#t348">348</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t349" href="#t349">349</a></span><span class="t"><span class="str">    Args:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t350" href="#t350">350</a></span><span class="t"><span class="str">        stage (str): The current stage of the step the level belongs to</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t351" href="#t351">351</a></span><span class="t"><span class="str">    """</span>&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t352" href="#t352">352</a></span><span class="t">    <span class="key">from</span> <span class="nam">mpi4py</span> <span class="key">import</span> <span class="nam">MPI</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t353" href="#t353">353</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t354" href="#t354">354</a></span><span class="t">    <span class="nam">L</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">level</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t355" href="#t355">355</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t356" href="#t356">356</a></span><span class="t">    <span class="com"># Check if we want to skip the residual computation to gain performance</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t357" href="#t357">357</a></span><span class="t">    <span class="com"># Keep in mind that skipping any residual computation is likely to give incorrect outputs of the residual!</span>&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t358" href="#t358">358</a></span><span class="t">    <span class="key">if</span> <span class="nam">stage</span> <span class="key">in</span> <span class="nam">self</span><span class="op">.</span><span class="nam">params</span><span class="op">.</span><span class="nam">skip_residual_computation</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t359" href="#t359">359</a></span><span class="t">        <span class="nam">L</span><span class="op">.</span><span class="nam">status</span><span class="op">.</span><span class="nam">residual</span> <span class="op">=</span> <span class="num">0.0</span> <span class="key">if</span> <span class="nam">L</span><span class="op">.</span><span class="nam">status</span><span class="op">.</span><span class="nam">residual</span> <span class="key">is</span> <span class="key">None</span> <span class="key">else</span> <span class="nam">L</span><span class="op">.</span><span class="nam">status</span><span class="op">.</span><span class="nam">residual</span>&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t360" href="#t360">360</a></span><span class="t">        <span class="key">return</span> <span class="key">None</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t361" href="#t361">361</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t362" href="#t362">362</a></span><span class="t">    <span class="com"># compute the residual for each node</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t363" href="#t363">363</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t364" href="#t364">364</a></span><span class="t">    <span class="com"># build QF(u)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t365" href="#t365">365</a></span><span class="t">    <span class="nam">res</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">integrate</span><span class="op">(</span><span class="nam">last_only</span><span class="op">=</span><span class="nam">L</span><span class="op">.</span><span class="nam">params</span><span class="op">.</span><span class="nam">residual_type</span><span class="op">[</span><span class="op">:</span><span class="num">4</span><span class="op">]</span> <span class="op">==</span> <span class="str">'last'</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t366" href="#t366">366</a></span><span class="t">    <span class="nam">mask</span> <span class="op">=</span> <span class="nam">L</span><span class="op">.</span><span class="nam">prob</span><span class="op">.</span><span class="nam">diff_mask</span>&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t367" href="#t367">367</a></span><span class="t">    <span class="nam">res</span><span class="op">[</span><span class="nam">mask</span><span class="op">]</span> <span class="op">+=</span> <span class="nam">L</span><span class="op">.</span><span class="nam">u</span><span class="op">[</span><span class="num">0</span><span class="op">]</span><span class="op">[</span><span class="nam">mask</span><span class="op">]</span> <span class="op">-</span> <span class="nam">L</span><span class="op">.</span><span class="nam">u</span><span class="op">[</span><span class="nam">self</span><span class="op">.</span><span class="nam">rank</span> <span class="op">+</span> <span class="num">1</span><span class="op">]</span><span class="op">[</span><span class="nam">mask</span><span class="op">]</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t368" href="#t368">368</a></span><span class="t">    <span class="com"># add tau if associated</span>&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t369" href="#t369">369</a></span><span class="t">    <span class="key">if</span> <span class="nam">L</span><span class="op">.</span><span class="nam">tau</span><span class="op">[</span><span class="nam">self</span><span class="op">.</span><span class="nam">rank</span><span class="op">]</span> <span class="key">is</span> <span class="key">not</span> <span class="key">None</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t370" href="#t370">370</a></span><span class="t">        <span class="nam">res</span> <span class="op">+=</span> <span class="nam">L</span><span class="op">.</span><span class="nam">tau</span><span class="op">[</span><span class="nam">self</span><span class="op">.</span><span class="nam">rank</span><span class="op">]</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t371" href="#t371">371</a></span><span class="t">    <span class="com"># use abs function from data type here</span>&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t372" href="#t372">372</a></span><span class="t">    <span class="nam">res_norm</span> <span class="op">=</span> <span class="nam">abs</span><span class="op">(</span><span class="nam">res</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t373" href="#t373">373</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t374" href="#t374">374</a></span><span class="t">    <span class="com"># find maximal residual over the nodes</span>&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t375" href="#t375">375</a></span><span class="t">    <span class="key">if</span> <span class="nam">L</span><span class="op">.</span><span class="nam">params</span><span class="op">.</span><span class="nam">residual_type</span> <span class="op">==</span> <span class="str">'full_abs'</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t376" href="#t376">376</a></span><span class="t">        <span class="nam">L</span><span class="op">.</span><span class="nam">status</span><span class="op">.</span><span class="nam">residual</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">comm</span><span class="op">.</span><span class="nam">allreduce</span><span class="op">(</span><span class="nam">res_norm</span><span class="op">,</span> <span class="nam">op</span><span class="op">=</span><span class="nam">MPI</span><span class="op">.</span><span class="nam">MAX</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t377" href="#t377">377</a></span><span class="t">    <span class="key">elif</span> <span class="nam">L</span><span class="op">.</span><span class="nam">params</span><span class="op">.</span><span class="nam">residual_type</span> <span class="op">==</span> <span class="str">'last_abs'</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t378" href="#t378">378</a></span><span class="t">        <span class="nam">L</span><span class="op">.</span><span class="nam">status</span><span class="op">.</span><span class="nam">residual</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">comm</span><span class="op">.</span><span class="nam">bcast</span><span class="op">(</span><span class="nam">res_norm</span><span class="op">,</span> <span class="nam">root</span><span class="op">=</span><span class="nam">self</span><span class="op">.</span><span class="nam">comm</span><span class="op">.</span><span class="nam">size</span> <span class="op">-</span> <span class="num">1</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t379" href="#t379">379</a></span><span class="t">    <span class="key">elif</span> <span class="nam">L</span><span class="op">.</span><span class="nam">params</span><span class="op">.</span><span class="nam">residual_type</span> <span class="op">==</span> <span class="str">'full_rel'</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t380" href="#t380">380</a></span><span class="t">        <span class="nam">L</span><span class="op">.</span><span class="nam">status</span><span class="op">.</span><span class="nam">residual</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">comm</span><span class="op">.</span><span class="nam">allreduce</span><span class="op">(</span><span class="nam">res_norm</span> <span class="op">/</span> <span class="nam">abs</span><span class="op">(</span><span class="nam">L</span><span class="op">.</span><span class="nam">u</span><span class="op">[</span><span class="num">0</span><span class="op">]</span><span class="op">)</span><span class="op">,</span> <span class="nam">op</span><span class="op">=</span><span class="nam">MPI</span><span class="op">.</span><span class="nam">MAX</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t381" href="#t381">381</a></span><span class="t">    <span class="key">elif</span> <span class="nam">L</span><span class="op">.</span><span class="nam">params</span><span class="op">.</span><span class="nam">residual_type</span> <span class="op">==</span> <span class="str">'last_rel'</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t382" href="#t382">382</a></span><span class="t">        <span class="nam">L</span><span class="op">.</span><span class="nam">status</span><span class="op">.</span><span class="nam">residual</span> <span class="op">=</span> <span class="nam">self</span><span class="op">.</span><span class="nam">comm</span><span class="op">.</span><span class="nam">bcast</span><span class="op">(</span><span class="nam">res_norm</span> <span class="op">/</span> <span class="nam">abs</span><span class="op">(</span><span class="nam">L</span><span class="op">.</span><span class="nam">u</span><span class="op">[</span><span class="num">0</span><span class="op">]</span><span class="op">)</span><span class="op">,</span> <span class="nam">root</span><span class="op">=</span><span class="nam">self</span><span class="op">.</span><span class="nam">comm</span><span class="op">.</span><span class="nam">size</span> <span class="op">-</span> <span class="num">1</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t383" href="#t383">383</a></span><span class="t">    <span class="key">else</span><span class="op">:</span>&nbsp;</span><span class="r"></span></p>
+    <p class="exc show_exc"><span class="n"><a id="t384" href="#t384">384</a></span><span class="t">        <span class="key">raise</span> <span class="nam">NotImplementedError</span><span class="op">(</span><span class="fst">f'</span><span class="fst">residual type \"</span><span class="op">{</span><span class="nam">L</span><span class="op">.</span><span class="nam">params</span><span class="op">.</span><span class="nam">residual_type</span><span class="op">}</span><span class="fst">\" not implemented!</span><span class="fst">'</span><span class="op">)</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t385" href="#t385">385</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t386" href="#t386">386</a></span><span class="t">    <span class="com"># indicate that the residual has seen the new values</span>&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t387" href="#t387">387</a></span><span class="t">    <span class="nam">L</span><span class="op">.</span><span class="nam">status</span><span class="op">.</span><span class="nam">updated</span> <span class="op">=</span> <span class="key">False</span>&nbsp;</span><span class="r"></span></p>
+    <p class="pln"><span class="n"><a id="t388" href="#t388">388</a></span><span class="t">&nbsp;</span><span class="r"></span></p>
+    <p class="mis show_mis"><span class="n"><a id="t389" href="#t389">389</a></span><span class="t">    <span class="key">return</span> <span class="key">None</span>&nbsp;</span><span class="r"></span></p>
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+<footer>
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+            created at 2024-09-20 16:55 +0000
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             &nbsp; &nbsp; &nbsp;
             <a class="nav" href="https://coverage.readthedocs.io/en/7.6.1">coverage.py v7.6.1</a>,
-            created at 2024-09-19 09:13 +0000
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diff --git a/coverage/z_c28af68f8f845bbd_check_iteration_estimator_py.html b/coverage/z_c28af68f8f845bbd_check_iteration_estimator_py.html
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-            created at 2024-09-19 09:13 +0000
+            created at 2024-09-20 16:55 +0000
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-            created at 2024-09-19 09:13 +0000
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diff --git a/coverage/z_c28af68f8f845bbd_crash_py.html b/coverage/z_c28af68f8f845bbd_crash_py.html
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             <a class="nav" href="https://coverage.readthedocs.io/en/7.6.1">coverage.py v7.6.1</a>,
-            created at 2024-09-19 09:13 +0000
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-            created at 2024-09-19 09:13 +0000
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             &nbsp; &nbsp; &nbsp;
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-            created at 2024-09-19 09:13 +0000
+            created at 2024-09-20 16:55 +0000
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             <a class="nav" href="z_c28af68f8f845bbd_estimate_embedded_error_py.html">&#xbb; next</a>
             &nbsp; &nbsp; &nbsp;
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-            created at 2024-09-19 09:13 +0000
+            created at 2024-09-20 16:55 +0000
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diff --git a/coverage/z_c28af68f8f845bbd_estimate_embedded_error_py.html b/coverage/z_c28af68f8f845bbd_estimate_embedded_error_py.html
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@@ -65,7 +65,7 @@ <h2>
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             &nbsp; &nbsp; &nbsp;
             <a class="nav" href="https://coverage.readthedocs.io/en/7.6.1">coverage.py v7.6.1</a>,
-            created at 2024-09-19 09:13 +0000
+            created at 2024-09-20 16:55 +0000
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             <a class="nav" href="z_c28af68f8f845bbd_estimate_extrapolation_error_py.html">&#xbb; next</a>
             &nbsp; &nbsp; &nbsp;
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-            created at 2024-09-19 09:13 +0000
+            created at 2024-09-20 16:55 +0000
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diff --git a/coverage/z_c28af68f8f845bbd_estimate_extrapolation_error_py.html b/coverage/z_c28af68f8f845bbd_estimate_extrapolation_error_py.html
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             &nbsp; &nbsp; &nbsp;
             <a class="nav" href="https://coverage.readthedocs.io/en/7.6.1">coverage.py v7.6.1</a>,
-            created at 2024-09-19 09:13 +0000
+            created at 2024-09-20 16:55 +0000
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             &nbsp; &nbsp; &nbsp;
             <a class="nav" href="https://coverage.readthedocs.io/en/7.6.1">coverage.py v7.6.1</a>,
-            created at 2024-09-19 09:13 +0000
+            created at 2024-09-20 16:55 +0000
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diff --git a/coverage/z_c28af68f8f845bbd_estimate_polynomial_error_py.html b/coverage/z_c28af68f8f845bbd_estimate_polynomial_error_py.html
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             &nbsp; &nbsp; &nbsp;
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-            created at 2024-09-19 09:13 +0000
+            created at 2024-09-20 16:55 +0000
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@@ -267,7 +267,7 @@ <h2>
             <a class="nav" href="z_c28af68f8f845bbd_hotrod_py.html">&#xbb; next</a>
             &nbsp; &nbsp; &nbsp;
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-            created at 2024-09-19 09:13 +0000
+            created at 2024-09-20 16:55 +0000
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diff --git a/coverage/z_c28af68f8f845bbd_hotrod_py.html b/coverage/z_c28af68f8f845bbd_hotrod_py.html
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@@ -65,7 +65,7 @@ <h2>
             <a id="nextFileLink" class="nav" href="z_c28af68f8f845bbd_inexactness_py.html">&#xbb; next</a>
             &nbsp; &nbsp; &nbsp;
             <a class="nav" href="https://coverage.readthedocs.io/en/7.6.1">coverage.py v7.6.1</a>,
-            created at 2024-09-19 09:13 +0000
+            created at 2024-09-20 16:55 +0000
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@@ -245,7 +245,7 @@ <h2>
             <a class="nav" href="z_c28af68f8f845bbd_inexactness_py.html">&#xbb; next</a>
             &nbsp; &nbsp; &nbsp;
             <a class="nav" href="https://coverage.readthedocs.io/en/7.6.1">coverage.py v7.6.1</a>,
-            created at 2024-09-19 09:13 +0000
+            created at 2024-09-20 16:55 +0000
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diff --git a/coverage/z_c28af68f8f845bbd_inexactness_py.html b/coverage/z_c28af68f8f845bbd_inexactness_py.html
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@@ -65,7 +65,7 @@ <h2>
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             &nbsp; &nbsp; &nbsp;
             <a class="nav" href="https://coverage.readthedocs.io/en/7.6.1">coverage.py v7.6.1</a>,
-            created at 2024-09-19 09:13 +0000
+            created at 2024-09-20 16:55 +0000
         </p>
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@@ -176,7 +176,7 @@ <h2>
             <a class="nav" href="z_c28af68f8f845bbd_interpolate_between_restarts_py.html">&#xbb; next</a>
             &nbsp; &nbsp; &nbsp;
             <a class="nav" href="https://coverage.readthedocs.io/en/7.6.1">coverage.py v7.6.1</a>,
-            created at 2024-09-19 09:13 +0000
+            created at 2024-09-20 16:55 +0000
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diff --git a/coverage/z_c28af68f8f845bbd_interpolate_between_restarts_py.html b/coverage/z_c28af68f8f845bbd_interpolate_between_restarts_py.html
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+++ b/coverage/z_c28af68f8f845bbd_interpolate_between_restarts_py.html
@@ -65,7 +65,7 @@ <h2>
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             &nbsp; &nbsp; &nbsp;
             <a class="nav" href="https://coverage.readthedocs.io/en/7.6.1">coverage.py v7.6.1</a>,
-            created at 2024-09-19 09:13 +0000
+            created at 2024-09-20 16:55 +0000
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-            created at 2024-09-19 09:13 +0000
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-            created at 2024-09-19 09:13 +0000
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-            created at 2024-09-19 09:13 +0000
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-            created at 2024-09-19 09:13 +0000
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-            created at 2024-09-19 09:13 +0000
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index e18fd62c1d..8cd30139d9 100644
--- a/coverage/z_e65ba0b64d945c43_penningtrap_run_work_precision_py.html
+++ b/coverage/z_e65ba0b64d945c43_penningtrap_run_work_precision_py.html
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+            created at 2024-09-20 16:55 +0000
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+            created at 2024-09-20 16:55 +0000
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+++ b/coverage/z_e65ba0b64d945c43_plot_helper_py.html
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+            created at 2024-09-20 16:55 +0000
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+            created at 2024-09-20 16:55 +0000
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index 55f8e692b1..a600739115 100644
--- a/coverage/z_e65ba0b64d945c43_stability_simulation_py.html
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@@ -65,7 +65,7 @@ <h2>
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+            created at 2024-09-20 16:55 +0000
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+            created at 2024-09-20 16:55 +0000
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index e58654b92f..f3c809c27b 100644
--- a/coverage/z_ecd017d7b94a3306_compare_to_matrixbased_py.html
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+            created at 2024-09-20 16:55 +0000
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index fe5fe2b1d3..34710bda86 100644
--- a/coverage/z_ecd017d7b94a3306_compare_to_propagator_py.html
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@@ -65,7 +65,7 @@ <h2>
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+            created at 2024-09-20 16:55 +0000
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+            created at 2024-09-20 16:55 +0000
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diff --git a/coverage/z_ecd017d7b94a3306_controller_matrix_nonMPI_py.html b/coverage/z_ecd017d7b94a3306_controller_matrix_nonMPI_py.html
index 279beaa854..9f1c27206a 100644
--- a/coverage/z_ecd017d7b94a3306_controller_matrix_nonMPI_py.html
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@@ -65,7 +65,7 @@ <h2>
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+            created at 2024-09-20 16:55 +0000
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+            created at 2024-09-20 16:55 +0000
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diff --git a/coverage/z_f092600e49a290b3_TransferVectorOfDCTVectors_py.html b/coverage/z_f092600e49a290b3_TransferVectorOfDCTVectors_py.html
index f54e3ee756..9d84f9dcb6 100644
--- a/coverage/z_f092600e49a290b3_TransferVectorOfDCTVectors_py.html
+++ b/coverage/z_f092600e49a290b3_TransferVectorOfDCTVectors_py.html
@@ -65,7 +65,7 @@ <h2>
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+            created at 2024-09-20 16:55 +0000
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+            created at 2024-09-20 16:55 +0000
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diff --git a/coverage/z_f092600e49a290b3_Transfer_DCT_Vector_py.html b/coverage/z_f092600e49a290b3_Transfer_DCT_Vector_py.html
index 9092891a09..d37895519f 100644
--- a/coverage/z_f092600e49a290b3_Transfer_DCT_Vector_py.html
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@@ -65,7 +65,7 @@ <h2>
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+            created at 2024-09-20 16:55 +0000
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+            created at 2024-09-20 16:55 +0000
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diff --git a/coverage/z_f7d64113b0673500_ac_fft_py.html b/coverage/z_f7d64113b0673500_ac_fft_py.html
index c8948d959c..7b8083dad9 100644
--- a/coverage/z_f7d64113b0673500_ac_fft_py.html
+++ b/coverage/z_f7d64113b0673500_ac_fft_py.html
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+            created at 2024-09-20 16:55 +0000
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+            created at 2024-09-20 16:55 +0000
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diff --git a/coverage/z_f7d64113b0673500_heat_py.html b/coverage/z_f7d64113b0673500_heat_py.html
index 43f8640c42..52b5461443 100644
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+++ b/coverage/z_f7d64113b0673500_heat_py.html
@@ -65,7 +65,7 @@ <h2>
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+            created at 2024-09-20 16:55 +0000
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+            created at 2024-09-20 16:55 +0000
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diff --git a/genindex.html b/genindex.html
index 6f24be6d23..44d3a3971d 100644
--- a/genindex.html
+++ b/genindex.html
@@ -151,8 +151,16 @@ <h2 id="A">A</h2>
       <li><a href="pySDC/implementations.problem_classes.boussinesq_helpers.helper_classes.html#implementations.problem_classes.boussinesq_helpers.helper_classes.logging.add">add() (logging method)</a>
 </li>
       <li><a href="pySDC/helpers.pysdc_helper.html#helpers.pysdc_helper.FrozenClass.add_attr">add_attr() (FrozenClass class method)</a>
+</li>
+      <li><a href="pySDC/helpers.spectral_helper.html#helpers.spectral_helper.SpectralHelper.add_axis">add_axis() (SpectralHelper method)</a>
+</li>
+      <li><a href="pySDC/helpers.spectral_helper.html#helpers.spectral_helper.SpectralHelper.add_BC">add_BC() (SpectralHelper method)</a>
+</li>
+      <li><a href="pySDC/helpers.spectral_helper.html#helpers.spectral_helper.SpectralHelper.add_component">add_component() (SpectralHelper method)</a>
 </li>
       <li><a href="pySDC/core.controller.html#core.controller.Controller.add_convergence_controller">add_convergence_controller() (Controller method)</a>
+</li>
+      <li><a href="pySDC/helpers.spectral_helper.html#helpers.spectral_helper.SpectralHelper.add_equation_lhs">add_equation_lhs() (SpectralHelper method)</a>
 </li>
       <li><a href="pySDC/core.controller.html#core.controller.Controller.add_hook">add_hook() (Controller method)</a>
 </li>
@@ -166,6 +174,8 @@ <h2 id="A">A</h2>
 </li>
       <li><a href="pySDC/implementations.problem_classes.AdvectionDiffusionEquation_1D_FFT.html#implementations.problem_classes.AdvectionDiffusionEquation_1D_FFT.advectiondiffusion1d_implicit">advectiondiffusion1d_implicit (class in implementations.problem_classes.AdvectionDiffusionEquation_1D_FFT)</a>
 </li>
+  </ul></td>
+  <td style="width: 33%; vertical-align: top;"><ul>
       <li><a href="pySDC/implementations.problem_classes.AdvectionEquation_ND_FD.html#implementations.problem_classes.AdvectionEquation_ND_FD.advectionNd">advectionNd (class in implementations.problem_classes.AdvectionEquation_ND_FD)</a>
 </li>
       <li><a href="pySDC/implementations.problem_classes.AllenCahn_2D_FFT.html#implementations.problem_classes.AllenCahn_2D_FFT.allencahn2d_imex">allencahn2d_imex (class in implementations.problem_classes.AllenCahn_2D_FFT)</a>
@@ -174,8 +184,6 @@ <h2 id="A">A</h2>
         <li><a href="pySDC/implementations.problem_classes.AllenCahn_2D_FFT_gpu.html#implementations.problem_classes.AllenCahn_2D_FFT_gpu.allencahn2d_imex">(class in implementations.problem_classes.AllenCahn_2D_FFT_gpu)</a>
 </li>
       </ul></li>
-  </ul></td>
-  <td style="width: 33%; vertical-align: top;"><ul>
       <li><a href="pySDC/implementations.problem_classes.AllenCahn_2D_FFT.html#implementations.problem_classes.AllenCahn_2D_FFT.allencahn2d_imex_stab">allencahn2d_imex_stab (class in implementations.problem_classes.AllenCahn_2D_FFT)</a>
 
       <ul>
@@ -269,6 +277,8 @@ <h2 id="A">A</h2>
       <li><a href="pySDC/helpers.pysdc_helper.html#helpers.pysdc_helper.FrozenClass.attrs">attrs (FrozenClass attribute)</a>
 </li>
       <li><a href="pySDC/implementations.problem_classes.Auzinger_implicit.html#implementations.problem_classes.Auzinger_implicit.auzinger">auzinger (class in implementations.problem_classes.Auzinger_implicit)</a>
+</li>
+      <li><a href="pySDC/helpers.spectral_helper.html#helpers.spectral_helper.SpectralHelper.axes">axes (SpectralHelper attribute)</a>
 </li>
   </ul></td>
 </tr></table>
@@ -315,8 +325,16 @@ <h2 id="B">B</h2>
 </li>
       </ul></li>
       <li><a href="pySDC/implementations.problem_classes.Boussinesq_2D_FD_imex.html#implementations.problem_classes.Boussinesq_2D_FD_imex.boussinesq_2d_imex.bc_hor">bc_hor (boussinesq_2d_imex attribute)</a>
+</li>
+      <li><a href="pySDC/helpers.spectral_helper.html#helpers.spectral_helper.SpectralHelper.BC_line_zero_matrix">BC_line_zero_matrix (SpectralHelper attribute)</a>
+</li>
+      <li><a href="pySDC/helpers.spectral_helper.html#helpers.spectral_helper.SpectralHelper.BC_mat">BC_mat (SpectralHelper attribute)</a>
+</li>
+      <li><a href="pySDC/helpers.spectral_helper.html#helpers.spectral_helper.SpectralHelper.BC_rhs_mask">BC_rhs_mask (SpectralHelper attribute)</a>
 </li>
       <li><a href="pySDC/implementations.problem_classes.Boussinesq_2D_FD_imex.html#implementations.problem_classes.Boussinesq_2D_FD_imex.boussinesq_2d_imex.bc_ver">bc_ver (boussinesq_2d_imex attribute)</a>
+</li>
+      <li><a href="pySDC/helpers.spectral_helper.html#helpers.spectral_helper.SpectralHelper.BC_zero_index">BC_zero_index (SpectralHelper attribute)</a>
 </li>
       <li><a href="pySDC/implementations.datatype_classes.cupy_mesh.html#implementations.datatype_classes.cupy_mesh.cupy_mesh.bcast">bcast() (cupy_mesh method)</a>
 
@@ -327,6 +345,8 @@ <h2 id="B">B</h2>
       <li><a href="pySDC/helpers.NCCL_communicator.html#helpers.NCCL_communicator.NCCLComm.Bcast">Bcast() (NCCLComm method)</a>
 </li>
       <li><a href="pySDC/implementations.datatype_classes.petsc_vec.html#implementations.datatype_classes.petsc_vec.petsc_vec.bcast">bcast() (petsc_vec method)</a>
+</li>
+      <li><a href="pySDC/helpers.spectral_helper.html#helpers.spectral_helper.SpectralHelper.BCs">BCs (SpectralHelper attribute)</a>
 </li>
       <li><a href="pySDC/implementations.problem_classes.acoustic_helpers.standard_integrators.html#implementations.problem_classes.acoustic_helpers.standard_integrators.bdf2">bdf2 (class in implementations.problem_classes.acoustic_helpers.standard_integrators)</a>
 
@@ -334,6 +354,8 @@ <h2 id="B">B</h2>
         <li><a href="pySDC/implementations.problem_classes.boussinesq_helpers.standard_integrators.html#implementations.problem_classes.boussinesq_helpers.standard_integrators.bdf2">(class in implementations.problem_classes.boussinesq_helpers.standard_integrators)</a>
 </li>
       </ul></li>
+  </ul></td>
+  <td style="width: 33%; vertical-align: top;"><ul>
       <li><a href="pySDC/implementations.sweeper_classes.Multistep.html#implementations.sweeper_classes.Multistep.AdamsBashforthExplicit1Step.beta">beta (AdamsBashforthExplicit1Step attribute)</a>
 
       <ul>
@@ -346,8 +368,6 @@ <h2 id="B">B</h2>
         <li><a href="pySDC/implementations.sweeper_classes.Multistep.html#implementations.sweeper_classes.Multistep.MultiStep.beta">(MultiStep attribute)</a>
 </li>
       </ul></li>
-  </ul></td>
-  <td style="width: 33%; vertical-align: top;"><ul>
       <li><a href="pySDC/helpers.transfer_helper.html#helpers.transfer_helper.border_padding">border_padding() (in module helpers.transfer_helper)</a>
 </li>
       <li><a href="pySDC/implementations.sweeper_classes.boris_2nd_order.html#implementations.sweeper_classes.boris_2nd_order.boris_2nd_order">boris_2nd_order (class in implementations.sweeper_classes.boris_2nd_order)</a>
@@ -361,6 +381,10 @@ <h2 id="B">B</h2>
       <li><a href="pySDC/implementations.problem_classes.BuckConverter.html#implementations.problem_classes.BuckConverter.buck_converter">buck_converter (class in implementations.problem_classes.BuckConverter)</a>
 </li>
       <li><a href="pySDC/implementations.problem_classes.PenningTrap_3D.html#implementations.problem_classes.PenningTrap_3D.penningtrap.build_f">build_f() (penningtrap method)</a>
+</li>
+      <li><a href="pySDC/implementations.problem_classes.Burgers.html#implementations.problem_classes.Burgers.Burgers1D">Burgers1D (class in implementations.problem_classes.Burgers)</a>
+</li>
+      <li><a href="pySDC/implementations.problem_classes.Burgers.html#implementations.problem_classes.Burgers.Burgers2D">Burgers2D (class in implementations.problem_classes.Burgers)</a>
 </li>
       <li><a href="pySDC/implementations.sweeper_classes.Runge_Kutta.html#implementations.sweeper_classes.Runge_Kutta.ButcherTableau">ButcherTableau (class in implementations.sweeper_classes.Runge_Kutta)</a>
 </li>
@@ -401,10 +425,18 @@ <h2 id="C">C</h2>
 <table style="width: 100%" class="indextable genindextable"><tr>
   <td style="width: 33%; vertical-align: top;"><ul>
       <li><a href="pySDC/implementations.sweeper_classes.Multistep.html#implementations.sweeper_classes.Multistep.Cache">Cache (class in implementations.sweeper_classes.Multistep)</a>
+</li>
+      <li><a href="pySDC/implementations.problem_classes.generic_spectral.html#implementations.problem_classes.generic_spectral.GenericSpectralLinear.cached_factorizations">cached_factorizations (GenericSpectralLinear attribute)</a>
 </li>
       <li><a href="pySDC/implementations.problem_classes.boussinesq_helpers.helper_classes.html#implementations.problem_classes.boussinesq_helpers.helper_classes.Callback">Callback (class in implementations.problem_classes.boussinesq_helpers.helper_classes)</a>
 </li>
       <li><a href="pySDC/implementations.sweeper_classes.Runge_Kutta.html#implementations.sweeper_classes.Runge_Kutta.Cash_Karp">Cash_Karp (class in implementations.sweeper_classes.Runge_Kutta)</a>
+</li>
+      <li><a href="pySDC/implementations.problem_classes.RayleighBenard.html#implementations.problem_classes.RayleighBenard.CFLLimit">CFLLimit (class in implementations.problem_classes.RayleighBenard)</a>
+</li>
+      <li><a href="pySDC/helpers.spectral_helper.html#helpers.spectral_helper.ChebychevHelper">ChebychevHelper (class in helpers.spectral_helper)</a>
+</li>
+      <li><a href="pySDC/helpers.spectral_helper.html#helpers.spectral_helper.SpectralHelper.check_BCs">check_BCs() (SpectralHelper method)</a>
 </li>
       <li><a href="pySDC/implementations.convergence_controller_classes.check_convergence.html#implementations.convergence_controller_classes.check_convergence.CheckConvergence.check_convergence">check_convergence() (CheckConvergence static method)</a>
 </li>
@@ -468,6 +500,8 @@ <h2 id="C">C</h2>
         <li><a href="pySDC/implementations.datatype_classes.cupy_mesh.html#implementations.datatype_classes.cupy_mesh.cupy_mesh.comm">(cupy_mesh attribute)</a>
 </li>
         <li><a href="pySDC/implementations.datatype_classes.mesh.html#id0">(mesh attribute)</a>, <a href="pySDC/implementations.datatype_classes.mesh.html#implementations.datatype_classes.mesh.mesh.comm">[1]</a>
+</li>
+        <li><a href="pySDC/helpers.spectral_helper.html#helpers.spectral_helper.SpectralHelper.comm">(SpectralHelper attribute)</a>
 </li>
         <li><a href="pySDC/implementations.sweeper_classes.generic_implicit_MPI.html#implementations.sweeper_classes.generic_implicit_MPI.SweeperMPI.comm">(SweeperMPI property)</a>
 </li>
@@ -496,8 +530,12 @@ <h2 id="C">C</h2>
         <li><a href="pySDC/implementations.datatype_classes.mesh.html#implementations.datatype_classes.mesh.imex_mesh.components">(imex_mesh attribute)</a>
 </li>
         <li><a href="pySDC/implementations.datatype_classes.mesh.html#implementations.datatype_classes.mesh.MultiComponentMesh.components">(MultiComponentMesh attribute)</a>
+</li>
+        <li><a href="pySDC/helpers.spectral_helper.html#helpers.spectral_helper.SpectralHelper.components">(SpectralHelper attribute)</a>
 </li>
       </ul></li>
+      <li><a href="pySDC/implementations.problem_classes.RayleighBenard.html#implementations.problem_classes.RayleighBenard.RayleighBenard.compute_buoyancy_generation">compute_buoyancy_generation() (RayleighBenard method)</a>
+</li>
       <li><a href="pySDC/implementations.sweeper_classes.boris_2nd_order.html#implementations.sweeper_classes.boris_2nd_order.boris_2nd_order.compute_end_point">compute_end_point() (boris_2nd_order method)</a>
 
       <ul>
@@ -530,6 +568,10 @@ <h2 id="C">C</h2>
       </ul></li>
   </ul></td>
   <td style="width: 33%; vertical-align: top;"><ul>
+      <li><a href="pySDC/implementations.problem_classes.RayleighBenard.html#implementations.problem_classes.RayleighBenard.CFLLimit.compute_max_step_size">compute_max_step_size() (CFLLimit static method)</a>
+</li>
+      <li><a href="pySDC/implementations.problem_classes.RayleighBenard.html#implementations.problem_classes.RayleighBenard.RayleighBenard.compute_Nusselt_numbers">compute_Nusselt_numbers() (RayleighBenard method)</a>
+</li>
       <li><a href="pySDC/implementations.convergence_controller_classes.adaptivity.html#implementations.convergence_controller_classes.adaptivity.AdaptivityBase.compute_optimal_step_size">compute_optimal_step_size() (AdaptivityBase method)</a>
 </li>
       <li><a href="pySDC/implementations.sweeper_classes.imex_1st_order_mass.html#implementations.sweeper_classes.imex_1st_order_mass.imex_1st_order_mass.compute_residual">compute_residual() (imex_1st_order_mass method)</a>
@@ -540,6 +582,18 @@ <h2 id="C">C</h2>
         <li><a href="pySDC/core.sweeper.html#core.sweeper.Sweeper.compute_residual">(Sweeper method)</a>
 </li>
         <li><a href="pySDC/implementations.sweeper_classes.generic_implicit_MPI.html#implementations.sweeper_classes.generic_implicit_MPI.SweeperMPI.compute_residual">(SweeperMPI method)</a>
+</li>
+      </ul></li>
+      <li><a href="pySDC/implementations.problem_classes.generic_spectral.html#implementations.problem_classes.generic_spectral.compute_residual_DAE">compute_residual_DAE() (in module implementations.problem_classes.generic_spectral)</a>
+</li>
+      <li><a href="pySDC/implementations.problem_classes.generic_spectral.html#implementations.problem_classes.generic_spectral.compute_residual_DAE_MPI">compute_residual_DAE_MPI() (in module implementations.problem_classes.generic_spectral)</a>
+</li>
+      <li><a href="pySDC/implementations.problem_classes.RayleighBenard.html#implementations.problem_classes.RayleighBenard.RayleighBenard.compute_viscous_dissipation">compute_viscous_dissipation() (RayleighBenard method)</a>
+</li>
+      <li><a href="pySDC/implementations.problem_classes.Burgers.html#implementations.problem_classes.Burgers.Burgers2D.compute_vorticity">compute_vorticity() (Burgers2D method)</a>
+
+      <ul>
+        <li><a href="pySDC/implementations.problem_classes.RayleighBenard.html#implementations.problem_classes.RayleighBenard.RayleighBenard.compute_vorticity">(RayleighBenard method)</a>
 </li>
       </ul></li>
       <li><a href="pySDC/core.step.html#core.step.Step.connect_levels">connect_levels() (Step method)</a>
@@ -559,6 +613,8 @@ <h2 id="C">C</h2>
       <li><a href="pySDC/core.convergence_controller.html#core.convergence_controller.ConvergenceController">ConvergenceController (class in core.convergence_controller)</a>
 </li>
       <li><a href="pySDC/core.errors.html#core.errors.ConvergenceError">ConvergenceError</a>
+</li>
+      <li><a href="pySDC/helpers.spectral_helper.html#helpers.spectral_helper.SpectralHelper.convert_operator_matrix_to_operator">convert_operator_matrix_to_operator() (SpectralHelper method)</a>
 </li>
       <li>
     core
@@ -680,6 +736,8 @@ <h2 id="D">D</h2>
       <li><a href="pySDC/core.errors.html#core.errors.DataError">DataError</a>
 </li>
       <li><a href="pySDC/implementations.problem_classes.AdvectionDiffusionEquation_1D_FFT.html#implementations.problem_classes.AdvectionDiffusionEquation_1D_FFT.advectiondiffusion1d_imex.ddx">ddx (advectiondiffusion1d_imex attribute)</a>
+</li>
+      <li><a href="pySDC/helpers.spectral_helper.html#helpers.spectral_helper.SpectralHelper.debug">debug (SpectralHelper attribute)</a>
 </li>
       <li><a href="pySDC/core.convergence_controller.html#core.convergence_controller.ConvergenceController.debug">debug() (ConvergenceController method)</a>
 </li>
@@ -707,6 +765,8 @@ <h2 id="D">D</h2>
         <li><a href="pySDC/implementations.convergence_controller_classes.adaptivity.html#implementations.convergence_controller_classes.adaptivity.AdaptivityPolynomialError.dependencies">(AdaptivityPolynomialError method)</a>
 </li>
         <li><a href="pySDC/implementations.convergence_controller_classes.basic_restarting.html#implementations.convergence_controller_classes.basic_restarting.BasicRestarting.dependencies">(BasicRestarting method)</a>
+</li>
+        <li><a href="pySDC/implementations.problem_classes.RayleighBenard.html#implementations.problem_classes.RayleighBenard.CFLLimit.dependencies">(CFLLimit method)</a>
 </li>
         <li><a href="pySDC/implementations.convergence_controller_classes.check_convergence.html#implementations.convergence_controller_classes.check_convergence.CheckConvergence.dependencies">(CheckConvergence method)</a>
 </li>
@@ -758,6 +818,8 @@ <h2 id="D">D</h2>
       <li><a href="pySDC/implementations.problem_classes.odeSystem.html#implementations.problem_classes.odeSystem.ProtheroRobinsonAutonomous.dgInv_LIN">dgInv_LIN() (ProtheroRobinsonAutonomous method)</a>
 </li>
       <li><a href="pySDC/implementations.problem_classes.odeSystem.html#implementations.problem_classes.odeSystem.ProtheroRobinsonAutonomous.dgInv_NONLIN">dgInv_NONLIN() (ProtheroRobinsonAutonomous method)</a>
+</li>
+      <li><a href="pySDC/implementations.problem_classes.generic_spectral.html#implementations.problem_classes.generic_spectral.GenericSpectralLinear.diff_mask">diff_mask (GenericSpectralLinear attribute)</a>
 </li>
       <li><a href="pySDC/implementations.problem_classes.acoustic_helpers.standard_integrators.html#implementations.problem_classes.acoustic_helpers.standard_integrators.dirk">dirk (class in implementations.problem_classes.acoustic_helpers.standard_integrators)</a>
 
@@ -777,6 +839,8 @@ <h2 id="D">D</h2>
         <li><a href="pySDC/core.step.html#core.step.Step.dt">(Step property)</a>
 </li>
       </ul></li>
+      <li><a href="pySDC/helpers.spectral_helper.html#helpers.spectral_helper.SpectralHelper.dtype">dtype (SpectralHelper attribute)</a>
+</li>
       <li><a href="pySDC/implementations.problem_classes.AcousticAdvection_1D_FD_imex.html#implementations.problem_classes.AcousticAdvection_1D_FD_imex.acoustic_1d_imex.dtype_f">dtype_f (acoustic_1d_imex attribute)</a>
 
       <ul>
@@ -819,6 +883,10 @@ <h2 id="D">D</h2>
         <li><a href="pySDC/implementations.problem_classes.Boussinesq_2D_FD_imex.html#implementations.problem_classes.Boussinesq_2D_FD_imex.boussinesq_2d_imex.dtype_f">(boussinesq_2d_imex attribute)</a>
 </li>
         <li><a href="pySDC/implementations.problem_classes.BuckConverter.html#implementations.problem_classes.BuckConverter.buck_converter.dtype_f">(buck_converter attribute)</a>
+</li>
+        <li><a href="pySDC/implementations.problem_classes.Burgers.html#implementations.problem_classes.Burgers.Burgers1D.dtype_f">(Burgers1D attribute)</a>
+</li>
+        <li><a href="pySDC/implementations.problem_classes.Burgers.html#implementations.problem_classes.Burgers.Burgers2D.dtype_f">(Burgers2D attribute)</a>
 </li>
         <li><a href="pySDC/implementations.problem_classes.odeSystem.html#implementations.problem_classes.odeSystem.ChemicalReaction3Var.dtype_f">(ChemicalReaction3Var attribute)</a>
 </li>
@@ -843,8 +911,16 @@ <h2 id="D">D</h2>
         <li><a href="pySDC/implementations.problem_classes.GrayScott_MPIFFT.html#implementations.problem_classes.GrayScott_MPIFFT.grayscott_mi_linear.dtype_f">(grayscott_mi_linear attribute)</a>
 </li>
         <li><a href="pySDC/implementations.problem_classes.HarmonicOscillator.html#implementations.problem_classes.HarmonicOscillator.harmonic_oscillator.dtype_f">(harmonic_oscillator attribute)</a>
+</li>
+        <li><a href="pySDC/implementations.problem_classes.HeatEquation_Chebychev.html#implementations.problem_classes.HeatEquation_Chebychev.Heat1DChebychev.dtype_f">(Heat1DChebychev attribute)</a>
+</li>
+        <li><a href="pySDC/implementations.problem_classes.HeatEquation_Chebychev.html#implementations.problem_classes.HeatEquation_Chebychev.Heat1DUltraspherical.dtype_f">(Heat1DUltraspherical attribute)</a>
 </li>
         <li><a href="pySDC/implementations.problem_classes.HeatEquation_2D_PETSc_forced.html#implementations.problem_classes.HeatEquation_2D_PETSc_forced.heat2d_petsc_forced.dtype_f">(heat2d_petsc_forced attribute)</a>
+</li>
+        <li><a href="pySDC/implementations.problem_classes.HeatEquation_Chebychev.html#implementations.problem_classes.HeatEquation_Chebychev.Heat2DChebychev.dtype_f">(Heat2DChebychev attribute)</a>
+</li>
+        <li><a href="pySDC/implementations.problem_classes.HeatEquation_Chebychev.html#implementations.problem_classes.HeatEquation_Chebychev.Heat2DUltraspherical.dtype_f">(Heat2DUltraspherical attribute)</a>
 </li>
         <li><a href="pySDC/implementations.problem_classes.HeatEquation_ND_FD.html#implementations.problem_classes.HeatEquation_ND_FD.heatNd_forced.dtype_f">(heatNd_forced attribute)</a>, <a href="pySDC/implementations.problem_classes.HeatEquation_ND_FD_CuPy.html#implementations.problem_classes.HeatEquation_ND_FD_CuPy.heatNd_forced.dtype_f">[1]</a>
 </li>
@@ -897,6 +973,8 @@ <h2 id="D">D</h2>
         <li><a href="pySDC/implementations.problem_classes.Quench.html#implementations.problem_classes.Quench.Quench.dtype_f">(Quench attribute)</a>
 </li>
         <li><a href="pySDC/implementations.problem_classes.Quench.html#implementations.problem_classes.Quench.QuenchIMEX.dtype_f">(QuenchIMEX attribute)</a>
+</li>
+        <li><a href="pySDC/implementations.problem_classes.RayleighBenard.html#implementations.problem_classes.RayleighBenard.RayleighBenard.dtype_f">(RayleighBenard attribute)</a>
 </li>
         <li><a href="pySDC/implementations.problem_classes.FastWaveSlowWave_0D.html#implementations.problem_classes.FastWaveSlowWave_0D.swfw_scalar.dtype_f">(swfw_scalar attribute)</a>
 </li>
@@ -931,6 +1009,10 @@ <h2 id="D">D</h2>
         <li><a href="pySDC/implementations.problem_classes.Boussinesq_2D_FD_imex.html#implementations.problem_classes.Boussinesq_2D_FD_imex.boussinesq_2d_imex.dtype_u">(boussinesq_2d_imex attribute)</a>
 </li>
         <li><a href="pySDC/implementations.problem_classes.BuckConverter.html#implementations.problem_classes.BuckConverter.buck_converter.dtype_u">(buck_converter attribute)</a>
+</li>
+        <li><a href="pySDC/implementations.problem_classes.Burgers.html#implementations.problem_classes.Burgers.Burgers1D.dtype_u">(Burgers1D attribute)</a>
+</li>
+        <li><a href="pySDC/implementations.problem_classes.Burgers.html#implementations.problem_classes.Burgers.Burgers2D.dtype_u">(Burgers2D attribute)</a>
 </li>
         <li><a href="pySDC/implementations.problem_classes.odeSystem.html#implementations.problem_classes.odeSystem.ChemicalReaction3Var.dtype_u">(ChemicalReaction3Var attribute)</a>
 </li>
@@ -951,8 +1033,16 @@ <h2 id="D">D</h2>
         <li><a href="pySDC/implementations.problem_classes.generic_ND_FD.html#implementations.problem_classes.generic_ND_FD.GenericNDimFinDiff.dtype_u">(GenericNDimFinDiff attribute)</a>
 </li>
         <li><a href="pySDC/implementations.problem_classes.HarmonicOscillator.html#implementations.problem_classes.HarmonicOscillator.harmonic_oscillator.dtype_u">(harmonic_oscillator attribute)</a>
+</li>
+        <li><a href="pySDC/implementations.problem_classes.HeatEquation_Chebychev.html#implementations.problem_classes.HeatEquation_Chebychev.Heat1DChebychev.dtype_u">(Heat1DChebychev attribute)</a>
+</li>
+        <li><a href="pySDC/implementations.problem_classes.HeatEquation_Chebychev.html#implementations.problem_classes.HeatEquation_Chebychev.Heat1DUltraspherical.dtype_u">(Heat1DUltraspherical attribute)</a>
 </li>
         <li><a href="pySDC/implementations.problem_classes.HeatEquation_2D_PETSc_forced.html#implementations.problem_classes.HeatEquation_2D_PETSc_forced.heat2d_petsc_forced.dtype_u">(heat2d_petsc_forced attribute)</a>
+</li>
+        <li><a href="pySDC/implementations.problem_classes.HeatEquation_Chebychev.html#implementations.problem_classes.HeatEquation_Chebychev.Heat2DChebychev.dtype_u">(Heat2DChebychev attribute)</a>
+</li>
+        <li><a href="pySDC/implementations.problem_classes.HeatEquation_Chebychev.html#implementations.problem_classes.HeatEquation_Chebychev.Heat2DUltraspherical.dtype_u">(Heat2DUltraspherical attribute)</a>
 </li>
         <li><a href="pySDC/implementations.problem_classes.HeatEquation_ND_FD_CuPy.html#implementations.problem_classes.HeatEquation_ND_FD_CuPy.heatNd_forced.dtype_u">(heatNd_forced attribute)</a>
 </li>
@@ -991,6 +1081,8 @@ <h2 id="D">D</h2>
         <li><a href="pySDC/implementations.problem_classes.odeSystem.html#implementations.problem_classes.odeSystem.ProtheroRobinsonAutonomous.dtype_u">(ProtheroRobinsonAutonomous attribute)</a>
 </li>
         <li><a href="pySDC/implementations.problem_classes.Quench.html#implementations.problem_classes.Quench.Quench.dtype_u">(Quench attribute)</a>
+</li>
+        <li><a href="pySDC/implementations.problem_classes.RayleighBenard.html#implementations.problem_classes.RayleighBenard.RayleighBenard.dtype_u">(RayleighBenard attribute)</a>
 </li>
         <li><a href="pySDC/implementations.problem_classes.FastWaveSlowWave_0D.html#implementations.problem_classes.FastWaveSlowWave_0D.swfw_scalar.dtype_u">(swfw_scalar attribute)</a>
 </li>
@@ -1139,6 +1231,10 @@ <h2 id="E">E</h2>
         <li><a href="pySDC/implementations.problem_classes.Brusselator.html#implementations.problem_classes.Brusselator.Brusselator.eval_f">(Brusselator method)</a>
 </li>
         <li><a href="pySDC/implementations.problem_classes.BuckConverter.html#implementations.problem_classes.BuckConverter.buck_converter.eval_f">(buck_converter method)</a>
+</li>
+        <li><a href="pySDC/implementations.problem_classes.Burgers.html#implementations.problem_classes.Burgers.Burgers1D.eval_f">(Burgers1D method)</a>
+</li>
+        <li><a href="pySDC/implementations.problem_classes.Burgers.html#implementations.problem_classes.Burgers.Burgers2D.eval_f">(Burgers2D method)</a>
 </li>
         <li><a href="pySDC/implementations.problem_classes.odeSystem.html#implementations.problem_classes.odeSystem.ChemicalReaction3Var.eval_f">(ChemicalReaction3Var method)</a>
 </li>
@@ -1171,8 +1267,16 @@ <h2 id="E">E</h2>
         <li><a href="pySDC/implementations.problem_classes.GrayScott_MPIFFT.html#implementations.problem_classes.GrayScott_MPIFFT.grayscott_mi_linear.eval_f">(grayscott_mi_linear method)</a>
 </li>
         <li><a href="pySDC/implementations.problem_classes.HarmonicOscillator.html#implementations.problem_classes.HarmonicOscillator.harmonic_oscillator.eval_f">(harmonic_oscillator method)</a>
+</li>
+        <li><a href="pySDC/implementations.problem_classes.HeatEquation_Chebychev.html#implementations.problem_classes.HeatEquation_Chebychev.Heat1DChebychev.eval_f">(Heat1DChebychev method)</a>
+</li>
+        <li><a href="pySDC/implementations.problem_classes.HeatEquation_Chebychev.html#implementations.problem_classes.HeatEquation_Chebychev.Heat1DUltraspherical.eval_f">(Heat1DUltraspherical method)</a>
 </li>
         <li><a href="pySDC/implementations.problem_classes.HeatEquation_2D_PETSc_forced.html#implementations.problem_classes.HeatEquation_2D_PETSc_forced.heat2d_petsc_forced.eval_f">(heat2d_petsc_forced method)</a>
+</li>
+        <li><a href="pySDC/implementations.problem_classes.HeatEquation_Chebychev.html#implementations.problem_classes.HeatEquation_Chebychev.Heat2DChebychev.eval_f">(Heat2DChebychev method)</a>
+</li>
+        <li><a href="pySDC/implementations.problem_classes.HeatEquation_Chebychev.html#implementations.problem_classes.HeatEquation_Chebychev.Heat2DUltraspherical.eval_f">(Heat2DUltraspherical method)</a>
 </li>
         <li><a href="pySDC/implementations.problem_classes.HeatEquation_ND_FD.html#implementations.problem_classes.HeatEquation_ND_FD.heatNd_forced.eval_f">(heatNd_forced method)</a>, <a href="pySDC/implementations.problem_classes.HeatEquation_ND_FD_CuPy.html#implementations.problem_classes.HeatEquation_ND_FD_CuPy.heatNd_forced.eval_f">[1]</a>
 </li>
@@ -1225,6 +1329,8 @@ <h2 id="E">E</h2>
         <li><a href="pySDC/implementations.problem_classes.Quench.html#implementations.problem_classes.Quench.Quench.eval_f">(Quench method)</a>
 </li>
         <li><a href="pySDC/implementations.problem_classes.Quench.html#implementations.problem_classes.Quench.QuenchIMEX.eval_f">(QuenchIMEX method)</a>
+</li>
+        <li><a href="pySDC/implementations.problem_classes.RayleighBenard.html#implementations.problem_classes.RayleighBenard.RayleighBenard.eval_f">(RayleighBenard method)</a>
 </li>
         <li><a href="pySDC/implementations.problem_classes.FastWaveSlowWave_0D.html#implementations.problem_classes.FastWaveSlowWave_0D.swfw_scalar.eval_f">(swfw_scalar method)</a>
 </li>
@@ -1368,10 +1474,30 @@ <h2 id="F">F</h2>
   </ul></td>
   <td style="width: 33%; vertical-align: top;"><ul>
       <li><a href="pySDC/implementations.problem_classes.generic_MPIFFT_Laplacian.html#implementations.problem_classes.generic_MPIFFT_Laplacian.IMEX_Laplacian_MPIFFT.fft_backend">fft_backend (IMEX_Laplacian_MPIFFT attribute)</a>
+
+      <ul>
+        <li><a href="pySDC/helpers.spectral_helper.html#helpers.spectral_helper.SpectralHelper.fft_backend">(SpectralHelper attribute)</a>
+</li>
+      </ul></li>
+      <li><a href="pySDC/helpers.spectral_helper.html#helpers.spectral_helper.SpectralHelper.fft_cache">fft_cache (SpectralHelper attribute)</a>
 </li>
       <li><a href="pySDC/implementations.problem_classes.generic_MPIFFT_Laplacian.html#implementations.problem_classes.generic_MPIFFT_Laplacian.IMEX_Laplacian_MPIFFT.fft_comm_backend">fft_comm_backend (IMEX_Laplacian_MPIFFT attribute)</a>
+
+      <ul>
+        <li><a href="pySDC/helpers.spectral_helper.html#helpers.spectral_helper.SpectralHelper.fft_comm_backend">(SpectralHelper attribute)</a>
+</li>
+      </ul></li>
+      <li><a href="pySDC/helpers.spectral_helper.html#helpers.spectral_helper.SpectralHelper.fft_lib">fft_lib (SpectralHelper attribute)</a>
+
+      <ul>
+        <li><a href="pySDC/helpers.spectral_helper.html#helpers.spectral_helper.SpectralHelper1D.fft_lib">(SpectralHelper1D attribute)</a>
+</li>
+      </ul></li>
+      <li><a href="pySDC/helpers.spectral_helper.html#helpers.spectral_helper.SpectralHelper.fft_obj">fft_obj (SpectralHelper attribute)</a>
 </li>
       <li><a href="pySDC/implementations.transfer_classes.TransferMesh_MPIFFT.html#implementations.transfer_classes.TransferMesh_MPIFFT.fft_to_fft">fft_to_fft (class in implementations.transfer_classes.TransferMesh_MPIFFT)</a>
+</li>
+      <li><a href="pySDC/helpers.spectral_helper.html#helpers.spectral_helper.FFTHelper">FFTHelper (class in helpers.spectral_helper)</a>
 </li>
       <li><a href="pySDC/implementations.datatype_classes.particles.html#implementations.datatype_classes.particles.fields">fields (class in implementations.datatype_classes.particles)</a>
 </li>
@@ -1434,6 +1560,8 @@ <h2 id="F">F</h2>
       <li><a href="pySDC/implementations.sweeper_classes.Runge_Kutta.html#implementations.sweeper_classes.Runge_Kutta.ForwardEuler">ForwardEuler (class in implementations.sweeper_classes.Runge_Kutta)</a>
 </li>
       <li><a href="pySDC/helpers.pysdc_helper.html#helpers.pysdc_helper.FrozenClass">FrozenClass (class in helpers.pysdc_helper)</a>
+</li>
+      <li><a href="pySDC/helpers.spectral_helper.html#helpers.spectral_helper.SpectralHelper.full_BCs">full_BCs (SpectralHelper attribute)</a>
 </li>
       <li><a href="pySDC/implementations.problem_classes.FullSolarSystem.html#implementations.problem_classes.FullSolarSystem.full_solar_system">full_solar_system (class in implementations.problem_classes.FullSolarSystem)</a>
 </li>
@@ -1516,6 +1644,8 @@ <h2 id="G">G</h2>
       <li><a href="pySDC/implementations.sweeper_classes.generic_implicit_MPI.html#implementations.sweeper_classes.generic_implicit_MPI.generic_implicit_MPI">generic_implicit_MPI (class in implementations.sweeper_classes.generic_implicit_MPI)</a>
 </li>
       <li><a href="pySDC/implementations.problem_classes.generic_ND_FD.html#implementations.problem_classes.generic_ND_FD.GenericNDimFinDiff">GenericNDimFinDiff (class in implementations.problem_classes.generic_ND_FD)</a>
+</li>
+      <li><a href="pySDC/implementations.problem_classes.generic_spectral.html#implementations.problem_classes.generic_spectral.GenericSpectralLinear">GenericSpectralLinear (class in implementations.problem_classes.generic_spectral)</a>
 </li>
       <li><a href="pySDC/helpers.pysdc_helper.html#helpers.pysdc_helper.FrozenClass.get">get() (FrozenClass method)</a>
 </li>
@@ -1527,9 +1657,41 @@ <h2 id="G">G</h2>
 </li>
       <li><a href="pySDC/helpers.problem_helper.html#helpers.problem_helper.get_1d_grid">get_1d_grid() (in module helpers.problem_helper)</a>
 </li>
+      <li><a href="pySDC/helpers.spectral_helper.html#helpers.spectral_helper.ChebychevHelper.get_1dgrid">get_1dgrid() (ChebychevHelper method)</a>
+
+      <ul>
+        <li><a href="pySDC/helpers.spectral_helper.html#helpers.spectral_helper.FFTHelper.get_1dgrid">(FFTHelper method)</a>
+</li>
+        <li><a href="pySDC/helpers.spectral_helper.html#helpers.spectral_helper.SpectralHelper1D.get_1dgrid">(SpectralHelper1D method)</a>
+</li>
+      </ul></li>
+      <li><a href="pySDC/helpers.spectral_helper.html#helpers.spectral_helper.SpectralHelper.get_aligned">get_aligned() (SpectralHelper method)</a>
+</li>
+      <li><a href="pySDC/helpers.spectral_helper.html#helpers.spectral_helper.ChebychevHelper.get_basis_change_matrix">get_basis_change_matrix() (ChebychevHelper method)</a>
+
+      <ul>
+        <li><a href="pySDC/helpers.spectral_helper.html#helpers.spectral_helper.SpectralHelper.get_basis_change_matrix">(SpectralHelper method)</a>
+</li>
+        <li><a href="pySDC/helpers.spectral_helper.html#helpers.spectral_helper.SpectralHelper1D.get_basis_change_matrix">(SpectralHelper1D method)</a>
+</li>
+        <li><a href="pySDC/helpers.spectral_helper.html#helpers.spectral_helper.UltrasphericalHelper.get_basis_change_matrix">(UltrasphericalHelper method)</a>
+</li>
+      </ul></li>
+      <li><a href="pySDC/helpers.spectral_helper.html#helpers.spectral_helper.ChebychevHelper.get_BC">get_BC() (ChebychevHelper method)</a>
+
+      <ul>
+        <li><a href="pySDC/helpers.spectral_helper.html#helpers.spectral_helper.FFTHelper.get_BC">(FFTHelper method)</a>
+</li>
+        <li><a href="pySDC/helpers.spectral_helper.html#helpers.spectral_helper.SpectralHelper.get_BC">(SpectralHelper method)</a>
+</li>
+        <li><a href="pySDC/helpers.spectral_helper.html#helpers.spectral_helper.SpectralHelper1D.get_BC">(SpectralHelper1D method)</a>
+</li>
+      </ul></li>
       <li><a href="pySDC/implementations.sweeper_classes.Runge_Kutta.html#implementations.sweeper_classes.Runge_Kutta.RungeKutta.get_Butcher_tableau">get_Butcher_tableau() (RungeKutta class method)</a>
 </li>
       <li><a href="pySDC/implementations.sweeper_classes.Runge_Kutta.html#implementations.sweeper_classes.Runge_Kutta.RungeKuttaIMEX.get_Butcher_tableau_explicit">get_Butcher_tableau_explicit() (RungeKuttaIMEX class method)</a>
+</li>
+      <li><a href="pySDC/helpers.spectral_helper.html#helpers.spectral_helper.ChebychevHelper.get_conv">get_conv() (ChebychevHelper method)</a>
 </li>
       <li><a href="pySDC/implementations.convergence_controller_classes.adaptivity.html#implementations.convergence_controller_classes.adaptivity.AdaptivityCollocation.get_convergence">get_convergence() (AdaptivityCollocation method)</a>
 
@@ -1549,20 +1711,66 @@ <h2 id="G">G</h2>
 
       <ul>
         <li><a href="pySDC/core.problem.html#core.problem.Problem.get_default_sweeper_class">(Problem class method)</a>
+</li>
+      </ul></li>
+      <li><a href="pySDC/helpers.spectral_helper.html#helpers.spectral_helper.ChebychevHelper.get_differentiation_matrix">get_differentiation_matrix() (ChebychevHelper method)</a>
+
+      <ul>
+        <li><a href="pySDC/helpers.spectral_helper.html#helpers.spectral_helper.FFTHelper.get_differentiation_matrix">(FFTHelper method)</a>
+</li>
+        <li><a href="pySDC/helpers.spectral_helper.html#helpers.spectral_helper.SpectralHelper.get_differentiation_matrix">(SpectralHelper method)</a>
+</li>
+        <li><a href="pySDC/helpers.spectral_helper.html#helpers.spectral_helper.SpectralHelper1D.get_differentiation_matrix">(SpectralHelper1D method)</a>
+</li>
+        <li><a href="pySDC/helpers.spectral_helper.html#helpers.spectral_helper.UltrasphericalHelper.get_differentiation_matrix">(UltrasphericalHelper method)</a>
+</li>
+      </ul></li>
+      <li><a href="pySDC/helpers.spectral_helper.html#helpers.spectral_helper.ChebychevHelper.get_Dirichlet_BC_row">get_Dirichlet_BC_row() (ChebychevHelper method)</a>
+</li>
+      <li><a href="pySDC/helpers.spectral_helper.html#helpers.spectral_helper.ChebychevHelper.get_Dirichlet_recombination_matrix">get_Dirichlet_recombination_matrix() (ChebychevHelper method)</a>
+
+      <ul>
+        <li><a href="pySDC/helpers.spectral_helper.html#helpers.spectral_helper.SpectralHelper.get_Dirichlet_recombination_matrix">(SpectralHelper method)</a>
 </li>
       </ul></li>
       <li><a href="pySDC/helpers.NCCL_communicator.html#helpers.NCCL_communicator.NCCLComm.get_dtype">get_dtype() (NCCLComm static method)</a>
 </li>
+      <li><a href="pySDC/helpers.spectral_helper.html#helpers.spectral_helper.SpectralHelper.get_empty_operator_matrix">get_empty_operator_matrix() (SpectralHelper method)</a>
+
+      <ul>
+        <li><a href="pySDC/helpers.spectral_helper.html#helpers.spectral_helper.SpectralHelper1D.get_empty_operator_matrix">(SpectralHelper1D method)</a>
+</li>
+      </ul></li>
       <li><a href="pySDC/implementations.convergence_controller_classes.estimate_extrapolation_error.html#implementations.convergence_controller_classes.estimate_extrapolation_error.EstimateExtrapolationErrorNonMPI.get_extrapolated_error">get_extrapolated_error() (EstimateExtrapolationErrorNonMPI method)</a>
 </li>
       <li><a href="pySDC/implementations.convergence_controller_classes.estimate_extrapolation_error.html#implementations.convergence_controller_classes.estimate_extrapolation_error.EstimateExtrapolationErrorNonMPI.get_extrapolated_solution">get_extrapolated_solution() (EstimateExtrapolationErrorNonMPI method)</a>
 </li>
       <li><a href="pySDC/implementations.convergence_controller_classes.estimate_extrapolation_error.html#implementations.convergence_controller_classes.estimate_extrapolation_error.EstimateExtrapolationErrorBase.get_extrapolation_coefficients">get_extrapolation_coefficients() (EstimateExtrapolationErrorBase method)</a>
+</li>
+      <li><a href="pySDC/helpers.spectral_helper.html#helpers.spectral_helper.SpectralHelper.get_fft">get_fft() (SpectralHelper method)</a>
+</li>
+      <li><a href="pySDC/helpers.spectral_helper.html#helpers.spectral_helper.ChebychevHelper.get_fft_shift">get_fft_shift() (ChebychevHelper method)</a>
+</li>
+      <li><a href="pySDC/helpers.spectral_helper.html#helpers.spectral_helper.ChebychevHelper.get_fft_shuffle">get_fft_shuffle() (ChebychevHelper method)</a>
+</li>
+      <li><a href="pySDC/helpers.spectral_helper.html#helpers.spectral_helper.ChebychevHelper.get_fft_utils">get_fft_utils() (ChebychevHelper method)</a>
 </li>
       <li><a href="pySDC/implementations.problem_classes.Brusselator.html#implementations.problem_classes.Brusselator.Brusselator.get_fig">get_fig() (Brusselator method)</a>
 
       <ul>
+        <li><a href="pySDC/implementations.problem_classes.Burgers.html#implementations.problem_classes.Burgers.Burgers1D.get_fig">(Burgers1D method)</a>
+</li>
+        <li><a href="pySDC/implementations.problem_classes.Burgers.html#implementations.problem_classes.Burgers.Burgers2D.get_fig">(Burgers2D method)</a>
+</li>
         <li><a href="pySDC/core.problem.html#core.problem.Problem.get_fig">(Problem method)</a>
+</li>
+        <li><a href="pySDC/implementations.problem_classes.RayleighBenard.html#implementations.problem_classes.RayleighBenard.RayleighBenard.get_fig">(RayleighBenard method)</a>
+</li>
+      </ul></li>
+      <li><a href="pySDC/helpers.spectral_helper.html#helpers.spectral_helper.SpectralHelper.get_filter_matrix">get_filter_matrix() (SpectralHelper method)</a>
+
+      <ul>
+        <li><a href="pySDC/helpers.spectral_helper.html#helpers.spectral_helper.SpectralHelper1D.get_filter_matrix">(SpectralHelper1D method)</a>
 </li>
       </ul></li>
       <li><a href="pySDC/helpers.problem_helper.html#helpers.problem_helper.get_finite_difference_matrix">get_finite_difference_matrix() (in module helpers.problem_helper)</a>
@@ -1573,6 +1781,14 @@ <h2 id="G">G</h2>
 
       <ul>
         <li><a href="pySDC/implementations.sweeper_classes.Runge_Kutta_Nystrom.html#implementations.sweeper_classes.Runge_Kutta_Nystrom.RungeKuttaNystrom.get_full_f">(RungeKuttaNystrom method)</a>
+</li>
+      </ul></li>
+      <li><a href="pySDC/helpers.spectral_helper.html#helpers.spectral_helper.SpectralHelper.get_grid">get_grid() (SpectralHelper method)</a>
+</li>
+      <li><a href="pySDC/helpers.spectral_helper.html#helpers.spectral_helper.SpectralHelper.get_Id">get_Id() (SpectralHelper method)</a>
+
+      <ul>
+        <li><a href="pySDC/helpers.spectral_helper.html#helpers.spectral_helper.SpectralHelper1D.get_Id">(SpectralHelper1D method)</a>
 </li>
       </ul></li>
       <li><a href="pySDC/implementations.convergence_controller_classes.basic_restarting.html#implementations.convergence_controller_classes.basic_restarting.BasicRestarting.get_implementation">get_implementation() (BasicRestarting class method)</a>
@@ -1581,6 +1797,28 @@ <h2 id="G">G</h2>
         <li><a href="pySDC/implementations.convergence_controller_classes.estimate_embedded_error.html#implementations.convergence_controller_classes.estimate_embedded_error.EstimateEmbeddedError.get_implementation">(EstimateEmbeddedError class method)</a>
 </li>
         <li><a href="pySDC/implementations.convergence_controller_classes.spread_step_sizes.html#implementations.convergence_controller_classes.spread_step_sizes.SpreadStepSizesBlockwise.get_implementation">(SpreadStepSizesBlockwise class method)</a>
+</li>
+      </ul></li>
+  </ul></td>
+  <td style="width: 33%; vertical-align: top;"><ul>
+      <li><a href="pySDC/helpers.spectral_helper.html#helpers.spectral_helper.ChebychevHelper.get_integ_BC_row">get_integ_BC_row() (ChebychevHelper method)</a>
+
+      <ul>
+        <li><a href="pySDC/helpers.spectral_helper.html#helpers.spectral_helper.FFTHelper.get_integ_BC_row">(FFTHelper method)</a>
+</li>
+      </ul></li>
+      <li><a href="pySDC/helpers.spectral_helper.html#helpers.spectral_helper.UltrasphericalHelper.get_integration_constant">get_integration_constant() (UltrasphericalHelper method)</a>
+</li>
+      <li><a href="pySDC/helpers.spectral_helper.html#helpers.spectral_helper.ChebychevHelper.get_integration_matrix">get_integration_matrix() (ChebychevHelper method)</a>
+
+      <ul>
+        <li><a href="pySDC/helpers.spectral_helper.html#helpers.spectral_helper.FFTHelper.get_integration_matrix">(FFTHelper method)</a>
+</li>
+        <li><a href="pySDC/helpers.spectral_helper.html#helpers.spectral_helper.SpectralHelper.get_integration_matrix">(SpectralHelper method)</a>
+</li>
+        <li><a href="pySDC/helpers.spectral_helper.html#helpers.spectral_helper.SpectralHelper1D.get_integration_matrix">(SpectralHelper1D method)</a>
+</li>
+        <li><a href="pySDC/helpers.spectral_helper.html#helpers.spectral_helper.UltrasphericalHelper.get_integration_matrix">(UltrasphericalHelper method)</a>
 </li>
       </ul></li>
       <li><a href="pySDC/implementations.problem_classes.PenningTrap_3D.html#implementations.problem_classes.PenningTrap_3D.penningtrap.get_interactions">get_interactions() (penningtrap method)</a>
@@ -1601,6 +1839,8 @@ <h2 id="G">G</h2>
         <li><a href="pySDC/implementations.convergence_controller_classes.adaptivity.html#implementations.convergence_controller_classes.adaptivity.AdaptivityResidual.get_local_error_estimate">(AdaptivityResidual method)</a>
 </li>
       </ul></li>
+      <li><a href="pySDC/helpers.spectral_helper.html#helpers.spectral_helper.SpectralHelper.get_local_slice_of_1D_matrix">get_local_slice_of_1D_matrix() (SpectralHelper method)</a>
+</li>
       <li><a href="pySDC/implementations.convergence_controller_classes.adaptivity.html#implementations.convergence_controller_classes.adaptivity.Adaptivity.get_new_step_size">get_new_step_size() (Adaptivity method)</a>
 
       <ul>
@@ -1615,6 +1855,8 @@ <h2 id="G">G</h2>
         <li><a href="pySDC/implementations.convergence_controller_classes.adaptivity.html#implementations.convergence_controller_classes.adaptivity.AdaptivityResidual.get_new_step_size">(AdaptivityResidual method)</a>
 </li>
         <li><a href="pySDC/implementations.convergence_controller_classes.adaptivity.html#implementations.convergence_controller_classes.adaptivity.AdaptivityRK.get_new_step_size">(AdaptivityRK method)</a>
+</li>
+        <li><a href="pySDC/implementations.problem_classes.RayleighBenard.html#implementations.problem_classes.RayleighBenard.CFLLimit.get_new_step_size">(CFLLimit method)</a>
 </li>
         <li><a href="pySDC/core.convergence_controller.html#core.convergence_controller.ConvergenceController.get_new_step_size">(ConvergenceController method)</a>
 </li>
@@ -1623,9 +1865,11 @@ <h2 id="G">G</h2>
         <li><a href="pySDC/implementations.convergence_controller_classes.step_size_limiter.html#implementations.convergence_controller_classes.step_size_limiter.StepSizeSlopeLimiter.get_new_step_size">(StepSizeSlopeLimiter method)</a>
 </li>
       </ul></li>
-  </ul></td>
-  <td style="width: 33%; vertical-align: top;"><ul>
       <li><a href="pySDC/implementations.problem_classes.Quench.html#implementations.problem_classes.Quench.Quench.get_non_linear_Jacobian">get_non_linear_Jacobian() (Quench method)</a>
+</li>
+      <li><a href="pySDC/helpers.spectral_helper.html#helpers.spectral_helper.ChebychevHelper.get_norm">get_norm() (ChebychevHelper method)</a>
+</li>
+      <li><a href="pySDC/helpers.spectral_helper.html#helpers.spectral_helper.FFTHelper.get_Nyquist_mode_index">get_Nyquist_mode_index() (FFTHelper method)</a>
 </li>
       <li><a href="pySDC/helpers.NCCL_communicator.html#helpers.NCCL_communicator.NCCLComm.get_op">get_op() (NCCLComm method)</a>
 </li>
@@ -1636,6 +1880,8 @@ <h2 id="G">G</h2>
       <li><a href="pySDC/core.sweeper.html#core.sweeper.Sweeper.get_Qdelta_explicit">get_Qdelta_explicit() (Sweeper method)</a>
 </li>
       <li><a href="pySDC/core.sweeper.html#core.sweeper.Sweeper.get_Qdelta_implicit">get_Qdelta_implicit() (Sweeper method)</a>
+</li>
+      <li><a href="pySDC/helpers.spectral_helper.html#helpers.spectral_helper.UltrasphericalHelper.get_S">get_S() (UltrasphericalHelper method)</a>
 </li>
       <li><a href="pySDC/implementations.sweeper_classes.imex_1st_order.html#implementations.sweeper_classes.imex_1st_order.imex_1st_order.get_scalar_problems_manysweep_mat">get_scalar_problems_manysweep_mat() (imex_1st_order method)</a>
 </li>
@@ -1699,6 +1945,18 @@ <h2 id="G">G</h2>
         <li><a href="pySDC/implementations.sweeper_classes.Runge_Kutta.html#implementations.sweeper_classes.Runge_Kutta.RungeKutta.get_update_order">(RungeKutta class method)</a>
 </li>
       </ul></li>
+      <li><a href="pySDC/helpers.spectral_helper.html#helpers.spectral_helper.ChebychevHelper.get_wavenumbers">get_wavenumbers() (ChebychevHelper method)</a>
+
+      <ul>
+        <li><a href="pySDC/helpers.spectral_helper.html#helpers.spectral_helper.FFTHelper.get_wavenumbers">(FFTHelper method)</a>
+</li>
+        <li><a href="pySDC/helpers.spectral_helper.html#helpers.spectral_helper.SpectralHelper.get_wavenumbers">(SpectralHelper method)</a>
+</li>
+        <li><a href="pySDC/helpers.spectral_helper.html#helpers.spectral_helper.SpectralHelper1D.get_wavenumbers">(SpectralHelper1D method)</a>
+</li>
+      </ul></li>
+      <li><a href="pySDC/helpers.spectral_helper.html#helpers.spectral_helper.SpectralHelper1D.get_zero">get_zero() (SpectralHelper1D method)</a>
+</li>
       <li><a href="pySDC/implementations.problem_classes.boussinesq_helpers.build2DFDMatrix.html#implementations.problem_classes.boussinesq_helpers.build2DFDMatrix.getBCHorizontal">getBCHorizontal() (in module implementations.problem_classes.boussinesq_helpers.build2DFDMatrix)</a>
 </li>
       <li><a href="pySDC/implementations.problem_classes.acoustic_helpers.buildFDMatrix.html#implementations.problem_classes.acoustic_helpers.buildFDMatrix.getBCLeft">getBCLeft() (in module implementations.problem_classes.acoustic_helpers.buildFDMatrix)</a>
@@ -1744,6 +2002,8 @@ <h2 id="G">G</h2>
       <li><a href="pySDC/implementations.problem_classes.acoustic_helpers.buildWave1DMatrix.html#implementations.problem_classes.acoustic_helpers.buildWave1DMatrix.getWaveBCLeft">getWaveBCLeft() (in module implementations.problem_classes.acoustic_helpers.buildWave1DMatrix)</a>
 </li>
       <li><a href="pySDC/implementations.problem_classes.acoustic_helpers.buildWave1DMatrix.html#implementations.problem_classes.acoustic_helpers.buildWave1DMatrix.getWaveBCRight">getWaveBCRight() (in module implementations.problem_classes.acoustic_helpers.buildWave1DMatrix)</a>
+</li>
+      <li><a href="pySDC/helpers.spectral_helper.html#helpers.spectral_helper.SpectralHelper.global_shape">global_shape (SpectralHelper attribute)</a>
 </li>
       <li><a href="pySDC/implementations.problem_classes.Boussinesq_2D_FD_imex.html#implementations.problem_classes.Boussinesq_2D_FD_imex.boussinesq_2d_imex.gmres_logger">gmres_logger (boussinesq_2d_imex attribute)</a>
 </li>
@@ -1770,8 +2030,16 @@ <h2 id="H">H</h2>
       <li><a href="pySDC/implementations.problem_classes.Boussinesq_2D_FD_imex.html#implementations.problem_classes.Boussinesq_2D_FD_imex.boussinesq_2d_imex.h">h (boussinesq_2d_imex attribute)</a>
 </li>
       <li><a href="pySDC/implementations.problem_classes.HarmonicOscillator.html#implementations.problem_classes.HarmonicOscillator.harmonic_oscillator">harmonic_oscillator (class in implementations.problem_classes.HarmonicOscillator)</a>
+</li>
+      <li><a href="pySDC/implementations.problem_classes.HeatEquation_Chebychev.html#implementations.problem_classes.HeatEquation_Chebychev.Heat1DChebychev">Heat1DChebychev (class in implementations.problem_classes.HeatEquation_Chebychev)</a>
+</li>
+      <li><a href="pySDC/implementations.problem_classes.HeatEquation_Chebychev.html#implementations.problem_classes.HeatEquation_Chebychev.Heat1DUltraspherical">Heat1DUltraspherical (class in implementations.problem_classes.HeatEquation_Chebychev)</a>
 </li>
       <li><a href="pySDC/implementations.problem_classes.HeatEquation_2D_PETSc_forced.html#implementations.problem_classes.HeatEquation_2D_PETSc_forced.heat2d_petsc_forced">heat2d_petsc_forced (class in implementations.problem_classes.HeatEquation_2D_PETSc_forced)</a>
+</li>
+      <li><a href="pySDC/implementations.problem_classes.HeatEquation_Chebychev.html#implementations.problem_classes.HeatEquation_Chebychev.Heat2DChebychev">Heat2DChebychev (class in implementations.problem_classes.HeatEquation_Chebychev)</a>
+</li>
+      <li><a href="pySDC/implementations.problem_classes.HeatEquation_Chebychev.html#implementations.problem_classes.HeatEquation_Chebychev.Heat2DUltraspherical">Heat2DUltraspherical (class in implementations.problem_classes.HeatEquation_Chebychev)</a>
 </li>
       <li><a href="pySDC/implementations.problem_classes.HeatEquation_ND_FD.html#implementations.problem_classes.HeatEquation_ND_FD.heatNd_forced">heatNd_forced (class in implementations.problem_classes.HeatEquation_ND_FD)</a>
 
@@ -1813,6 +2081,8 @@ <h2 id="H">H</h2>
         <li><a href="pySDC/helpers.problem_helper.html#module-helpers.problem_helper">module</a>
 </li>
       </ul></li>
+  </ul></td>
+  <td style="width: 33%; vertical-align: top;"><ul>
       <li>
     helpers.pysdc_helper
 
@@ -1820,13 +2090,18 @@ <h2 id="H">H</h2>
         <li><a href="pySDC/helpers.pysdc_helper.html#module-helpers.pysdc_helper">module</a>
 </li>
       </ul></li>
-  </ul></td>
-  <td style="width: 33%; vertical-align: top;"><ul>
       <li>
     helpers.setup_helper
 
       <ul>
         <li><a href="pySDC/helpers.setup_helper.html#module-helpers.setup_helper">module</a>
+</li>
+      </ul></li>
+      <li>
+    helpers.spectral_helper
+
+      <ul>
+        <li><a href="pySDC/helpers.spectral_helper.html#module-helpers.spectral_helper">module</a>
 </li>
       </ul></li>
       <li>
@@ -2289,6 +2564,13 @@ <h2 id="I">I</h2>
 
       <ul>
         <li><a href="pySDC/implementations.problem_classes.BuckConverter.html#module-implementations.problem_classes.BuckConverter">module</a>
+</li>
+      </ul></li>
+      <li>
+    implementations.problem_classes.Burgers
+
+      <ul>
+        <li><a href="pySDC/implementations.problem_classes.Burgers.html#module-implementations.problem_classes.Burgers">module</a>
 </li>
       </ul></li>
       <li>
@@ -2326,8 +2608,6 @@ <h2 id="I">I</h2>
         <li><a href="pySDC/implementations.problem_classes.GeneralizedFisher_1D_FD_implicit.html#module-implementations.problem_classes.GeneralizedFisher_1D_FD_implicit">module</a>
 </li>
       </ul></li>
-  </ul></td>
-  <td style="width: 33%; vertical-align: top;"><ul>
       <li>
     implementations.problem_classes.GeneralizedFisher_1D_PETSc
 
@@ -2347,6 +2627,15 @@ <h2 id="I">I</h2>
 
       <ul>
         <li><a href="pySDC/implementations.problem_classes.generic_ND_FD.html#module-implementations.problem_classes.generic_ND_FD">module</a>
+</li>
+      </ul></li>
+  </ul></td>
+  <td style="width: 33%; vertical-align: top;"><ul>
+      <li>
+    implementations.problem_classes.generic_spectral
+
+      <ul>
+        <li><a href="pySDC/implementations.problem_classes.generic_spectral.html#module-implementations.problem_classes.generic_spectral">module</a>
 </li>
       </ul></li>
       <li>
@@ -2389,6 +2678,13 @@ <h2 id="I">I</h2>
 
       <ul>
         <li><a href="pySDC/implementations.problem_classes.HeatEquation_2D_PETSc_forced.html#module-implementations.problem_classes.HeatEquation_2D_PETSc_forced">module</a>
+</li>
+      </ul></li>
+      <li>
+    implementations.problem_classes.HeatEquation_Chebychev
+
+      <ul>
+        <li><a href="pySDC/implementations.problem_classes.HeatEquation_Chebychev.html#module-implementations.problem_classes.HeatEquation_Chebychev">module</a>
 </li>
       </ul></li>
       <li>
@@ -2487,6 +2783,13 @@ <h2 id="I">I</h2>
 
       <ul>
         <li><a href="pySDC/implementations.problem_classes.Quench.html#module-implementations.problem_classes.Quench">module</a>
+</li>
+      </ul></li>
+      <li>
+    implementations.problem_classes.RayleighBenard
+
+      <ul>
+        <li><a href="pySDC/implementations.problem_classes.RayleighBenard.html#module-implementations.problem_classes.RayleighBenard">module</a>
 </li>
       </ul></li>
       <li>
@@ -2681,6 +2984,12 @@ <h2 id="I">I</h2>
       <li><a href="pySDC/implementations.sweeper_classes.Runge_Kutta.html#implementations.sweeper_classes.Runge_Kutta.ImplicitMidpointMethod">ImplicitMidpointMethod (class in implementations.sweeper_classes.Runge_Kutta)</a>
 </li>
       <li><a href="pySDC/core.hooks.html#core.hooks.Hooks.increment_stats">increment_stats() (Hooks method)</a>
+</li>
+      <li><a href="pySDC/helpers.spectral_helper.html#helpers.spectral_helper.SpectralHelper.index">index() (SpectralHelper method)</a>
+</li>
+      <li><a href="pySDC/helpers.spectral_helper.html#helpers.spectral_helper.SpectralHelper.init">init (SpectralHelper attribute)</a>
+</li>
+      <li><a href="pySDC/helpers.spectral_helper.html#helpers.spectral_helper.SpectralHelper.init_forward">init_forward (SpectralHelper attribute)</a>
 </li>
       <li><a href="pySDC/core.step.html#core.step.Step.init_step">init_step() (Step method)</a>
 </li>
@@ -2765,6 +3074,16 @@ <h2 id="I">I</h2>
 </li>
       </ul></li>
       <li><a href="pySDC/core.hooks.html#core.hooks.Entry.iter">iter (Entry attribute)</a>
+</li>
+      <li><a href="pySDC/helpers.spectral_helper.html#helpers.spectral_helper.ChebychevHelper.itransform">itransform() (ChebychevHelper method)</a>
+
+      <ul>
+        <li><a href="pySDC/helpers.spectral_helper.html#helpers.spectral_helper.FFTHelper.itransform">(FFTHelper method)</a>
+</li>
+        <li><a href="pySDC/helpers.spectral_helper.html#helpers.spectral_helper.SpectralHelper.itransform">(SpectralHelper method)</a>
+</li>
+      </ul></li>
+      <li><a href="pySDC/helpers.spectral_helper.html#helpers.spectral_helper.SpectralHelper.itransform_single_component">itransform_single_component() (SpectralHelper method)</a>
 </li>
   </ul></td>
 </tr></table>
@@ -2847,7 +3166,13 @@ <h2 id="L">L</h2>
 <table style="width: 100%" class="indextable genindextable"><tr>
   <td style="width: 33%; vertical-align: top;"><ul>
       <li><a href="pySDC/implementations.problem_classes.AllenCahn_Temp_MPIFFT.html#implementations.problem_classes.AllenCahn_Temp_MPIFFT.allencahn_temp_imex.L">L (allencahn_temp_imex attribute)</a>
+
+      <ul>
+        <li><a href="pySDC/implementations.problem_classes.generic_spectral.html#implementations.problem_classes.generic_spectral.GenericSpectralLinear.L">(GenericSpectralLinear attribute)</a>
 </li>
+        <li><a href="pySDC/helpers.spectral_helper.html#helpers.spectral_helper.SpectralHelper1D.L">(SpectralHelper1D attribute)</a>
+</li>
+      </ul></li>
       <li><a href="pySDC/implementations.problem_classes.AdvectionDiffusionEquation_1D_FFT.html#implementations.problem_classes.AdvectionDiffusionEquation_1D_FFT.advectiondiffusion1d_imex.lap">lap (advectiondiffusion1d_imex attribute)</a>
 
       <ul>
@@ -2878,6 +3203,16 @@ <h2 id="L">L</h2>
 </li>
       <li><a href="pySDC/implementations.problem_classes.AllenCahn_2D_FD.html#implementations.problem_classes.AllenCahn_2D_FD.allencahn_fullyimplicit.lin_ncalls">lin_ncalls (allencahn_fullyimplicit attribute)</a>, <a href="pySDC/implementations.problem_classes.AllenCahn_2D_FD_gpu.html#implementations.problem_classes.AllenCahn_2D_FD_gpu.allencahn_fullyimplicit.lin_ncalls">[1]</a>
 </li>
+      <li><a href="pySDC/helpers.spectral_helper.html#helpers.spectral_helper.SpectralHelper.linalg">linalg (SpectralHelper attribute)</a>
+
+      <ul>
+        <li><a href="pySDC/helpers.spectral_helper.html#helpers.spectral_helper.SpectralHelper1D.linalg">(SpectralHelper1D attribute)</a>
+</li>
+      </ul></li>
+      <li><a href="pySDC/helpers.spectral_helper.html#helpers.spectral_helper.SpectralHelper.local_slice">local_slice (SpectralHelper attribute)</a>
+</li>
+  </ul></td>
+  <td style="width: 33%; vertical-align: top;"><ul>
       <li><a href="pySDC/implementations.problem_classes.GeneralizedFisher_1D_PETSc.html#implementations.problem_classes.GeneralizedFisher_1D_PETSc.Fisher_full.localX">localX (Fisher_full attribute)</a>
 
       <ul>
@@ -2892,9 +3227,11 @@ <h2 id="L">L</h2>
         <li><a href="pySDC/implementations.problem_classes.GrayScott_2D_PETSc_periodic.html#implementations.problem_classes.GrayScott_2D_PETSc_periodic.petsc_grayscott_multiimplicit.localX">(petsc_grayscott_multiimplicit attribute)</a>
 </li>
       </ul></li>
-  </ul></td>
-  <td style="width: 33%; vertical-align: top;"><ul>
       <li><a href="pySDC/core.convergence_controller.html#core.convergence_controller.ConvergenceController.log">log() (ConvergenceController method)</a>
+</li>
+      <li><a href="pySDC/implementations.problem_classes.RayleighBenard.html#implementations.problem_classes.RayleighBenard.LogAnalysisVariables">LogAnalysisVariables (class in implementations.problem_classes.RayleighBenard)</a>
+</li>
+      <li><a href="pySDC/implementations.problem_classes.RayleighBenard.html#implementations.problem_classes.RayleighBenard.LogCFL">LogCFL (class in implementations.problem_classes.RayleighBenard)</a>
 </li>
       <li><a href="pySDC/implementations.transfer_classes.BaseTransfer_mass.html#implementations.transfer_classes.BaseTransfer_mass.base_transfer_mass.logger">logger (base_transfer_mass attribute)</a>
 
@@ -2940,6 +3277,8 @@ <h2 id="M">M</h2>
         <li><a href="pySDC/implementations.problem_classes.VorticityVelocity_2D_FEniCS_periodic.html#implementations.problem_classes.VorticityVelocity_2D_FEniCS_periodic.fenics_vortex_2d.M">(fenics_vortex_2d attribute)</a>
 </li>
         <li><a href="pySDC/implementations.problem_classes.VorticityVelocity_2D_FEniCS_periodic.html#implementations.problem_classes.VorticityVelocity_2D_FEniCS_periodic.fenics_vortex_2d_mass.M">(fenics_vortex_2d_mass attribute)</a>
+</li>
+        <li><a href="pySDC/implementations.problem_classes.generic_spectral.html#implementations.problem_classes.generic_spectral.GenericSpectralLinear.M">(GenericSpectralLinear attribute)</a>
 </li>
       </ul></li>
       <li><a href="pySDC/implementations.datatype_classes.particles.html#implementations.datatype_classes.particles.fields.magn">magn (fields attribute)</a>
@@ -3063,6 +3402,8 @@ <h2 id="M">M</h2>
         <li><a href="pySDC/helpers.pysdc_helper.html#module-helpers.pysdc_helper">helpers.pysdc_helper</a>
 </li>
         <li><a href="pySDC/helpers.setup_helper.html#module-helpers.setup_helper">helpers.setup_helper</a>
+</li>
+        <li><a href="pySDC/helpers.spectral_helper.html#module-helpers.spectral_helper">helpers.spectral_helper</a>
 </li>
         <li><a href="pySDC/helpers.stats_helper.html#module-helpers.stats_helper">helpers.stats_helper</a>
 </li>
@@ -3179,6 +3520,8 @@ <h2 id="M">M</h2>
         <li><a href="pySDC/implementations.problem_classes.Brusselator.html#module-implementations.problem_classes.Brusselator">implementations.problem_classes.Brusselator</a>
 </li>
         <li><a href="pySDC/implementations.problem_classes.BuckConverter.html#module-implementations.problem_classes.BuckConverter">implementations.problem_classes.BuckConverter</a>
+</li>
+        <li><a href="pySDC/implementations.problem_classes.Burgers.html#module-implementations.problem_classes.Burgers">implementations.problem_classes.Burgers</a>
 </li>
         <li><a href="pySDC/implementations.problem_classes.DiscontinuousTestODE.html#module-implementations.problem_classes.DiscontinuousTestODE">implementations.problem_classes.DiscontinuousTestODE</a>
 </li>
@@ -3195,6 +3538,8 @@ <h2 id="M">M</h2>
         <li><a href="pySDC/implementations.problem_classes.generic_MPIFFT_Laplacian.html#module-implementations.problem_classes.generic_MPIFFT_Laplacian">implementations.problem_classes.generic_MPIFFT_Laplacian</a>
 </li>
         <li><a href="pySDC/implementations.problem_classes.generic_ND_FD.html#module-implementations.problem_classes.generic_ND_FD">implementations.problem_classes.generic_ND_FD</a>
+</li>
+        <li><a href="pySDC/implementations.problem_classes.generic_spectral.html#module-implementations.problem_classes.generic_spectral">implementations.problem_classes.generic_spectral</a>
 </li>
         <li><a href="pySDC/implementations.problem_classes.GrayScott_1D_FEniCS_implicit.html#module-implementations.problem_classes.GrayScott_1D_FEniCS_implicit">implementations.problem_classes.GrayScott_1D_FEniCS_implicit</a>
 </li>
@@ -3207,6 +3552,8 @@ <h2 id="M">M</h2>
         <li><a href="pySDC/implementations.problem_classes.HeatEquation_1D_FEniCS_matrix_forced.html#module-implementations.problem_classes.HeatEquation_1D_FEniCS_matrix_forced">implementations.problem_classes.HeatEquation_1D_FEniCS_matrix_forced</a>
 </li>
         <li><a href="pySDC/implementations.problem_classes.HeatEquation_2D_PETSc_forced.html#module-implementations.problem_classes.HeatEquation_2D_PETSc_forced">implementations.problem_classes.HeatEquation_2D_PETSc_forced</a>
+</li>
+        <li><a href="pySDC/implementations.problem_classes.HeatEquation_Chebychev.html#module-implementations.problem_classes.HeatEquation_Chebychev">implementations.problem_classes.HeatEquation_Chebychev</a>
 </li>
         <li><a href="pySDC/implementations.problem_classes.HeatEquation_ND_FD.html#module-implementations.problem_classes.HeatEquation_ND_FD">implementations.problem_classes.HeatEquation_ND_FD</a>
 </li>
@@ -3235,6 +3582,8 @@ <h2 id="M">M</h2>
         <li><a href="pySDC/implementations.problem_classes.polynomial_test_problem.html#module-implementations.problem_classes.polynomial_test_problem">implementations.problem_classes.polynomial_test_problem</a>
 </li>
         <li><a href="pySDC/implementations.problem_classes.Quench.html#module-implementations.problem_classes.Quench">implementations.problem_classes.Quench</a>
+</li>
+        <li><a href="pySDC/implementations.problem_classes.RayleighBenard.html#module-implementations.problem_classes.RayleighBenard">implementations.problem_classes.RayleighBenard</a>
 </li>
         <li><a href="pySDC/implementations.problem_classes.TestEquation_0D.html#module-implementations.problem_classes.TestEquation_0D">implementations.problem_classes.TestEquation_0D</a>
 </li>
@@ -3308,8 +3657,14 @@ <h2 id="N">N</h2>
 <table style="width: 100%" class="indextable genindextable"><tr>
   <td style="width: 33%; vertical-align: top;"><ul>
       <li><a href="pySDC/implementations.problem_classes.Boussinesq_2D_FD_imex.html#implementations.problem_classes.Boussinesq_2D_FD_imex.boussinesq_2d_imex.N">N (boussinesq_2d_imex attribute)</a>
+
+      <ul>
+        <li><a href="pySDC/helpers.spectral_helper.html#helpers.spectral_helper.SpectralHelper1D.N">(SpectralHelper1D attribute)</a>
 </li>
+      </ul></li>
       <li><a href="pySDC/helpers.NCCL_communicator.html#helpers.NCCL_communicator.NCCLComm">NCCLComm (class in helpers.NCCL_communicator)</a>
+</li>
+      <li><a href="pySDC/helpers.spectral_helper.html#helpers.spectral_helper.SpectralHelper.ncomponents">ncomponents (SpectralHelper property)</a>
 </li>
       <li><a href="pySDC/implementations.problem_classes.generic_ND_FD.html#implementations.problem_classes.generic_ND_FD.GenericNDimFinDiff.ndim">ndim (GenericNDimFinDiff property)</a>
 
@@ -3319,6 +3674,8 @@ <h2 id="N">N</h2>
         <li><a href="pySDC/implementations.problem_classes.GrayScott_MPIFFT.html#implementations.problem_classes.GrayScott_MPIFFT.grayscott_mi_diffusion.ndim">(grayscott_mi_diffusion attribute)</a>
 </li>
         <li><a href="pySDC/implementations.problem_classes.HeatEquation_ND_FD_CuPy.html#implementations.problem_classes.HeatEquation_ND_FD_CuPy.heatNd_forced.ndim">(heatNd_forced property)</a>
+</li>
+        <li><a href="pySDC/helpers.spectral_helper.html#helpers.spectral_helper.SpectralHelper.ndim">(SpectralHelper property)</a>
 </li>
       </ul></li>
       <li><a href="pySDC/helpers.plot_helper.html#helpers.plot_helper.newfig">newfig() (in module helpers.plot_helper)</a>
@@ -3486,11 +3843,19 @@ <h2 id="P">P</h2>
 </li>
       </ul></li>
       <li><a href="pySDC/implementations.problem_classes.Piline.html#implementations.problem_classes.Piline.piline">piline (class in implementations.problem_classes.Piline)</a>
+</li>
+      <li><a href="pySDC/implementations.problem_classes.generic_spectral.html#implementations.problem_classes.generic_spectral.GenericSpectralLinear.Pl">Pl (GenericSpectralLinear attribute)</a>
 </li>
       <li><a href="pySDC/implementations.problem_classes.Brusselator.html#implementations.problem_classes.Brusselator.Brusselator.plot">plot() (Brusselator method)</a>
 
       <ul>
+        <li><a href="pySDC/implementations.problem_classes.Burgers.html#implementations.problem_classes.Burgers.Burgers1D.plot">(Burgers1D method)</a>
+</li>
+        <li><a href="pySDC/implementations.problem_classes.Burgers.html#implementations.problem_classes.Burgers.Burgers2D.plot">(Burgers2D method)</a>
+</li>
         <li><a href="pySDC/core.problem.html#core.problem.Problem.plot">(Problem method)</a>
+</li>
+        <li><a href="pySDC/implementations.problem_classes.RayleighBenard.html#implementations.problem_classes.RayleighBenard.RayleighBenard.plot">(RayleighBenard method)</a>
 </li>
       </ul></li>
       <li><a href="pySDC/implementations.problem_classes.polynomial_test_problem.html#implementations.problem_classes.polynomial_test_problem.polynomial_testequation">polynomial_testequation (class in implementations.problem_classes.polynomial_test_problem)</a>
@@ -3554,10 +3919,18 @@ <h2 id="P">P</h2>
   </ul></td>
   <td style="width: 33%; vertical-align: top;"><ul>
       <li><a href="pySDC/core.hooks.html#core.hooks.Hooks.post_step">post_step() (Hooks method)</a>
+
+      <ul>
+        <li><a href="pySDC/implementations.problem_classes.RayleighBenard.html#implementations.problem_classes.RayleighBenard.LogAnalysisVariables.post_step">(LogAnalysisVariables method)</a>
 </li>
+        <li><a href="pySDC/implementations.problem_classes.RayleighBenard.html#implementations.problem_classes.RayleighBenard.LogCFL.post_step">(LogCFL method)</a>
+</li>
+      </ul></li>
       <li><a href="pySDC/core.convergence_controller.html#core.convergence_controller.ConvergenceController.post_step_processing">post_step_processing() (ConvergenceController method)</a>
 </li>
       <li><a href="pySDC/core.hooks.html#core.hooks.Hooks.post_sweep">post_sweep() (Hooks method)</a>
+</li>
+      <li><a href="pySDC/implementations.problem_classes.generic_spectral.html#implementations.problem_classes.generic_spectral.GenericSpectralLinear.Pr">Pr (GenericSpectralLinear attribute)</a>
 </li>
       <li><a href="pySDC/core.hooks.html#core.hooks.Hooks.pre_comm">pre_comm() (Hooks method)</a>
 </li>
@@ -3673,6 +4046,12 @@ <h2 id="P">P</h2>
         <li><a href="pySDC/implementations.transfer_classes.TransferMesh_FFT2D.html#implementations.transfer_classes.TransferMesh_FFT2D.mesh_to_mesh_fft2d.Pspace">(mesh_to_mesh_fft2d attribute)</a>
 </li>
       </ul></li>
+      <li><a href="pySDC/helpers.spectral_helper.html#helpers.spectral_helper.SpectralHelper.put_BCs_in_matrix">put_BCs_in_matrix() (SpectralHelper method)</a>
+</li>
+      <li><a href="pySDC/helpers.spectral_helper.html#helpers.spectral_helper.SpectralHelper.put_BCs_in_rhs">put_BCs_in_rhs() (SpectralHelper method)</a>
+</li>
+      <li><a href="pySDC/helpers.spectral_helper.html#helpers.spectral_helper.SpectralHelper.put_BCs_in_rhs_hat">put_BCs_in_rhs_hat() (SpectralHelper method)</a>
+</li>
   </ul></td>
 </tr></table>
 
@@ -3727,6 +4106,8 @@ <h2 id="R">R</h2>
         <li><a href="pySDC/implementations.sweeper_classes.generic_implicit_MPI.html#implementations.sweeper_classes.generic_implicit_MPI.SweeperMPI.rank">(SweeperMPI property)</a>
 </li>
       </ul></li>
+      <li><a href="pySDC/implementations.problem_classes.RayleighBenard.html#implementations.problem_classes.RayleighBenard.RayleighBenard">RayleighBenard (class in implementations.problem_classes.RayleighBenard)</a>
+</li>
       <li><a href="pySDC/core.errors.html#core.errors.ReadOnlyError">ReadOnlyError</a>
 </li>
       <li><a href="pySDC/implementations.controller_classes.controller_MPI.html#implementations.controller_classes.controller_MPI.controller_MPI.recv">recv() (controller_MPI method)</a>
@@ -3748,6 +4129,8 @@ <h2 id="R">R</h2>
       <li><a href="pySDC/helpers.NCCL_communicator.html#helpers.NCCL_communicator.NCCLComm.Reduce">Reduce() (NCCLComm method)</a>
 </li>
       <li><a href="pySDC/core.common.html#core.common.RegisterParams">RegisterParams (class in core.common)</a>
+</li>
+      <li><a href="pySDC/helpers.spectral_helper.html#helpers.spectral_helper.SpectralHelper.remove_BC">remove_BC() (SpectralHelper method)</a>
 </li>
       <li><a href="pySDC/implementations.convergence_controller_classes.basic_restarting.html#implementations.convergence_controller_classes.basic_restarting.BasicRestartingNonMPI.reset_buffers_nonMPI">reset_buffers_nonMPI() (BasicRestartingNonMPI method)</a>
 
@@ -3818,6 +4201,8 @@ <h2 id="R">R</h2>
 </li>
       </ul></li>
       <li><a href="pySDC/implementations.transfer_classes.TransferMesh_FFT.html#implementations.transfer_classes.TransferMesh_FFT.mesh_to_mesh_fft.rfft_object_coarse">rfft_object_coarse (mesh_to_mesh_fft attribute)</a>
+</li>
+      <li><a href="pySDC/helpers.spectral_helper.html#helpers.spectral_helper.SpectralHelper.rhs_BCs_hat">rhs_BCs_hat (SpectralHelper attribute)</a>
 </li>
       <li><a href="pySDC/implementations.datatype_classes.fenics_mesh.html#implementations.datatype_classes.fenics_mesh.rhs_fenics_mesh">rhs_fenics_mesh (class in implementations.datatype_classes.fenics_mesh)</a>
 </li>
@@ -3909,6 +4294,8 @@ <h2 id="S">S</h2>
         <li><a href="pySDC/implementations.convergence_controller_classes.adaptivity.html#implementations.convergence_controller_classes.adaptivity.AdaptivityRK.setup">(AdaptivityRK method)</a>
 </li>
         <li><a href="pySDC/implementations.convergence_controller_classes.basic_restarting.html#implementations.convergence_controller_classes.basic_restarting.BasicRestarting.setup">(BasicRestarting method)</a>
+</li>
+        <li><a href="pySDC/implementations.problem_classes.RayleighBenard.html#implementations.problem_classes.RayleighBenard.CFLLimit.setup">(CFLLimit method)</a>
 </li>
         <li><a href="pySDC/implementations.convergence_controller_classes.check_convergence.html#implementations.convergence_controller_classes.check_convergence.CheckConvergence.setup">(CheckConvergence method)</a>
 </li>
@@ -3951,15 +4338,31 @@ <h2 id="S">S</h2>
         <li><a href="pySDC/implementations.convergence_controller_classes.store_uold.html#implementations.convergence_controller_classes.store_uold.StoreUOld.setup">(StoreUOld method)</a>
 </li>
       </ul></li>
+      <li><a href="pySDC/helpers.spectral_helper.html#helpers.spectral_helper.SpectralHelper.setup_BCs">setup_BCs() (SpectralHelper method)</a>
+</li>
       <li><a href="pySDC/core.controller.html#core.controller.Controller.setup_convergence_controllers">setup_convergence_controllers() (Controller method)</a>
 </li>
-      <li><a href="pySDC/implementations.problem_classes.generic_MPIFFT_Laplacian.html#implementations.problem_classes.generic_MPIFFT_Laplacian.IMEX_Laplacian_MPIFFT.setup_GPU">setup_GPU() (IMEX_Laplacian_MPIFFT class method)</a>
+      <li><a href="pySDC/helpers.spectral_helper.html#helpers.spectral_helper.SpectralHelper.setup_fft">setup_fft() (SpectralHelper method)</a>
+</li>
+      <li><a href="pySDC/implementations.problem_classes.generic_spectral.html#implementations.problem_classes.generic_spectral.GenericSpectralLinear.setup_GPU">setup_GPU() (GenericSpectralLinear class method)</a>
 
       <ul>
+        <li><a href="pySDC/implementations.problem_classes.generic_MPIFFT_Laplacian.html#implementations.problem_classes.generic_MPIFFT_Laplacian.IMEX_Laplacian_MPIFFT.setup_GPU">(IMEX_Laplacian_MPIFFT class method)</a>
+</li>
+        <li><a href="pySDC/helpers.spectral_helper.html#helpers.spectral_helper.SpectralHelper.setup_GPU">(SpectralHelper class method)</a>
+</li>
+        <li><a href="pySDC/helpers.spectral_helper.html#helpers.spectral_helper.SpectralHelper1D.setup_GPU">(SpectralHelper1D class method)</a>
+</li>
         <li><a href="pySDC/implementations.problem_classes.TestEquation_0D.html#implementations.problem_classes.TestEquation_0D.testequation0d.setup_GPU">(testequation0d class method)</a>
 </li>
       </ul></li>
+      <li><a href="pySDC/implementations.problem_classes.generic_spectral.html#implementations.problem_classes.generic_spectral.GenericSpectralLinear.setup_L">setup_L() (GenericSpectralLinear method)</a>
+</li>
+      <li><a href="pySDC/implementations.problem_classes.generic_spectral.html#implementations.problem_classes.generic_spectral.GenericSpectralLinear.setup_M">setup_M() (GenericSpectralLinear method)</a>
+</li>
       <li><a href="pySDC/helpers.plot_helper.html#helpers.plot_helper.setup_mpl">setup_mpl() (in module helpers.plot_helper)</a>
+</li>
+      <li><a href="pySDC/implementations.problem_classes.generic_spectral.html#implementations.problem_classes.generic_spectral.GenericSpectralLinear.setup_preconditioner">setup_preconditioner() (GenericSpectralLinear method)</a>
 </li>
       <li><a href="pySDC/implementations.convergence_controller_classes.adaptive_collocation.html#implementations.convergence_controller_classes.adaptive_collocation.AdaptiveCollocation.setup_status_variables">setup_status_variables() (AdaptiveCollocation method)</a>
 
@@ -3969,6 +4372,8 @@ <h2 id="S">S</h2>
         <li><a href="pySDC/implementations.convergence_controller_classes.adaptivity.html#implementations.convergence_controller_classes.adaptivity.AdaptivityResidual.setup_status_variables">(AdaptivityResidual method)</a>
 </li>
         <li><a href="pySDC/implementations.convergence_controller_classes.basic_restarting.html#implementations.convergence_controller_classes.basic_restarting.BasicRestarting.setup_status_variables">(BasicRestarting method)</a>
+</li>
+        <li><a href="pySDC/implementations.problem_classes.RayleighBenard.html#implementations.problem_classes.RayleighBenard.CFLLimit.setup_status_variables">(CFLLimit method)</a>
 </li>
         <li><a href="pySDC/implementations.convergence_controller_classes.check_iteration_estimator.html#implementations.convergence_controller_classes.check_iteration_estimator.CheckIterationEstimatorNonMPI.setup_status_variables">(CheckIterationEstimatorNonMPI method)</a>
 </li>
@@ -3988,6 +4393,8 @@ <h2 id="S">S</h2>
 </li>
       </ul></li>
       <li><a href="pySDC/core.sweeper.html#core.sweeper.Sweeper.setupGenerator">setupGenerator() (Sweeper method)</a>
+</li>
+      <li><a href="pySDC/helpers.spectral_helper.html#helpers.spectral_helper.SpectralHelper.shape">shape (SpectralHelper property)</a>
 </li>
       <li><a href="pySDC/helpers.visualization_tools.html#helpers.visualization_tools.show_residual_across_simulation">show_residual_across_simulation() (in module helpers.visualization_tools)</a>
 </li>
@@ -4067,6 +4474,8 @@ <h2 id="S">S</h2>
         <li><a href="pySDC/implementations.problem_classes.GeneralizedFisher_1D_FD_implicit.html#implementations.problem_classes.GeneralizedFisher_1D_FD_implicit.generalized_fisher.solve_system">(generalized_fisher method)</a>
 </li>
         <li><a href="pySDC/implementations.problem_classes.generic_ND_FD.html#implementations.problem_classes.generic_ND_FD.GenericNDimFinDiff.solve_system">(GenericNDimFinDiff method)</a>
+</li>
+        <li><a href="pySDC/implementations.problem_classes.generic_spectral.html#implementations.problem_classes.generic_spectral.GenericSpectralLinear.solve_system">(GenericSpectralLinear method)</a>
 </li>
         <li><a href="pySDC/implementations.problem_classes.GrayScott_MPIFFT.html#implementations.problem_classes.GrayScott_MPIFFT.grayscott_imex_diffusion.solve_system">(grayscott_imex_diffusion method)</a>
 </li>
@@ -4153,7 +4562,21 @@ <h2 id="S">S</h2>
 </li>
       <li><a href="pySDC/core.space_transfer.html#core.space_transfer.SpaceTransfer">SpaceTransfer (class in core.space_transfer)</a>
 </li>
+      <li><a href="pySDC/helpers.spectral_helper.html#helpers.spectral_helper.SpectralHelper.sparse_lib">sparse_lib (SpectralHelper attribute)</a>
+
+      <ul>
+        <li><a href="pySDC/helpers.spectral_helper.html#helpers.spectral_helper.SpectralHelper1D.sparse_lib">(SpectralHelper1D attribute)</a>
+</li>
+      </ul></li>
       <li><a href="pySDC/implementations.problem_classes.AllenCahn_Temp_MPIFFT.html#implementations.problem_classes.AllenCahn_Temp_MPIFFT.allencahn_temp_imex.spectral">spectral (allencahn_temp_imex attribute)</a>
+
+      <ul>
+        <li><a href="pySDC/implementations.problem_classes.generic_spectral.html#implementations.problem_classes.generic_spectral.GenericSpectralLinear.spectral">(GenericSpectralLinear attribute)</a>
+</li>
+      </ul></li>
+      <li><a href="pySDC/helpers.spectral_helper.html#helpers.spectral_helper.SpectralHelper">SpectralHelper (class in helpers.spectral_helper)</a>
+</li>
+      <li><a href="pySDC/helpers.spectral_helper.html#helpers.spectral_helper.SpectralHelper1D">SpectralHelper1D (class in helpers.spectral_helper)</a>
 </li>
       <li><a href="pySDC/implementations.problem_classes.boussinesq_helpers.standard_integrators.html#implementations.problem_classes.boussinesq_helpers.standard_integrators.SplitExplicit">SplitExplicit (class in implementations.problem_classes.boussinesq_helpers.standard_integrators)</a>
 </li>
@@ -4266,6 +4689,16 @@ <h2 id="T">T</h2>
       <li><a href="pySDC/core.step.html#core.step.Step.transfer">transfer() (Step method)</a>
 </li>
       <li><a href="pySDC/core.errors.html#core.errors.TransferError">TransferError</a>
+</li>
+      <li><a href="pySDC/helpers.spectral_helper.html#helpers.spectral_helper.ChebychevHelper.transform">transform() (ChebychevHelper method)</a>
+
+      <ul>
+        <li><a href="pySDC/helpers.spectral_helper.html#helpers.spectral_helper.FFTHelper.transform">(FFTHelper method)</a>
+</li>
+        <li><a href="pySDC/helpers.spectral_helper.html#helpers.spectral_helper.SpectralHelper.transform">(SpectralHelper method)</a>
+</li>
+      </ul></li>
+      <li><a href="pySDC/helpers.spectral_helper.html#helpers.spectral_helper.SpectralHelper.transform_single_component">transform_single_component() (SpectralHelper method)</a>
 </li>
       <li><a href="pySDC/implementations.problem_classes.acoustic_helpers.standard_integrators.html#implementations.problem_classes.acoustic_helpers.standard_integrators.trapezoidal">trapezoidal (class in implementations.problem_classes.acoustic_helpers.standard_integrators)</a>
 
@@ -4319,6 +4752,10 @@ <h2 id="U">U</h2>
         <li><a href="pySDC/implementations.problem_classes.Brusselator.html#implementations.problem_classes.Brusselator.Brusselator.u_exact">(Brusselator method)</a>
 </li>
         <li><a href="pySDC/implementations.problem_classes.BuckConverter.html#implementations.problem_classes.BuckConverter.buck_converter.u_exact">(buck_converter method)</a>
+</li>
+        <li><a href="pySDC/implementations.problem_classes.Burgers.html#implementations.problem_classes.Burgers.Burgers1D.u_exact">(Burgers1D method)</a>
+</li>
+        <li><a href="pySDC/implementations.problem_classes.Burgers.html#implementations.problem_classes.Burgers.Burgers2D.u_exact">(Burgers2D method)</a>
 </li>
         <li><a href="pySDC/implementations.problem_classes.odeSystem.html#implementations.problem_classes.odeSystem.ChemicalReaction3Var.u_exact">(ChemicalReaction3Var method)</a>
 </li>
@@ -4341,8 +4778,16 @@ <h2 id="U">U</h2>
         <li><a href="pySDC/implementations.problem_classes.GrayScott_MPIFFT.html#implementations.problem_classes.GrayScott_MPIFFT.grayscott_imex_diffusion.u_exact">(grayscott_imex_diffusion method)</a>
 </li>
         <li><a href="pySDC/implementations.problem_classes.HarmonicOscillator.html#implementations.problem_classes.HarmonicOscillator.harmonic_oscillator.u_exact">(harmonic_oscillator method)</a>
+</li>
+        <li><a href="pySDC/implementations.problem_classes.HeatEquation_Chebychev.html#implementations.problem_classes.HeatEquation_Chebychev.Heat1DChebychev.u_exact">(Heat1DChebychev method)</a>
+</li>
+        <li><a href="pySDC/implementations.problem_classes.HeatEquation_Chebychev.html#implementations.problem_classes.HeatEquation_Chebychev.Heat1DUltraspherical.u_exact">(Heat1DUltraspherical method)</a>
 </li>
         <li><a href="pySDC/implementations.problem_classes.HeatEquation_2D_PETSc_forced.html#implementations.problem_classes.HeatEquation_2D_PETSc_forced.heat2d_petsc_forced.u_exact">(heat2d_petsc_forced method)</a>
+</li>
+        <li><a href="pySDC/implementations.problem_classes.HeatEquation_Chebychev.html#implementations.problem_classes.HeatEquation_Chebychev.Heat2DChebychev.u_exact">(Heat2DChebychev method)</a>
+</li>
+        <li><a href="pySDC/implementations.problem_classes.HeatEquation_Chebychev.html#implementations.problem_classes.HeatEquation_Chebychev.Heat2DUltraspherical.u_exact">(Heat2DUltraspherical method)</a>
 </li>
         <li><a href="pySDC/implementations.problem_classes.HeatEquation_ND_FD.html#implementations.problem_classes.HeatEquation_ND_FD.heatNd_forced.u_exact">(heatNd_forced method)</a>, <a href="pySDC/implementations.problem_classes.HeatEquation_ND_FD_CuPy.html#implementations.problem_classes.HeatEquation_ND_FD_CuPy.heatNd_forced.u_exact">[1]</a>
 </li>
@@ -4381,6 +4826,8 @@ <h2 id="U">U</h2>
         <li><a href="pySDC/implementations.problem_classes.Quench.html#implementations.problem_classes.Quench.Quench.u_exact">(Quench method)</a>
 </li>
         <li><a href="pySDC/implementations.problem_classes.Quench.html#implementations.problem_classes.Quench.QuenchIMEX.u_exact">(QuenchIMEX method)</a>
+</li>
+        <li><a href="pySDC/implementations.problem_classes.RayleighBenard.html#implementations.problem_classes.RayleighBenard.RayleighBenard.u_exact">(RayleighBenard method)</a>
 </li>
         <li><a href="pySDC/implementations.problem_classes.FastWaveSlowWave_0D.html#implementations.problem_classes.FastWaveSlowWave_0D.swfw_scalar.u_exact">(swfw_scalar method)</a>
 </li>
@@ -4392,16 +4839,24 @@ <h2 id="U">U</h2>
   </ul></td>
   <td style="width: 33%; vertical-align: top;"><ul>
       <li><a href="pySDC/core.problem.html#core.problem.Problem.u_init">u_init (Problem property)</a>
+
+      <ul>
+        <li><a href="pySDC/helpers.spectral_helper.html#helpers.spectral_helper.SpectralHelper.u_init">(SpectralHelper property)</a>
 </li>
+      </ul></li>
       <li><a href="pySDC/implementations.problem_classes.HarmonicOscillator.html#implementations.problem_classes.HarmonicOscillator.harmonic_oscillator.u_init">u_init() (harmonic_oscillator method)</a>
 
       <ul>
         <li><a href="pySDC/implementations.problem_classes.PenningTrap_3D.html#implementations.problem_classes.PenningTrap_3D.penningtrap.u_init">(penningtrap method)</a>
 </li>
       </ul></li>
+      <li><a href="pySDC/helpers.spectral_helper.html#helpers.spectral_helper.SpectralHelper.u_init_forward">u_init_forward (SpectralHelper property)</a>
+</li>
       <li><a href="pySDC/core.level.html#core.level.Level.uend">uend (Level attribute)</a>
 </li>
       <li><a href="pySDC/implementations.problem_classes.AllenCahn_1D_FD.html#implementations.problem_classes.AllenCahn_1D_FD.allencahn_front_fullyimplicit.uext">uext (allencahn_front_fullyimplicit attribute)</a>
+</li>
+      <li><a href="pySDC/helpers.spectral_helper.html#helpers.spectral_helper.UltrasphericalHelper">UltrasphericalHelper (class in helpers.spectral_helper)</a>
 </li>
       <li><a href="pySDC/implementations.problem_classes.boussinesq_helpers.unflatten.html#implementations.problem_classes.boussinesq_helpers.unflatten.unflatten">unflatten() (in module implementations.problem_classes.boussinesq_helpers.unflatten)</a>
 </li>
@@ -4443,6 +4898,12 @@ <h2 id="U">U</h2>
       </ul></li>
       <li><a href="pySDC/core.sweeper.html#core.sweeper.Sweeper.updateVariableCoeffs">updateVariableCoeffs() (Sweeper method)</a>
 </li>
+      <li><a href="pySDC/helpers.spectral_helper.html#helpers.spectral_helper.SpectralHelper.useGPU">useGPU (SpectralHelper attribute)</a>
+
+      <ul>
+        <li><a href="pySDC/helpers.spectral_helper.html#helpers.spectral_helper.SpectralHelper1D.useGPU">(SpectralHelper1D attribute)</a>
+</li>
+      </ul></li>
   </ul></td>
 </tr></table>
 
@@ -4461,6 +4922,8 @@ <h2 id="V">V</h2>
         <li><a href="pySDC/implementations.problem_classes.VorticityVelocity_2D_FEniCS_periodic.html#implementations.problem_classes.VorticityVelocity_2D_FEniCS_periodic.fenics_vortex_2d.V">(fenics_vortex_2d attribute)</a>
 </li>
         <li><a href="pySDC/implementations.problem_classes.VorticityVelocity_2D_FEniCS_periodic.html#implementations.problem_classes.VorticityVelocity_2D_FEniCS_periodic.fenics_vortex_2d_mass.V">(fenics_vortex_2d_mass attribute)</a>
+</li>
+        <li><a href="pySDC/helpers.spectral_helper.html#helpers.spectral_helper.SpectralHelper.V">(SpectralHelper property)</a>
 </li>
       </ul></li>
   </ul></td>
@@ -4549,6 +5012,8 @@ <h2 id="W">W</h2>
         <li><a href="pySDC/implementations.problem_classes.Battery.html#implementations.problem_classes.Battery.battery_implicit.work_counters">(battery_implicit attribute)</a>
 </li>
         <li><a href="pySDC/implementations.problem_classes.DiscontinuousTestODE.html#implementations.problem_classes.DiscontinuousTestODE.DiscontinuousTestODE.work_counters">(DiscontinuousTestODE attribute)</a>
+</li>
+        <li><a href="pySDC/implementations.problem_classes.generic_spectral.html#implementations.problem_classes.generic_spectral.GenericSpectralLinear.work_counters">(GenericSpectralLinear attribute)</a>
 </li>
         <li><a href="pySDC/implementations.problem_classes.Van_der_Pol_implicit.html#implementations.problem_classes.Van_der_Pol_implicit.vanderpol.work_counters">(vanderpol attribute)</a>
 </li>
@@ -4577,12 +5042,20 @@ <h2 id="X">X</h2>
         <li><a href="pySDC/implementations.problem_classes.NonlinearSchroedinger_MPIFFT.html#implementations.problem_classes.NonlinearSchroedinger_MPIFFT.nonlinearschroedinger_imex.X">(nonlinearschroedinger_imex attribute)</a>
 </li>
       </ul></li>
+      <li><a href="pySDC/helpers.spectral_helper.html#helpers.spectral_helper.SpectralHelper1D.x0">x0 (SpectralHelper1D attribute)</a>
+</li>
+      <li><a href="pySDC/helpers.spectral_helper.html#helpers.spectral_helper.SpectralHelper1D.x1">x1 (SpectralHelper1D attribute)</a>
+</li>
       <li><a href="pySDC/implementations.problem_classes.generic_MPIFFT_Laplacian.html#implementations.problem_classes.generic_MPIFFT_Laplacian.IMEX_Laplacian_MPIFFT.xp">xp (IMEX_Laplacian_MPIFFT attribute)</a>
 
       <ul>
         <li><a href="pySDC/implementations.datatype_classes.mesh.html#implementations.datatype_classes.mesh.mesh.xp">(mesh attribute)</a>
 </li>
         <li><a href="pySDC/implementations.problem_classes.polynomial_test_problem.html#implementations.problem_classes.polynomial_test_problem.polynomial_testequation.xp">(polynomial_testequation attribute)</a>
+</li>
+        <li><a href="pySDC/helpers.spectral_helper.html#helpers.spectral_helper.SpectralHelper.xp">(SpectralHelper attribute)</a>
+</li>
+        <li><a href="pySDC/helpers.spectral_helper.html#helpers.spectral_helper.SpectralHelper1D.xp">(SpectralHelper1D attribute)</a>
 </li>
         <li><a href="pySDC/implementations.problem_classes.TestEquation_0D.html#implementations.problem_classes.TestEquation_0D.testequation0d.xp">(testequation0d attribute)</a>
 </li>
diff --git a/objects.inv b/objects.inv
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diff --git a/projects/Hamiltonian.html b/projects/Hamiltonian.html
index 07003efc00..6fa0134d84 100644
--- a/projects/Hamiltonian.html
+++ b/projects/Hamiltonian.html
@@ -866,7 +866,7 @@ <h2>Fermi-Pasta-Ulam-Tsingou problem<a class="headerlink" href="#fermi-pasta-ula
    <span class="n">Range</span> <span class="n">of</span> <span class="n">values</span> <span class="k">for</span> <span class="n">number</span> <span class="n">of</span> <span class="n">iterations</span><span class="p">:</span>  <span class="mi">7</span> 
    <span class="n">Position</span> <span class="n">of</span> <span class="nb">max</span><span class="o">/</span><span class="nb">min</span> <span class="n">number</span> <span class="n">of</span> <span class="n">iterations</span><span class="p">:</span> <span class="mi">475</span> <span class="o">--</span>  <span class="mi">0</span>
    <span class="n">Std</span> <span class="ow">and</span> <span class="n">var</span> <span class="k">for</span> <span class="n">number</span> <span class="n">of</span> <span class="n">iterations</span><span class="p">:</span> <span class="mf">2.10</span> <span class="o">--</span> <span class="mf">4.42</span>
-<span class="o">...</span> <span class="n">took</span> <span class="mf">38.5197</span> <span class="n">seconds</span> <span class="n">to</span> <span class="n">run</span> <span class="n">this</span><span class="o">.</span>
+<span class="o">...</span> <span class="n">took</span> <span class="mf">38.4061</span> <span class="n">seconds</span> <span class="n">to</span> <span class="n">run</span> <span class="n">this</span><span class="o">.</span>
 </pre></div>
 </div>
 <a class="reference internal image-reference" href="../_images/fput_hamiltonian.png"><img alt="../_images/fput_hamiltonian.png" src="../_images/fput_hamiltonian.png" style="width: 383.0px; height: 212.0px;" />
diff --git a/projects/doc_fput.html b/projects/doc_fput.html
index 7e22d4c0c2..ad7d68edc6 100644
--- a/projects/doc_fput.html
+++ b/projects/doc_fput.html
@@ -284,7 +284,7 @@ <h3>Navigation</h3>
    <span class="n">Range</span> <span class="n">of</span> <span class="n">values</span> <span class="k">for</span> <span class="n">number</span> <span class="n">of</span> <span class="n">iterations</span><span class="p">:</span>  <span class="mi">7</span> 
    <span class="n">Position</span> <span class="n">of</span> <span class="nb">max</span><span class="o">/</span><span class="nb">min</span> <span class="n">number</span> <span class="n">of</span> <span class="n">iterations</span><span class="p">:</span> <span class="mi">475</span> <span class="o">--</span>  <span class="mi">0</span>
    <span class="n">Std</span> <span class="ow">and</span> <span class="n">var</span> <span class="k">for</span> <span class="n">number</span> <span class="n">of</span> <span class="n">iterations</span><span class="p">:</span> <span class="mf">2.10</span> <span class="o">--</span> <span class="mf">4.42</span>
-<span class="o">...</span> <span class="n">took</span> <span class="mf">38.5197</span> <span class="n">seconds</span> <span class="n">to</span> <span class="n">run</span> <span class="n">this</span><span class="o">.</span>
+<span class="o">...</span> <span class="n">took</span> <span class="mf">38.4061</span> <span class="n">seconds</span> <span class="n">to</span> <span class="n">run</span> <span class="n">this</span><span class="o">.</span>
 </pre></div>
 </div>
 <a class="reference internal image-reference" href="../_images/fput_hamiltonian.png"><img alt="../_images/fput_hamiltonian.png" src="../_images/fput_hamiltonian.png" style="width: 383.0px; height: 212.0px;" />
diff --git a/projects/parallelSDC_reloaded.html b/projects/parallelSDC_reloaded.html
index f1d3105722..020adf93d5 100644
--- a/projects/parallelSDC_reloaded.html
+++ b/projects/parallelSDC_reloaded.html
@@ -52,6 +52,14 @@ <h2>Figures for the manuscript<a class="headerlink" href="#figures-for-the-manus
 <p>See the <cite>scripts</cite> folder, all python scripts have the <cite>fig0[…]</cite> prefix, and each script generate several figures
 from the manuscript.
 One can run everything with the <cite>run.sh</cite> script, and crop the pdf figures to the format used in the manuscript with the <cite>crop.sh</cite> script.</p>
+<p>Note that most scripts generate error vs cost figure using a cost modeling of the simulation that counts the
+number of Newton iterations and RHS evaluation.
+This can be switched to a estimation of the cost using runtime measurements by :</p>
+<ul class="simple">
+<li><p>modifying the return statement in the <cite>getCost</cite> function by <cite>return tComp</cite></p></li>
+<li><p>changing the <cite>xlabel=”Cost”</cite> in the <cite>plt.gca().set</cite> call into <cite>xlabel=”Computation time [s]”</cite></p></li>
+<li><p>commenting the <cite>xlim</cite> argument in the <cite>plt.gca().set</cite> call</p></li>
+</ul>
 </section>
 <section id="experimental-scripts">
 <h2>Experimental scripts<a class="headerlink" href="#experimental-scripts" title="Link to this heading">¶</a></h2>
diff --git a/py-modindex.html b/py-modindex.html
index b35369d41c..4646c56495 100644
--- a/py-modindex.html
+++ b/py-modindex.html
@@ -151,6 +151,11 @@ <h1>Python Module Index</h1>
        <td>&#160;&#160;&#160;
        <a href="pySDC/helpers.setup_helper.html#module-helpers.setup_helper"><code class="xref">helpers.setup_helper</code></a></td><td>
        <em></em></td></tr>
+     <tr class="cg-2">
+       <td></td>
+       <td>&#160;&#160;&#160;
+       <a href="pySDC/helpers.spectral_helper.html#module-helpers.spectral_helper"><code class="xref">helpers.spectral_helper</code></a></td><td>
+       <em></em></td></tr>
      <tr class="cg-2">
        <td></td>
        <td>&#160;&#160;&#160;
@@ -445,6 +450,11 @@ <h1>Python Module Index</h1>
        <td>&#160;&#160;&#160;
        <a href="pySDC/implementations.problem_classes.BuckConverter.html#module-implementations.problem_classes.BuckConverter"><code class="xref">implementations.problem_classes.BuckConverter</code></a></td><td>
        <em></em></td></tr>
+     <tr class="cg-3">
+       <td></td>
+       <td>&#160;&#160;&#160;
+       <a href="pySDC/implementations.problem_classes.Burgers.html#module-implementations.problem_classes.Burgers"><code class="xref">implementations.problem_classes.Burgers</code></a></td><td>
+       <em></em></td></tr>
      <tr class="cg-3">
        <td></td>
        <td>&#160;&#160;&#160;
@@ -485,6 +495,11 @@ <h1>Python Module Index</h1>
        <td>&#160;&#160;&#160;
        <a href="pySDC/implementations.problem_classes.generic_ND_FD.html#module-implementations.problem_classes.generic_ND_FD"><code class="xref">implementations.problem_classes.generic_ND_FD</code></a></td><td>
        <em></em></td></tr>
+     <tr class="cg-3">
+       <td></td>
+       <td>&#160;&#160;&#160;
+       <a href="pySDC/implementations.problem_classes.generic_spectral.html#module-implementations.problem_classes.generic_spectral"><code class="xref">implementations.problem_classes.generic_spectral</code></a></td><td>
+       <em></em></td></tr>
      <tr class="cg-3">
        <td></td>
        <td>&#160;&#160;&#160;
@@ -515,6 +530,11 @@ <h1>Python Module Index</h1>
        <td>&#160;&#160;&#160;
        <a href="pySDC/implementations.problem_classes.HeatEquation_2D_PETSc_forced.html#module-implementations.problem_classes.HeatEquation_2D_PETSc_forced"><code class="xref">implementations.problem_classes.HeatEquation_2D_PETSc_forced</code></a></td><td>
        <em></em></td></tr>
+     <tr class="cg-3">
+       <td></td>
+       <td>&#160;&#160;&#160;
+       <a href="pySDC/implementations.problem_classes.HeatEquation_Chebychev.html#module-implementations.problem_classes.HeatEquation_Chebychev"><code class="xref">implementations.problem_classes.HeatEquation_Chebychev</code></a></td><td>
+       <em></em></td></tr>
      <tr class="cg-3">
        <td></td>
        <td>&#160;&#160;&#160;
@@ -585,6 +605,11 @@ <h1>Python Module Index</h1>
        <td>&#160;&#160;&#160;
        <a href="pySDC/implementations.problem_classes.Quench.html#module-implementations.problem_classes.Quench"><code class="xref">implementations.problem_classes.Quench</code></a></td><td>
        <em></em></td></tr>
+     <tr class="cg-3">
+       <td></td>
+       <td>&#160;&#160;&#160;
+       <a href="pySDC/implementations.problem_classes.RayleighBenard.html#module-implementations.problem_classes.RayleighBenard"><code class="xref">implementations.problem_classes.RayleighBenard</code></a></td><td>
+       <em></em></td></tr>
      <tr class="cg-3">
        <td></td>
        <td>&#160;&#160;&#160;
diff --git a/pySDC/helpers.html b/pySDC/helpers.html
index a93e7f4d8e..ecea29958a 100644
--- a/pySDC/helpers.html
+++ b/pySDC/helpers.html
@@ -77,6 +77,14 @@ <h2>Submodules<a class="headerlink" href="#submodules" title="Link to this headi
 <li class="toctree-l2"><a class="reference internal" href="helpers.setup_helper.html#helpers.setup_helper.generate_description"><code class="docutils literal notranslate"><span class="pre">generate_description()</span></code></a></li>
 </ul>
 </li>
+<li class="toctree-l1"><a class="reference internal" href="helpers.spectral_helper.html">helpers.spectral_helper module</a><ul>
+<li class="toctree-l2"><a class="reference internal" href="helpers.spectral_helper.html#helpers.spectral_helper.ChebychevHelper"><code class="docutils literal notranslate"><span class="pre">ChebychevHelper</span></code></a></li>
+<li class="toctree-l2"><a class="reference internal" href="helpers.spectral_helper.html#helpers.spectral_helper.FFTHelper"><code class="docutils literal notranslate"><span class="pre">FFTHelper</span></code></a></li>
+<li class="toctree-l2"><a class="reference internal" href="helpers.spectral_helper.html#helpers.spectral_helper.SpectralHelper"><code class="docutils literal notranslate"><span class="pre">SpectralHelper</span></code></a></li>
+<li class="toctree-l2"><a class="reference internal" href="helpers.spectral_helper.html#helpers.spectral_helper.SpectralHelper1D"><code class="docutils literal notranslate"><span class="pre">SpectralHelper1D</span></code></a></li>
+<li class="toctree-l2"><a class="reference internal" href="helpers.spectral_helper.html#helpers.spectral_helper.UltrasphericalHelper"><code class="docutils literal notranslate"><span class="pre">UltrasphericalHelper</span></code></a></li>
+</ul>
+</li>
 <li class="toctree-l1"><a class="reference internal" href="helpers.stats_helper.html">helpers.stats_helper module</a><ul>
 <li class="toctree-l2"><a class="reference internal" href="helpers.stats_helper.html#helpers.stats_helper.filter_stats"><code class="docutils literal notranslate"><span class="pre">filter_stats()</span></code></a></li>
 <li class="toctree-l2"><a class="reference internal" href="helpers.stats_helper.html#helpers.stats_helper.get_list_of_types"><code class="docutils literal notranslate"><span class="pre">get_list_of_types()</span></code></a></li>
diff --git a/pySDC/helpers.setup_helper.html b/pySDC/helpers.setup_helper.html
index f470fa609f..89219b7fec 100644
--- a/pySDC/helpers.setup_helper.html
+++ b/pySDC/helpers.setup_helper.html
@@ -16,7 +16,7 @@
     
     <link rel="index" title="Index" href="../genindex.html" />
     <link rel="search" title="Search" href="../search.html" />
-    <link rel="next" title="helpers.stats_helper module" href="helpers.stats_helper.html" />
+    <link rel="next" title="helpers.spectral_helper module" href="helpers.spectral_helper.html" />
     <link rel="prev" title="helpers.pysdc_helper module" href="helpers.pysdc_helper.html" /> 
   </head><body>
     <div class="related" role="navigation" aria-label="Related">
@@ -29,7 +29,7 @@ <h3>Navigation</h3>
           <a href="../py-modindex.html" title="Python Module Index"
              >modules</a> |</li>
         <li class="right" >
-          <a href="helpers.stats_helper.html" title="helpers.stats_helper module"
+          <a href="helpers.spectral_helper.html" title="helpers.spectral_helper module"
              accesskey="N">next</a> |</li>
         <li class="right" >
           <a href="helpers.pysdc_helper.html" title="helpers.pysdc_helper module"
@@ -91,8 +91,8 @@ <h4>Previous topic</h4>
   </div>
   <div>
     <h4>Next topic</h4>
-    <p class="topless"><a href="helpers.stats_helper.html"
-                          title="next chapter">helpers.stats_helper module</a></p>
+    <p class="topless"><a href="helpers.spectral_helper.html"
+                          title="next chapter">helpers.spectral_helper module</a></p>
   </div>
   <div role="note" aria-label="source link">
     <h3>This Page</h3>
@@ -125,7 +125,7 @@ <h3>Navigation</h3>
           <a href="../py-modindex.html" title="Python Module Index"
              >modules</a> |</li>
         <li class="right" >
-          <a href="helpers.stats_helper.html" title="helpers.stats_helper module"
+          <a href="helpers.spectral_helper.html" title="helpers.spectral_helper module"
              >next</a> |</li>
         <li class="right" >
           <a href="helpers.pysdc_helper.html" title="helpers.pysdc_helper module"
diff --git a/pySDC/helpers.spectral_helper.html b/pySDC/helpers.spectral_helper.html
new file mode 100644
index 0000000000..e01b6b8d58
--- /dev/null
+++ b/pySDC/helpers.spectral_helper.html
@@ -0,0 +1,1693 @@
+<!DOCTYPE html>
+
+<html lang="en" data-content_root="../">
+  <head>
+    <meta charset="utf-8" />
+    <meta name="viewport" content="width=device-width, initial-scale=1.0" /><meta name="viewport" content="width=device-width, initial-scale=1" />
+
+    <title>helpers.spectral_helper module &#8212; pySDC 5.5.1 documentation</title>
+    <link rel="stylesheet" type="text/css" href="../_static/pygments.css?v=fa44fd50" />
+    <link rel="stylesheet" type="text/css" href="../_static/classic.css?v=def86cc0" />
+    <link rel="stylesheet" type="text/css" href="../_static/custom.css?v=c40552b6" />
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+  <section id="module-helpers.spectral_helper">
+<span id="helpers-spectral-helper-module"></span><h1>helpers.spectral_helper module<a class="headerlink" href="#module-helpers.spectral_helper" title="Link to this heading">¶</a></h1>
+<dl class="py class">
+<dt class="sig sig-object py" id="helpers.spectral_helper.ChebychevHelper">
+<em class="property"><span class="pre">class</span><span class="w"> </span></em><span class="sig-name descname"><span class="pre">ChebychevHelper</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="o"><span class="pre">*</span></span><span class="n"><span class="pre">args</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">transform_type</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">'fft'</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">x0</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">-1</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">x1</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">1</span></span></em>, <em class="sig-param"><span class="o"><span class="pre">**</span></span><span class="n"><span class="pre">kwargs</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../_modules/helpers/spectral_helper.html#ChebychevHelper"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#helpers.spectral_helper.ChebychevHelper" title="Link to this definition">¶</a></dt>
+<dd><p>Bases: <a class="reference internal" href="#helpers.spectral_helper.SpectralHelper1D" title="helpers.spectral_helper.SpectralHelper1D"><code class="xref py py-class docutils literal notranslate"><span class="pre">SpectralHelper1D</span></code></a></p>
+<p>The Chebychev base consists of special kinds of polynomials, with the main advantage that you can easily transform
+between physical and spectral space by discrete cosine transform.
+The differentiation in the Chebychev T base is dense, but can be preconditioned to yield a differentiation operator
+that moves to Chebychev U basis during differentiation, which is sparse. When using this technique, problems need to
+be formulated in first order formulation.</p>
+<p>This implementation is largely based on the Dedalus paper (arXiv:1905.10388).</p>
+<dl class="py method">
+<dt class="sig sig-object py" id="helpers.spectral_helper.ChebychevHelper.get_1dgrid">
+<span class="sig-name descname"><span class="pre">get_1dgrid</span></span><span class="sig-paren">(</span><span class="sig-paren">)</span><a class="reference internal" href="../_modules/helpers/spectral_helper.html#ChebychevHelper.get_1dgrid"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#helpers.spectral_helper.ChebychevHelper.get_1dgrid" title="Link to this definition">¶</a></dt>
+<dd><p>Generates a 1D grid with Chebychev points. These are clustered at the boundary. You need this kind of grid to
+use discrete cosine transformation (DCT) to get the Chebychev representation. If you want a different grid, you
+need to do an affine transformation before any Chebychev business.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Returns<span class="colon">:</span></dt>
+<dd class="field-odd"><p>1D grid</p>
+</dd>
+<dt class="field-even">Return type<span class="colon">:</span></dt>
+<dd class="field-even"><p>numpy.ndarray</p>
+</dd>
+</dl>
+</dd></dl>
+
+<dl class="py method">
+<dt class="sig sig-object py" id="helpers.spectral_helper.ChebychevHelper.get_BC">
+<span class="sig-name descname"><span class="pre">get_BC</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">kind</span></span></em>, <em class="sig-param"><span class="o"><span class="pre">**</span></span><span class="n"><span class="pre">kwargs</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../_modules/helpers/spectral_helper.html#ChebychevHelper.get_BC"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#helpers.spectral_helper.ChebychevHelper.get_BC" title="Link to this definition">¶</a></dt>
+<dd><p>Get boundary condition row for boundary bordering. <cite>kwargs</cite> will be passed on to implementations of the BC of
+the kind you choose. Specifically, <cite>x</cite> for <cite>‘dirichlet’</cite> boundary condition, which is the coordinate at which to
+set the BC.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><p><strong>kind</strong> (<em>'integral'</em><em> or </em><em>'dirichlet'</em>) – Kind of boundary condition you want</p>
+</dd>
+</dl>
+</dd></dl>
+
+<dl class="py method">
+<dt class="sig sig-object py" id="helpers.spectral_helper.ChebychevHelper.get_Dirichlet_BC_row">
+<span class="sig-name descname"><span class="pre">get_Dirichlet_BC_row</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">x</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../_modules/helpers/spectral_helper.html#ChebychevHelper.get_Dirichlet_BC_row"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#helpers.spectral_helper.ChebychevHelper.get_Dirichlet_BC_row" title="Link to this definition">¶</a></dt>
+<dd><p>Get a row for generating Dirichlet BCs at x with T polynomials.
+It returns the values of the T polynomials at x.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><p><strong>x</strong> (<em>float</em>) – Position of the boundary condition</p>
+</dd>
+<dt class="field-even">Returns<span class="colon">:</span></dt>
+<dd class="field-even"><p>Row to put into a matrix</p>
+</dd>
+<dt class="field-odd">Return type<span class="colon">:</span></dt>
+<dd class="field-odd"><p>self.xp.ndarray</p>
+</dd>
+</dl>
+</dd></dl>
+
+<dl class="py method">
+<dt class="sig sig-object py" id="helpers.spectral_helper.ChebychevHelper.get_Dirichlet_recombination_matrix">
+<span class="sig-name descname"><span class="pre">get_Dirichlet_recombination_matrix</span></span><span class="sig-paren">(</span><span class="sig-paren">)</span><a class="reference internal" href="../_modules/helpers/spectral_helper.html#ChebychevHelper.get_Dirichlet_recombination_matrix"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#helpers.spectral_helper.ChebychevHelper.get_Dirichlet_recombination_matrix" title="Link to this definition">¶</a></dt>
+<dd><p>Get matrix for Dirichlet recombination, which changes the basis to have sparse boundary conditions.
+This makes for a good right preconditioner.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Returns<span class="colon">:</span></dt>
+<dd class="field-odd"><p>Sparse conversion matrix</p>
+</dd>
+<dt class="field-even">Return type<span class="colon">:</span></dt>
+<dd class="field-even"><p>scipy.sparse</p>
+</dd>
+</dl>
+</dd></dl>
+
+<dl class="py method">
+<dt class="sig sig-object py" id="helpers.spectral_helper.ChebychevHelper.get_basis_change_matrix">
+<span class="sig-name descname"><span class="pre">get_basis_change_matrix</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">conv</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">'T2T'</span></span></em>, <em class="sig-param"><span class="o"><span class="pre">**</span></span><span class="n"><span class="pre">kwargs</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../_modules/helpers/spectral_helper.html#ChebychevHelper.get_basis_change_matrix"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#helpers.spectral_helper.ChebychevHelper.get_basis_change_matrix" title="Link to this definition">¶</a></dt>
+<dd><p>As the differentiation matrix in Chebychev-T base is dense but is sparse when simultaneously changing base to
+Chebychev-U, you may need a basis change matrix to transfer the other matrices as well. This function returns a
+conversion matrix from <cite>ChebychevHelper.get_conv</cite>. Not that <cite>**kwargs</cite> are used to absorb arguments for other
+bases, see documentation of <cite>SpectralHelper1D.get_basis_change_matrix</cite>.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><p><strong>conv</strong> (<em>str</em>) – Conversion code, i.e. T2U</p>
+</dd>
+<dt class="field-even">Returns<span class="colon">:</span></dt>
+<dd class="field-even"><p>Sparse conversion matrix</p>
+</dd>
+</dl>
+</dd></dl>
+
+<dl class="py method">
+<dt class="sig sig-object py" id="helpers.spectral_helper.ChebychevHelper.get_conv">
+<span class="sig-name descname"><span class="pre">get_conv</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">name</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">N</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../_modules/helpers/spectral_helper.html#ChebychevHelper.get_conv"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#helpers.spectral_helper.ChebychevHelper.get_conv" title="Link to this definition">¶</a></dt>
+<dd><dl class="simple">
+<dt>Get conversion matrix between different kinds of polynomials. The supported kinds are</dt><dd><ul class="simple">
+<li><p>T: Chebychev polynomials of first kind</p></li>
+<li><p>U: Chebychev polynomials of second kind</p></li>
+<li><p>D: Dirichlet recombination.</p></li>
+</ul>
+</dd>
+</dl>
+<p>You get the desired matrix by choosing a name as <code class="docutils literal notranslate"><span class="pre">A2B</span></code>. I.e. <code class="docutils literal notranslate"><span class="pre">T2U</span></code> for the conversion matrix from T to U.
+Once generates matrices are cached. So feel free to call the method as often as you like.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><ul class="simple">
+<li><p><strong>name</strong> (<em>str</em>) – Conversion code, e.g. ‘T2U’</p></li>
+<li><p><strong>N</strong> (<em>int</em>) – Size of the matrix (optional)</p></li>
+</ul>
+</dd>
+<dt class="field-even">Returns<span class="colon">:</span></dt>
+<dd class="field-even"><p>Sparse conversion matrix</p>
+</dd>
+<dt class="field-odd">Return type<span class="colon">:</span></dt>
+<dd class="field-odd"><p>scipy.sparse</p>
+</dd>
+</dl>
+</dd></dl>
+
+<dl class="py method">
+<dt class="sig sig-object py" id="helpers.spectral_helper.ChebychevHelper.get_differentiation_matrix">
+<span class="sig-name descname"><span class="pre">get_differentiation_matrix</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">p</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">1</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../_modules/helpers/spectral_helper.html#ChebychevHelper.get_differentiation_matrix"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#helpers.spectral_helper.ChebychevHelper.get_differentiation_matrix" title="Link to this definition">¶</a></dt>
+<dd><p>Keep in mind that the T2T differentiation matrix is dense.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><p><strong>p</strong> (<em>int</em>) – Derivative you want to compute</p>
+</dd>
+<dt class="field-even">Returns<span class="colon">:</span></dt>
+<dd class="field-even"><p>Differentiation matrix</p>
+</dd>
+<dt class="field-odd">Return type<span class="colon">:</span></dt>
+<dd class="field-odd"><p>numpy.ndarray</p>
+</dd>
+</dl>
+</dd></dl>
+
+<dl class="py method">
+<dt class="sig sig-object py" id="helpers.spectral_helper.ChebychevHelper.get_fft_shift">
+<span class="sig-name descname"><span class="pre">get_fft_shift</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">forward</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">N</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../_modules/helpers/spectral_helper.html#ChebychevHelper.get_fft_shift"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#helpers.spectral_helper.ChebychevHelper.get_fft_shift" title="Link to this definition">¶</a></dt>
+<dd><p>As described in the docstring for <cite>get_fft_shuffle</cite>, we need to rotate in the complex plane in order to use FFT for DCT.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><ul class="simple">
+<li><p><strong>forward</strong> (<em>bool</em>) – Whether you want the rotation for forward transform or backward transform</p></li>
+<li><p><strong>N</strong> (<em>int</em>) – size of the grid</p></li>
+</ul>
+</dd>
+<dt class="field-even">Returns<span class="colon">:</span></dt>
+<dd class="field-even"><p>Rotation</p>
+</dd>
+<dt class="field-odd">Return type<span class="colon">:</span></dt>
+<dd class="field-odd"><p>self.xp.array</p>
+</dd>
+</dl>
+</dd></dl>
+
+<dl class="py method">
+<dt class="sig sig-object py" id="helpers.spectral_helper.ChebychevHelper.get_fft_shuffle">
+<span class="sig-name descname"><span class="pre">get_fft_shuffle</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">forward</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">N</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../_modules/helpers/spectral_helper.html#ChebychevHelper.get_fft_shuffle"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#helpers.spectral_helper.ChebychevHelper.get_fft_shuffle" title="Link to this definition">¶</a></dt>
+<dd><p>In order to more easily parallelize using distributed FFT libraries, we express the DCT via an FFT following
+doi.org/10.1109/TASSP.1980.1163351. The idea is based on reshuffling the data to be periodic and rotating it
+in the complex plane. This function returns a mask to do the shuffling.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><ul class="simple">
+<li><p><strong>forward</strong> (<em>bool</em>) – Whether you want the shuffle for forward transform or backward transform</p></li>
+<li><p><strong>N</strong> (<em>int</em>) – size of the grid</p></li>
+</ul>
+</dd>
+<dt class="field-even">Returns<span class="colon">:</span></dt>
+<dd class="field-even"><p>Use as mask</p>
+</dd>
+<dt class="field-odd">Return type<span class="colon">:</span></dt>
+<dd class="field-odd"><p>self.xp.array</p>
+</dd>
+</dl>
+</dd></dl>
+
+<dl class="py method">
+<dt class="sig sig-object py" id="helpers.spectral_helper.ChebychevHelper.get_fft_utils">
+<span class="sig-name descname"><span class="pre">get_fft_utils</span></span><span class="sig-paren">(</span><span class="sig-paren">)</span><a class="reference internal" href="../_modules/helpers/spectral_helper.html#ChebychevHelper.get_fft_utils"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#helpers.spectral_helper.ChebychevHelper.get_fft_utils" title="Link to this definition">¶</a></dt>
+<dd><p>Get the required utilities for using FFT to do DCT as described in the docstring for <cite>get_fft_shuffle</cite> and keep
+them cached.</p>
+</dd></dl>
+
+<dl class="py method">
+<dt class="sig sig-object py" id="helpers.spectral_helper.ChebychevHelper.get_integ_BC_row">
+<span class="sig-name descname"><span class="pre">get_integ_BC_row</span></span><span class="sig-paren">(</span><span class="sig-paren">)</span><a class="reference internal" href="../_modules/helpers/spectral_helper.html#ChebychevHelper.get_integ_BC_row"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#helpers.spectral_helper.ChebychevHelper.get_integ_BC_row" title="Link to this definition">¶</a></dt>
+<dd><p>Get a row for generating integral BCs with T polynomials.
+It returns the values of the integrals of T polynomials over the entire interval.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Returns<span class="colon">:</span></dt>
+<dd class="field-odd"><p>Row to put into a matrix</p>
+</dd>
+<dt class="field-even">Return type<span class="colon">:</span></dt>
+<dd class="field-even"><p>self.xp.ndarray</p>
+</dd>
+</dl>
+</dd></dl>
+
+<dl class="py method">
+<dt class="sig sig-object py" id="helpers.spectral_helper.ChebychevHelper.get_integration_matrix">
+<span class="sig-name descname"><span class="pre">get_integration_matrix</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">lbnd</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">0</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../_modules/helpers/spectral_helper.html#ChebychevHelper.get_integration_matrix"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#helpers.spectral_helper.ChebychevHelper.get_integration_matrix" title="Link to this definition">¶</a></dt>
+<dd><p>Get matrix for integration</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><p><strong>lbnd</strong> (<em>float</em>) – Lower bound for integration, only 0 is currently implemented</p>
+</dd>
+<dt class="field-even">Returns<span class="colon">:</span></dt>
+<dd class="field-even"><p>Sparse integration matrix</p>
+</dd>
+</dl>
+</dd></dl>
+
+<dl class="py method">
+<dt class="sig sig-object py" id="helpers.spectral_helper.ChebychevHelper.get_norm">
+<span class="sig-name descname"><span class="pre">get_norm</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">N</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../_modules/helpers/spectral_helper.html#ChebychevHelper.get_norm"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#helpers.spectral_helper.ChebychevHelper.get_norm" title="Link to this definition">¶</a></dt>
+<dd><p>Get normalization for converting Chebychev coefficients and DCT</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><p><strong>N</strong> (<em>int</em><em>, </em><em>optional</em>) – Resolution</p>
+</dd>
+<dt class="field-even">Returns<span class="colon">:</span></dt>
+<dd class="field-even"><p>Normalization</p>
+</dd>
+<dt class="field-odd">Return type<span class="colon">:</span></dt>
+<dd class="field-odd"><p>self.xp.array</p>
+</dd>
+</dl>
+</dd></dl>
+
+<dl class="py method">
+<dt class="sig sig-object py" id="helpers.spectral_helper.ChebychevHelper.get_wavenumbers">
+<span class="sig-name descname"><span class="pre">get_wavenumbers</span></span><span class="sig-paren">(</span><span class="sig-paren">)</span><a class="reference internal" href="../_modules/helpers/spectral_helper.html#ChebychevHelper.get_wavenumbers"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#helpers.spectral_helper.ChebychevHelper.get_wavenumbers" title="Link to this definition">¶</a></dt>
+<dd><p>Get the domain in spectral space</p>
+</dd></dl>
+
+<dl class="py method">
+<dt class="sig sig-object py" id="helpers.spectral_helper.ChebychevHelper.itransform">
+<span class="sig-name descname"><span class="pre">itransform</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">u</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">axis</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">-1</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../_modules/helpers/spectral_helper.html#ChebychevHelper.itransform"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#helpers.spectral_helper.ChebychevHelper.itransform" title="Link to this definition">¶</a></dt>
+<dd><p>1D inverse DCT along axis.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><ul class="simple">
+<li><p><strong>u</strong> – Data you want to transform</p></li>
+<li><p><strong>axis</strong> (<em>int</em>) – Axis you want to transform along</p></li>
+</ul>
+</dd>
+<dt class="field-even">Returns<span class="colon">:</span></dt>
+<dd class="field-even"><p>Data in physical space</p>
+</dd>
+</dl>
+</dd></dl>
+
+<dl class="py method">
+<dt class="sig sig-object py" id="helpers.spectral_helper.ChebychevHelper.transform">
+<span class="sig-name descname"><span class="pre">transform</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">u</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">axis</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">-1</span></span></em>, <em class="sig-param"><span class="o"><span class="pre">**</span></span><span class="n"><span class="pre">kwargs</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../_modules/helpers/spectral_helper.html#ChebychevHelper.transform"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#helpers.spectral_helper.ChebychevHelper.transform" title="Link to this definition">¶</a></dt>
+<dd><p>1D DCT along axis. <cite>kwargs</cite> will be passed on to the FFT library.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><ul class="simple">
+<li><p><strong>u</strong> – Data you want to transform</p></li>
+<li><p><strong>axis</strong> (<em>int</em>) – Axis you want to transform along</p></li>
+</ul>
+</dd>
+<dt class="field-even">Returns<span class="colon">:</span></dt>
+<dd class="field-even"><p>Data in spectral space</p>
+</dd>
+</dl>
+</dd></dl>
+
+</dd></dl>
+
+<dl class="py class">
+<dt class="sig sig-object py" id="helpers.spectral_helper.FFTHelper">
+<em class="property"><span class="pre">class</span><span class="w"> </span></em><span class="sig-name descname"><span class="pre">FFTHelper</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="o"><span class="pre">*</span></span><span class="n"><span class="pre">args</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">x0</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">0</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">x1</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">6.283185307179586</span></span></em>, <em class="sig-param"><span class="o"><span class="pre">**</span></span><span class="n"><span class="pre">kwargs</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../_modules/helpers/spectral_helper.html#FFTHelper"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#helpers.spectral_helper.FFTHelper" title="Link to this definition">¶</a></dt>
+<dd><p>Bases: <a class="reference internal" href="#helpers.spectral_helper.SpectralHelper1D" title="helpers.spectral_helper.SpectralHelper1D"><code class="xref py py-class docutils literal notranslate"><span class="pre">SpectralHelper1D</span></code></a></p>
+<dl class="py method">
+<dt class="sig sig-object py" id="helpers.spectral_helper.FFTHelper.get_1dgrid">
+<span class="sig-name descname"><span class="pre">get_1dgrid</span></span><span class="sig-paren">(</span><span class="sig-paren">)</span><a class="reference internal" href="../_modules/helpers/spectral_helper.html#FFTHelper.get_1dgrid"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#helpers.spectral_helper.FFTHelper.get_1dgrid" title="Link to this definition">¶</a></dt>
+<dd><p>We use equally spaced points including the left boundary and not including the right one, which is the left boundary.</p>
+</dd></dl>
+
+<dl class="py method">
+<dt class="sig sig-object py" id="helpers.spectral_helper.FFTHelper.get_BC">
+<span class="sig-name descname"><span class="pre">get_BC</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">kind</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../_modules/helpers/spectral_helper.html#FFTHelper.get_BC"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#helpers.spectral_helper.FFTHelper.get_BC" title="Link to this definition">¶</a></dt>
+<dd><p>Get a sort of boundary condition. You can use <cite>kind=integral</cite>, to fix the integral, or you can use <cite>kind=Nyquist</cite>.
+The latter is not really a boundary condition, but is used to set the Nyquist mode to some value, preferably zero.
+You should set the Nyquist mode zero when the solution in physical space is real and the resolution is even.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><p><strong>kind</strong> (<em>'integral'</em><em> or </em><em>'nyquist'</em>) – Kind of BC</p>
+</dd>
+<dt class="field-even">Returns<span class="colon">:</span></dt>
+<dd class="field-even"><p>Boundary condition row</p>
+</dd>
+<dt class="field-odd">Return type<span class="colon">:</span></dt>
+<dd class="field-odd"><p>self.xp.ndarray</p>
+</dd>
+</dl>
+</dd></dl>
+
+<dl class="py method">
+<dt class="sig sig-object py" id="helpers.spectral_helper.FFTHelper.get_Nyquist_mode_index">
+<span class="sig-name descname"><span class="pre">get_Nyquist_mode_index</span></span><span class="sig-paren">(</span><span class="sig-paren">)</span><a class="reference internal" href="../_modules/helpers/spectral_helper.html#FFTHelper.get_Nyquist_mode_index"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#helpers.spectral_helper.FFTHelper.get_Nyquist_mode_index" title="Link to this definition">¶</a></dt>
+<dd><p>Compute the index of the Nyquist mode, i.e. the mode with the lowest wavenumber, which doesn’t have a positive
+counterpart for even resolution. This means real waves of this wave number cannot be properly resolved and you
+are best advised to set this mode zero if representing real functions on even-resolution grids is what you’re
+after.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Returns<span class="colon">:</span></dt>
+<dd class="field-odd"><p>Index of the Nyquist mode</p>
+</dd>
+<dt class="field-even">Return type<span class="colon">:</span></dt>
+<dd class="field-even"><p>int</p>
+</dd>
+</dl>
+</dd></dl>
+
+<dl class="py method">
+<dt class="sig sig-object py" id="helpers.spectral_helper.FFTHelper.get_differentiation_matrix">
+<span class="sig-name descname"><span class="pre">get_differentiation_matrix</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">p</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">1</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../_modules/helpers/spectral_helper.html#FFTHelper.get_differentiation_matrix"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#helpers.spectral_helper.FFTHelper.get_differentiation_matrix" title="Link to this definition">¶</a></dt>
+<dd><p>This matrix is diagonal, allowing to invert concurrently.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><p><strong>p</strong> (<em>int</em>) – Order of the derivative</p>
+</dd>
+<dt class="field-even">Returns<span class="colon">:</span></dt>
+<dd class="field-even"><p>sparse differentiation matrix</p>
+</dd>
+</dl>
+</dd></dl>
+
+<dl class="py method">
+<dt class="sig sig-object py" id="helpers.spectral_helper.FFTHelper.get_integ_BC_row">
+<span class="sig-name descname"><span class="pre">get_integ_BC_row</span></span><span class="sig-paren">(</span><span class="sig-paren">)</span><a class="reference internal" href="../_modules/helpers/spectral_helper.html#FFTHelper.get_integ_BC_row"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#helpers.spectral_helper.FFTHelper.get_integ_BC_row" title="Link to this definition">¶</a></dt>
+<dd><p>Only the 0-mode has non-zero integral with FFT basis in periodic BCs</p>
+</dd></dl>
+
+<dl class="py method">
+<dt class="sig sig-object py" id="helpers.spectral_helper.FFTHelper.get_integration_matrix">
+<span class="sig-name descname"><span class="pre">get_integration_matrix</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">p</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">1</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../_modules/helpers/spectral_helper.html#FFTHelper.get_integration_matrix"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#helpers.spectral_helper.FFTHelper.get_integration_matrix" title="Link to this definition">¶</a></dt>
+<dd><p>Get integration matrix to compute <cite>p</cite>-th integral over the entire domain.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><p><strong>p</strong> (<em>int</em>) – Order of integral you want to compute</p>
+</dd>
+<dt class="field-even">Returns<span class="colon">:</span></dt>
+<dd class="field-even"><p>sparse integration matrix</p>
+</dd>
+</dl>
+</dd></dl>
+
+<dl class="py method">
+<dt class="sig sig-object py" id="helpers.spectral_helper.FFTHelper.get_wavenumbers">
+<span class="sig-name descname"><span class="pre">get_wavenumbers</span></span><span class="sig-paren">(</span><span class="sig-paren">)</span><a class="reference internal" href="../_modules/helpers/spectral_helper.html#FFTHelper.get_wavenumbers"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#helpers.spectral_helper.FFTHelper.get_wavenumbers" title="Link to this definition">¶</a></dt>
+<dd><p>Be careful that this ordering is very unintuitive.</p>
+</dd></dl>
+
+<dl class="py method">
+<dt class="sig sig-object py" id="helpers.spectral_helper.FFTHelper.itransform">
+<span class="sig-name descname"><span class="pre">itransform</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">u</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">axis</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">-1</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../_modules/helpers/spectral_helper.html#FFTHelper.itransform"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#helpers.spectral_helper.FFTHelper.itransform" title="Link to this definition">¶</a></dt>
+<dd><p>Inverse 1D FFT.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><ul class="simple">
+<li><p><strong>u</strong> – Data you want to transform</p></li>
+<li><p><strong>axis</strong> (<em>int</em>) – Axis you want to transform along</p></li>
+</ul>
+</dd>
+<dt class="field-even">Returns<span class="colon">:</span></dt>
+<dd class="field-even"><p>transformed data</p>
+</dd>
+</dl>
+</dd></dl>
+
+<dl class="py method">
+<dt class="sig sig-object py" id="helpers.spectral_helper.FFTHelper.transform">
+<span class="sig-name descname"><span class="pre">transform</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">u</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">axis</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">-1</span></span></em>, <em class="sig-param"><span class="o"><span class="pre">**</span></span><span class="n"><span class="pre">kwargs</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../_modules/helpers/spectral_helper.html#FFTHelper.transform"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#helpers.spectral_helper.FFTHelper.transform" title="Link to this definition">¶</a></dt>
+<dd><p>1D FFT along axis. <cite>kwargs</cite> are passed on to the FFT library.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><ul class="simple">
+<li><p><strong>u</strong> – Data you want to transform</p></li>
+<li><p><strong>axis</strong> (<em>int</em>) – Axis you want to transform along</p></li>
+</ul>
+</dd>
+<dt class="field-even">Returns<span class="colon">:</span></dt>
+<dd class="field-even"><p>transformed data</p>
+</dd>
+</dl>
+</dd></dl>
+
+</dd></dl>
+
+<dl class="py class">
+<dt class="sig sig-object py" id="helpers.spectral_helper.SpectralHelper">
+<em class="property"><span class="pre">class</span><span class="w"> </span></em><span class="sig-name descname"><span class="pre">SpectralHelper</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">comm</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">useGPU</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">False</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">debug</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">False</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../_modules/helpers/spectral_helper.html#SpectralHelper"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#helpers.spectral_helper.SpectralHelper" title="Link to this definition">¶</a></dt>
+<dd><p>Bases: <code class="xref py py-class docutils literal notranslate"><span class="pre">object</span></code></p>
+<dl class="simple">
+<dt>This class has three functions:</dt><dd><ul class="simple">
+<li><p>Easily assemble matrices containing multiple equations</p></li>
+<li><p>Direct product of 1D bases to solve problems in more dimensions</p></li>
+<li><p>Distribute the FFTs to facilitate concurrency.</p></li>
+</ul>
+</dd>
+</dl>
+<dl class="py attribute">
+<dt class="sig sig-object py" id="helpers.spectral_helper.SpectralHelper.comm">
+<span class="sig-name descname"><span class="pre">comm</span></span><a class="headerlink" href="#helpers.spectral_helper.SpectralHelper.comm" title="Link to this definition">¶</a></dt>
+<dd><p>MPI communicator</p>
+<dl class="field-list simple">
+<dt class="field-odd">Type<span class="colon">:</span></dt>
+<dd class="field-odd"><p>mpi4py.Intracomm</p>
+</dd>
+</dl>
+</dd></dl>
+
+<dl class="py attribute">
+<dt class="sig sig-object py" id="helpers.spectral_helper.SpectralHelper.debug">
+<span class="sig-name descname"><span class="pre">debug</span></span><a class="headerlink" href="#helpers.spectral_helper.SpectralHelper.debug" title="Link to this definition">¶</a></dt>
+<dd><p>Perform additional checks at extra computational cost</p>
+<dl class="field-list simple">
+<dt class="field-odd">Type<span class="colon">:</span></dt>
+<dd class="field-odd"><p>bool</p>
+</dd>
+</dl>
+</dd></dl>
+
+<dl class="py attribute">
+<dt class="sig sig-object py" id="helpers.spectral_helper.SpectralHelper.useGPU">
+<span class="sig-name descname"><span class="pre">useGPU</span></span><a class="headerlink" href="#helpers.spectral_helper.SpectralHelper.useGPU" title="Link to this definition">¶</a></dt>
+<dd><p>Whether to use GPUs</p>
+<dl class="field-list simple">
+<dt class="field-odd">Type<span class="colon">:</span></dt>
+<dd class="field-odd"><p>bool</p>
+</dd>
+</dl>
+</dd></dl>
+
+<dl class="py attribute">
+<dt class="sig sig-object py" id="helpers.spectral_helper.SpectralHelper.axes">
+<span class="sig-name descname"><span class="pre">axes</span></span><a class="headerlink" href="#helpers.spectral_helper.SpectralHelper.axes" title="Link to this definition">¶</a></dt>
+<dd><p>List of 1D bases</p>
+<dl class="field-list simple">
+<dt class="field-odd">Type<span class="colon">:</span></dt>
+<dd class="field-odd"><p>list</p>
+</dd>
+</dl>
+</dd></dl>
+
+<dl class="py attribute">
+<dt class="sig sig-object py" id="helpers.spectral_helper.SpectralHelper.components">
+<span class="sig-name descname"><span class="pre">components</span></span><a class="headerlink" href="#helpers.spectral_helper.SpectralHelper.components" title="Link to this definition">¶</a></dt>
+<dd><p>List of strings of the names of components in the equations</p>
+<dl class="field-list simple">
+<dt class="field-odd">Type<span class="colon">:</span></dt>
+<dd class="field-odd"><p>list</p>
+</dd>
+</dl>
+</dd></dl>
+
+<dl class="py attribute">
+<dt class="sig sig-object py" id="helpers.spectral_helper.SpectralHelper.full_BCs">
+<span class="sig-name descname"><span class="pre">full_BCs</span></span><a class="headerlink" href="#helpers.spectral_helper.SpectralHelper.full_BCs" title="Link to this definition">¶</a></dt>
+<dd><p>List of Dictionaries containing all information about the boundary conditions</p>
+<dl class="field-list simple">
+<dt class="field-odd">Type<span class="colon">:</span></dt>
+<dd class="field-odd"><p>list</p>
+</dd>
+</dl>
+</dd></dl>
+
+<dl class="py attribute">
+<dt class="sig sig-object py" id="helpers.spectral_helper.SpectralHelper.BC_mat">
+<span class="sig-name descname"><span class="pre">BC_mat</span></span><a class="headerlink" href="#helpers.spectral_helper.SpectralHelper.BC_mat" title="Link to this definition">¶</a></dt>
+<dd><p>List of lists of sparse matrices to put BCs into and eventually assemble the BC matrix from</p>
+<dl class="field-list simple">
+<dt class="field-odd">Type<span class="colon">:</span></dt>
+<dd class="field-odd"><p>list</p>
+</dd>
+</dl>
+</dd></dl>
+
+<dl class="py attribute">
+<dt class="sig sig-object py" id="helpers.spectral_helper.SpectralHelper.BCs">
+<span class="sig-name descname"><span class="pre">BCs</span></span><a class="headerlink" href="#helpers.spectral_helper.SpectralHelper.BCs" title="Link to this definition">¶</a></dt>
+<dd><p>Matrix containing only the BCs</p>
+<dl class="field-list simple">
+<dt class="field-odd">Type<span class="colon">:</span></dt>
+<dd class="field-odd"><p>sparse matrix</p>
+</dd>
+</dl>
+</dd></dl>
+
+<dl class="py attribute">
+<dt class="sig sig-object py" id="helpers.spectral_helper.SpectralHelper.fft_cache">
+<span class="sig-name descname"><span class="pre">fft_cache</span></span><a class="headerlink" href="#helpers.spectral_helper.SpectralHelper.fft_cache" title="Link to this definition">¶</a></dt>
+<dd><p>Cache FFTs of various shapes here to facilitate padding and so on</p>
+<dl class="field-list simple">
+<dt class="field-odd">Type<span class="colon">:</span></dt>
+<dd class="field-odd"><p>dict</p>
+</dd>
+</dl>
+</dd></dl>
+
+<dl class="py attribute">
+<dt class="sig sig-object py" id="helpers.spectral_helper.SpectralHelper.BC_rhs_mask">
+<span class="sig-name descname"><span class="pre">BC_rhs_mask</span></span><a class="headerlink" href="#helpers.spectral_helper.SpectralHelper.BC_rhs_mask" title="Link to this definition">¶</a></dt>
+<dd><p>Mask values that contain boundary conditions in the right hand side</p>
+<dl class="field-list simple">
+<dt class="field-odd">Type<span class="colon">:</span></dt>
+<dd class="field-odd"><p>self.xp.ndarray</p>
+</dd>
+</dl>
+</dd></dl>
+
+<dl class="py attribute">
+<dt class="sig sig-object py" id="helpers.spectral_helper.SpectralHelper.BC_zero_index">
+<span class="sig-name descname"><span class="pre">BC_zero_index</span></span><a class="headerlink" href="#helpers.spectral_helper.SpectralHelper.BC_zero_index" title="Link to this definition">¶</a></dt>
+<dd><p>Indeces of rows in the matrix that are replaced by BCs</p>
+<dl class="field-list simple">
+<dt class="field-odd">Type<span class="colon">:</span></dt>
+<dd class="field-odd"><p>self.xp.ndarray</p>
+</dd>
+</dl>
+</dd></dl>
+
+<dl class="py attribute">
+<dt class="sig sig-object py" id="helpers.spectral_helper.SpectralHelper.BC_line_zero_matrix">
+<span class="sig-name descname"><span class="pre">BC_line_zero_matrix</span></span><a class="headerlink" href="#helpers.spectral_helper.SpectralHelper.BC_line_zero_matrix" title="Link to this definition">¶</a></dt>
+<dd><p>Matrix that zeros rows where we can then add the BCs in using <cite>BCs</cite></p>
+<dl class="field-list simple">
+<dt class="field-odd">Type<span class="colon">:</span></dt>
+<dd class="field-odd"><p>sparse matrix</p>
+</dd>
+</dl>
+</dd></dl>
+
+<dl class="py attribute">
+<dt class="sig sig-object py" id="helpers.spectral_helper.SpectralHelper.rhs_BCs_hat">
+<span class="sig-name descname"><span class="pre">rhs_BCs_hat</span></span><a class="headerlink" href="#helpers.spectral_helper.SpectralHelper.rhs_BCs_hat" title="Link to this definition">¶</a></dt>
+<dd><p>Boundary conditions in spectral space</p>
+<dl class="field-list simple">
+<dt class="field-odd">Type<span class="colon">:</span></dt>
+<dd class="field-odd"><p>self.xp.ndarray</p>
+</dd>
+</dl>
+</dd></dl>
+
+<dl class="py attribute">
+<dt class="sig sig-object py" id="helpers.spectral_helper.SpectralHelper.global_shape">
+<span class="sig-name descname"><span class="pre">global_shape</span></span><a class="headerlink" href="#helpers.spectral_helper.SpectralHelper.global_shape" title="Link to this definition">¶</a></dt>
+<dd><p>Global shape of the solution as in <cite>mpi4py-fft</cite></p>
+<dl class="field-list simple">
+<dt class="field-odd">Type<span class="colon">:</span></dt>
+<dd class="field-odd"><p>tuple</p>
+</dd>
+</dl>
+</dd></dl>
+
+<dl class="py attribute">
+<dt class="sig sig-object py" id="helpers.spectral_helper.SpectralHelper.local_slice">
+<span class="sig-name descname"><span class="pre">local_slice</span></span><a class="headerlink" href="#helpers.spectral_helper.SpectralHelper.local_slice" title="Link to this definition">¶</a></dt>
+<dd><p>Local slice of the solution as in <cite>mpi4py-fft</cite></p>
+<dl class="field-list simple">
+<dt class="field-odd">Type<span class="colon">:</span></dt>
+<dd class="field-odd"><p>slice</p>
+</dd>
+</dl>
+</dd></dl>
+
+<dl class="py attribute">
+<dt class="sig sig-object py" id="helpers.spectral_helper.SpectralHelper.fft_obj">
+<span class="sig-name descname"><span class="pre">fft_obj</span></span><a class="headerlink" href="#helpers.spectral_helper.SpectralHelper.fft_obj" title="Link to this definition">¶</a></dt>
+<dd><p>When using distributed FFTs, this will be a parallel transform object from <cite>mpi4py-fft</cite></p>
+</dd></dl>
+
+<dl class="py attribute">
+<dt class="sig sig-object py" id="helpers.spectral_helper.SpectralHelper.init">
+<span class="sig-name descname"><span class="pre">init</span></span><a class="headerlink" href="#helpers.spectral_helper.SpectralHelper.init" title="Link to this definition">¶</a></dt>
+<dd><p>This is the same <cite>init</cite> that is used throughout the problem classes</p>
+<dl class="field-list simple">
+<dt class="field-odd">Type<span class="colon">:</span></dt>
+<dd class="field-odd"><p>tuple</p>
+</dd>
+</dl>
+</dd></dl>
+
+<dl class="py attribute">
+<dt class="sig sig-object py" id="helpers.spectral_helper.SpectralHelper.init_forward">
+<span class="sig-name descname"><span class="pre">init_forward</span></span><a class="headerlink" href="#helpers.spectral_helper.SpectralHelper.init_forward" title="Link to this definition">¶</a></dt>
+<dd><p>This is the equivalent of <cite>init</cite> in spectral space</p>
+<dl class="field-list simple">
+<dt class="field-odd">Type<span class="colon">:</span></dt>
+<dd class="field-odd"><p>tuple</p>
+</dd>
+</dl>
+</dd></dl>
+
+<dl class="py property">
+<dt class="sig sig-object py" id="helpers.spectral_helper.SpectralHelper.V">
+<em class="property"><span class="pre">property</span><span class="w"> </span></em><span class="sig-name descname"><span class="pre">V</span></span><a class="headerlink" href="#helpers.spectral_helper.SpectralHelper.V" title="Link to this definition">¶</a></dt>
+<dd><p>Get domain volume</p>
+</dd></dl>
+
+<dl class="py method">
+<dt class="sig sig-object py" id="helpers.spectral_helper.SpectralHelper.add_BC">
+<span class="sig-name descname"><span class="pre">add_BC</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">component</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">equation</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">axis</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">kind</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">v</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">line</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">-1</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">scalar</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">False</span></span></em>, <em class="sig-param"><span class="o"><span class="pre">**</span></span><span class="n"><span class="pre">kwargs</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../_modules/helpers/spectral_helper.html#SpectralHelper.add_BC"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#helpers.spectral_helper.SpectralHelper.add_BC" title="Link to this definition">¶</a></dt>
+<dd><p>Add a BC to the matrix. Note that you need to convert the list of lists of BCs that this method generates to a
+single sparse matrix by calling <cite>setup_BCs</cite> after adding/removing all BCs.
+Forward arguments for the boundary conditions using <cite>kwargs</cite>. Refer to documentation of 1D bases for details.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><ul class="simple">
+<li><p><strong>component</strong> (<em>str</em>) – Name of the component the BC should act on</p></li>
+<li><p><strong>equation</strong> (<em>str</em>) – Name of the equation for the component you want to put the BC in</p></li>
+<li><p><strong>axis</strong> (<em>int</em>) – Axis you want to add the BC to</p></li>
+<li><p><strong>kind</strong> (<em>str</em>) – kind of BC, e.g. Dirichlet</p></li>
+<li><p><strong>v</strong> – Value of the BC</p></li>
+<li><p><strong>line</strong> (<em>int</em>) – Line you want the BC to go in</p></li>
+<li><p><strong>scalar</strong> (<em>bool</em>) – Put the BC in all space positions in the other direction</p></li>
+</ul>
+</dd>
+</dl>
+</dd></dl>
+
+<dl class="py method">
+<dt class="sig sig-object py" id="helpers.spectral_helper.SpectralHelper.add_axis">
+<span class="sig-name descname"><span class="pre">add_axis</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">base</span></span></em>, <em class="sig-param"><span class="o"><span class="pre">*</span></span><span class="n"><span class="pre">args</span></span></em>, <em class="sig-param"><span class="o"><span class="pre">**</span></span><span class="n"><span class="pre">kwargs</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../_modules/helpers/spectral_helper.html#SpectralHelper.add_axis"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#helpers.spectral_helper.SpectralHelper.add_axis" title="Link to this definition">¶</a></dt>
+<dd><p>Add an axis to the domain by deciding on suitable 1D base.
+Arguments to the bases are forwarded using <cite>*args</cite> and <cite>**kwargs</cite>. Please refer to the documentation of the 1D
+bases for possible arguments.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><p><strong>base</strong> (<em>str</em>) – 1D spectral method</p>
+</dd>
+</dl>
+</dd></dl>
+
+<dl class="py method">
+<dt class="sig sig-object py" id="helpers.spectral_helper.SpectralHelper.add_component">
+<span class="sig-name descname"><span class="pre">add_component</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">name</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../_modules/helpers/spectral_helper.html#SpectralHelper.add_component"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#helpers.spectral_helper.SpectralHelper.add_component" title="Link to this definition">¶</a></dt>
+<dd><p>Add solution component(s).</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><p><strong>name</strong> (<em>str</em><em> or </em><em>list</em><em> of </em><em>strings</em>) – Name(s) of component(s)</p>
+</dd>
+</dl>
+</dd></dl>
+
+<dl class="py method">
+<dt class="sig sig-object py" id="helpers.spectral_helper.SpectralHelper.add_equation_lhs">
+<span class="sig-name descname"><span class="pre">add_equation_lhs</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">A</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">equation</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">relations</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../_modules/helpers/spectral_helper.html#SpectralHelper.add_equation_lhs"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#helpers.spectral_helper.SpectralHelper.add_equation_lhs" title="Link to this definition">¶</a></dt>
+<dd><p>Add the left hand part (that you want to solve implicitly) of an equation to a list of lists of sparse matrices
+that you will convert to an operator later.</p>
+<p>Example:
+Setup linear operator <cite>L</cite> for 1D heat equation using Chebychev method in first order form and T-to-U
+preconditioning:</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><ul class="simple">
+<li><p><strong>A</strong> (<em>list</em><em> of </em><em>lists</em><em> of </em><em>sparse matrices</em>) – The operator to be</p></li>
+<li><p><strong>equation</strong> (<em>str</em>) – The equation of the component you want this in</p></li>
+<li><p><strong>relations</strong> – (dict): Relations between quantities</p></li>
+</ul>
+</dd>
+</dl>
+</dd></dl>
+
+<dl class="py method">
+<dt class="sig sig-object py" id="helpers.spectral_helper.SpectralHelper.check_BCs">
+<span class="sig-name descname"><span class="pre">check_BCs</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">u</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../_modules/helpers/spectral_helper.html#SpectralHelper.check_BCs"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#helpers.spectral_helper.SpectralHelper.check_BCs" title="Link to this definition">¶</a></dt>
+<dd><p>Check that the solution satisfies the boundary conditions</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><p><strong>u</strong> – The solution you want to check</p>
+</dd>
+</dl>
+</dd></dl>
+
+<dl class="py method">
+<dt class="sig sig-object py" id="helpers.spectral_helper.SpectralHelper.convert_operator_matrix_to_operator">
+<span class="sig-name descname"><span class="pre">convert_operator_matrix_to_operator</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">M</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../_modules/helpers/spectral_helper.html#SpectralHelper.convert_operator_matrix_to_operator"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#helpers.spectral_helper.SpectralHelper.convert_operator_matrix_to_operator" title="Link to this definition">¶</a></dt>
+<dd><p>Promote the list of lists of sparse matrices to a single sparse matrix that can be used as linear operator.
+See documentation of <cite>SpectralHelper.add_equation_lhs</cite> for an example.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><p><strong>M</strong> (<em>list</em><em> of </em><em>lists</em><em> of </em><em>sparse matrices</em>) – The operator to be</p>
+</dd>
+<dt class="field-even">Returns<span class="colon">:</span></dt>
+<dd class="field-even"><p>sparse linear operator</p>
+</dd>
+</dl>
+</dd></dl>
+
+<dl class="py attribute">
+<dt class="sig sig-object py" id="helpers.spectral_helper.SpectralHelper.dtype">
+<span class="sig-name descname"><span class="pre">dtype</span></span><a class="headerlink" href="#helpers.spectral_helper.SpectralHelper.dtype" title="Link to this definition">¶</a></dt>
+<dd><p>alias of <code class="xref py py-class docutils literal notranslate"><span class="pre">mesh</span></code></p>
+</dd></dl>
+
+<dl class="py attribute">
+<dt class="sig sig-object py" id="helpers.spectral_helper.SpectralHelper.fft_backend">
+<span class="sig-name descname"><span class="pre">fft_backend</span></span><em class="property"><span class="w"> </span><span class="p"><span class="pre">=</span></span><span class="w"> </span><span class="pre">'fftw'</span></em><a class="headerlink" href="#helpers.spectral_helper.SpectralHelper.fft_backend" title="Link to this definition">¶</a></dt>
+<dd></dd></dl>
+
+<dl class="py attribute">
+<dt class="sig sig-object py" id="helpers.spectral_helper.SpectralHelper.fft_comm_backend">
+<span class="sig-name descname"><span class="pre">fft_comm_backend</span></span><em class="property"><span class="w"> </span><span class="p"><span class="pre">=</span></span><span class="w"> </span><span class="pre">'MPI'</span></em><a class="headerlink" href="#helpers.spectral_helper.SpectralHelper.fft_comm_backend" title="Link to this definition">¶</a></dt>
+<dd></dd></dl>
+
+<dl class="py attribute">
+<dt class="sig sig-object py" id="helpers.spectral_helper.SpectralHelper.fft_lib">
+<span class="sig-name descname"><span class="pre">fft_lib</span></span><em class="property"><span class="w"> </span><span class="p"><span class="pre">=</span></span><span class="w"> </span><span class="pre">&lt;module</span> <span class="pre">'scipy.fft'</span> <span class="pre">from</span> <span class="pre">'/home/runner/micromamba/envs/pySDC/lib/python3.12/site-packages/scipy/fft/__init__.py'&gt;</span></em><a class="headerlink" href="#helpers.spectral_helper.SpectralHelper.fft_lib" title="Link to this definition">¶</a></dt>
+<dd></dd></dl>
+
+<dl class="py method">
+<dt class="sig sig-object py" id="helpers.spectral_helper.SpectralHelper.get_BC">
+<span class="sig-name descname"><span class="pre">get_BC</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">axis</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">kind</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">line</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">-1</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">scalar</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">False</span></span></em>, <em class="sig-param"><span class="o"><span class="pre">**</span></span><span class="n"><span class="pre">kwargs</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../_modules/helpers/spectral_helper.html#SpectralHelper.get_BC"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#helpers.spectral_helper.SpectralHelper.get_BC" title="Link to this definition">¶</a></dt>
+<dd><p>Use this method for boundary bordering. It gets the respective matrix row and embeds it into a matrix.
+Pay attention that if you have multiple BCs in a single equation, you need to put them in different lines.
+Typically, the last line that does not contain a BC is the best choice.
+Forward arguments for the boundary conditions using <cite>kwargs</cite>. Refer to documentation of 1D bases for details.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><ul class="simple">
+<li><p><strong>axis</strong> (<em>int</em>) – Axis you want to add the BC to</p></li>
+<li><p><strong>kind</strong> (<em>str</em>) – kind of BC, e.g. Dirichlet</p></li>
+<li><p><strong>line</strong> (<em>int</em>) – Line you want the BC to go in</p></li>
+<li><p><strong>scalar</strong> (<em>bool</em>) – Put the BC in all space positions in the other direction</p></li>
+</ul>
+</dd>
+<dt class="field-even">Returns<span class="colon">:</span></dt>
+<dd class="field-even"><p>sparse matrix containing the BC</p>
+</dd>
+</dl>
+</dd></dl>
+
+<dl class="py method">
+<dt class="sig sig-object py" id="helpers.spectral_helper.SpectralHelper.get_Dirichlet_recombination_matrix">
+<span class="sig-name descname"><span class="pre">get_Dirichlet_recombination_matrix</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">axis</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">-1</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../_modules/helpers/spectral_helper.html#SpectralHelper.get_Dirichlet_recombination_matrix"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#helpers.spectral_helper.SpectralHelper.get_Dirichlet_recombination_matrix" title="Link to this definition">¶</a></dt>
+<dd><p>Get Dirichlet recombination matrix along axis. Not that it only makes sense in directions discretized with variations of Chebychev bases.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><p><strong>axis</strong> (<em>int</em>) – Axis you discretized with Chebychev</p>
+</dd>
+<dt class="field-even">Returns<span class="colon">:</span></dt>
+<dd class="field-even"><p>sparse matrix</p>
+</dd>
+</dl>
+</dd></dl>
+
+<dl class="py method">
+<dt class="sig sig-object py" id="helpers.spectral_helper.SpectralHelper.get_Id">
+<span class="sig-name descname"><span class="pre">get_Id</span></span><span class="sig-paren">(</span><span class="sig-paren">)</span><a class="reference internal" href="../_modules/helpers/spectral_helper.html#SpectralHelper.get_Id"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#helpers.spectral_helper.SpectralHelper.get_Id" title="Link to this definition">¶</a></dt>
+<dd><p>Get identity matrix</p>
+<dl class="field-list simple">
+<dt class="field-odd">Returns<span class="colon">:</span></dt>
+<dd class="field-odd"><p>sparse identity matrix</p>
+</dd>
+</dl>
+</dd></dl>
+
+<dl class="py method">
+<dt class="sig sig-object py" id="helpers.spectral_helper.SpectralHelper.get_aligned">
+<span class="sig-name descname"><span class="pre">get_aligned</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">u</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">axis_in</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">axis_out</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">fft</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">forward</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">False</span></span></em>, <em class="sig-param"><span class="o"><span class="pre">**</span></span><span class="n"><span class="pre">kwargs</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../_modules/helpers/spectral_helper.html#SpectralHelper.get_aligned"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#helpers.spectral_helper.SpectralHelper.get_aligned" title="Link to this definition">¶</a></dt>
+<dd><p>Realign the data along the axis when using distributed FFTs. <cite>kwargs</cite> will be used to get the correct PFFT
+object from <cite>mpi4py-fft</cite>, which has suitable transfer classes for the shape of data. Hence, they should include
+shape especially, if applicable.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><ul class="simple">
+<li><p><strong>u</strong> – The solution</p></li>
+<li><p><strong>axis_in</strong> (<em>int</em>) – Current alignment</p></li>
+<li><p><strong>axis_out</strong> (<em>int</em>) – New alignment</p></li>
+<li><p><strong>fft</strong> (<em>mpi4py_fft.PFFT</em>) – parallel FFT object</p></li>
+<li><p><strong>forward</strong> (<em>bool</em>) – Whether the input is in spectral space or not</p></li>
+</ul>
+</dd>
+<dt class="field-even">Returns<span class="colon">:</span></dt>
+<dd class="field-even"><p>solution aligned on <cite>axis_in</cite></p>
+</dd>
+</dl>
+</dd></dl>
+
+<dl class="py method">
+<dt class="sig sig-object py" id="helpers.spectral_helper.SpectralHelper.get_basis_change_matrix">
+<span class="sig-name descname"><span class="pre">get_basis_change_matrix</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">axes</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="o"><span class="pre">**</span></span><span class="n"><span class="pre">kwargs</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../_modules/helpers/spectral_helper.html#SpectralHelper.get_basis_change_matrix"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#helpers.spectral_helper.SpectralHelper.get_basis_change_matrix" title="Link to this definition">¶</a></dt>
+<dd><p>Some spectral bases do a change between bases while differentiating. This method returns matrices that changes the basis to whatever you want.
+Refer to the methods of the same name of the 1D bases to learn what parameters you need to pass here as <cite>kwargs</cite>.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><p><strong>axes</strong> (<em>tuple</em>) – Axes along which to change basis.</p>
+</dd>
+<dt class="field-even">Returns<span class="colon">:</span></dt>
+<dd class="field-even"><p>sparse basis change matrix</p>
+</dd>
+</dl>
+</dd></dl>
+
+<dl class="py method">
+<dt class="sig sig-object py" id="helpers.spectral_helper.SpectralHelper.get_differentiation_matrix">
+<span class="sig-name descname"><span class="pre">get_differentiation_matrix</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">axes</span></span></em>, <em class="sig-param"><span class="o"><span class="pre">**</span></span><span class="n"><span class="pre">kwargs</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../_modules/helpers/spectral_helper.html#SpectralHelper.get_differentiation_matrix"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#helpers.spectral_helper.SpectralHelper.get_differentiation_matrix" title="Link to this definition">¶</a></dt>
+<dd><p>Get differentiation matrix along specified axis. <cite>kwargs</cite> are forwarded to the 1D base implementation.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><p><strong>axes</strong> (<em>tuple</em>) – Axes along which to differentiate.</p>
+</dd>
+<dt class="field-even">Returns<span class="colon">:</span></dt>
+<dd class="field-even"><p>sparse differentiation matrix</p>
+</dd>
+</dl>
+</dd></dl>
+
+<dl class="py method">
+<dt class="sig sig-object py" id="helpers.spectral_helper.SpectralHelper.get_empty_operator_matrix">
+<span class="sig-name descname"><span class="pre">get_empty_operator_matrix</span></span><span class="sig-paren">(</span><span class="sig-paren">)</span><a class="reference internal" href="../_modules/helpers/spectral_helper.html#SpectralHelper.get_empty_operator_matrix"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#helpers.spectral_helper.SpectralHelper.get_empty_operator_matrix" title="Link to this definition">¶</a></dt>
+<dd><p>Return a matrix of operators to be filled with the connections between the solution components.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Returns<span class="colon">:</span></dt>
+<dd class="field-odd"><p>list containing sparse zeros</p>
+</dd>
+</dl>
+</dd></dl>
+
+<dl class="py method">
+<dt class="sig sig-object py" id="helpers.spectral_helper.SpectralHelper.get_fft">
+<span class="sig-name descname"><span class="pre">get_fft</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">axes</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">direction</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">'object'</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">padding</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">shape</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../_modules/helpers/spectral_helper.html#SpectralHelper.get_fft"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#helpers.spectral_helper.SpectralHelper.get_fft" title="Link to this definition">¶</a></dt>
+<dd><p>When using MPI, we use <cite>PFFT</cite> objects generated by mpi4py-fft</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><ul class="simple">
+<li><p><strong>axes</strong> (<em>tuple</em>) – Axes you want to transform over</p></li>
+<li><p><strong>direction</strong> (<em>str</em>) – use “forward” or “backward” to get functions for performing the transforms or “object” to get the PFFT object</p></li>
+<li><p><strong>padding</strong> (<em>tuple</em>) – Padding for dealiasing</p></li>
+<li><p><strong>shape</strong> (<em>tuple</em>) – Shape of the transform</p></li>
+</ul>
+</dd>
+<dt class="field-even">Returns<span class="colon">:</span></dt>
+<dd class="field-even"><p>transform</p>
+</dd>
+</dl>
+</dd></dl>
+
+<dl class="py method">
+<dt class="sig sig-object py" id="helpers.spectral_helper.SpectralHelper.get_filter_matrix">
+<span class="sig-name descname"><span class="pre">get_filter_matrix</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">axis</span></span></em>, <em class="sig-param"><span class="o"><span class="pre">**</span></span><span class="n"><span class="pre">kwargs</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../_modules/helpers/spectral_helper.html#SpectralHelper.get_filter_matrix"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#helpers.spectral_helper.SpectralHelper.get_filter_matrix" title="Link to this definition">¶</a></dt>
+<dd><p>Get bandpass filter along <cite>axis</cite>. See the documentation <cite>get_filter_matrix</cite> in the 1D bases for what kwargs are
+admissible.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Returns<span class="colon">:</span></dt>
+<dd class="field-odd"><p>sparse bandpass matrix</p>
+</dd>
+</dl>
+</dd></dl>
+
+<dl class="py method">
+<dt class="sig sig-object py" id="helpers.spectral_helper.SpectralHelper.get_grid">
+<span class="sig-name descname"><span class="pre">get_grid</span></span><span class="sig-paren">(</span><span class="sig-paren">)</span><a class="reference internal" href="../_modules/helpers/spectral_helper.html#SpectralHelper.get_grid"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#helpers.spectral_helper.SpectralHelper.get_grid" title="Link to this definition">¶</a></dt>
+<dd><p>Get grid in physical space</p>
+</dd></dl>
+
+<dl class="py method">
+<dt class="sig sig-object py" id="helpers.spectral_helper.SpectralHelper.get_integration_matrix">
+<span class="sig-name descname"><span class="pre">get_integration_matrix</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">axes</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../_modules/helpers/spectral_helper.html#SpectralHelper.get_integration_matrix"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#helpers.spectral_helper.SpectralHelper.get_integration_matrix" title="Link to this definition">¶</a></dt>
+<dd><p>Get integration matrix to integrate along specified axis.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><p><strong>axes</strong> (<em>tuple</em>) – Axes along which to integrate over.</p>
+</dd>
+<dt class="field-even">Returns<span class="colon">:</span></dt>
+<dd class="field-even"><p>sparse integration matrix</p>
+</dd>
+</dl>
+</dd></dl>
+
+<dl class="py method">
+<dt class="sig sig-object py" id="helpers.spectral_helper.SpectralHelper.get_local_slice_of_1D_matrix">
+<span class="sig-name descname"><span class="pre">get_local_slice_of_1D_matrix</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">M</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">axis</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../_modules/helpers/spectral_helper.html#SpectralHelper.get_local_slice_of_1D_matrix"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#helpers.spectral_helper.SpectralHelper.get_local_slice_of_1D_matrix" title="Link to this definition">¶</a></dt>
+<dd><p>Get the local version of a 1D matrix. When using distributed FFTs, each rank will carry only a subset of modes,
+which you can sort out via the <cite>SpectralHelper.local_slice</cite> attribute. When constructing a 1D matrix, you can
+use this method to get the part corresponding to the modes carried by this rank.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><ul class="simple">
+<li><p><strong>M</strong> (<em>sparse matrix</em>) – Global 1D matrix you want to get the local version of</p></li>
+<li><p><strong>axis</strong> (<em>int</em>) – Direction in which you want the local version. You will get the global matrix in other directions. This means slab decomposition only.</p></li>
+</ul>
+</dd>
+<dt class="field-even">Returns<span class="colon">:</span></dt>
+<dd class="field-even"><p>sparse local matrix</p>
+</dd>
+</dl>
+</dd></dl>
+
+<dl class="py method">
+<dt class="sig sig-object py" id="helpers.spectral_helper.SpectralHelper.get_wavenumbers">
+<span class="sig-name descname"><span class="pre">get_wavenumbers</span></span><span class="sig-paren">(</span><span class="sig-paren">)</span><a class="reference internal" href="../_modules/helpers/spectral_helper.html#SpectralHelper.get_wavenumbers"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#helpers.spectral_helper.SpectralHelper.get_wavenumbers" title="Link to this definition">¶</a></dt>
+<dd><p>Get grid in spectral space</p>
+</dd></dl>
+
+<dl class="py method">
+<dt class="sig sig-object py" id="helpers.spectral_helper.SpectralHelper.index">
+<span class="sig-name descname"><span class="pre">index</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">name</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../_modules/helpers/spectral_helper.html#SpectralHelper.index"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#helpers.spectral_helper.SpectralHelper.index" title="Link to this definition">¶</a></dt>
+<dd><p>Get the index of component <cite>name</cite>.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><p><strong>name</strong> (<em>str</em><em> or </em><em>list</em><em> of </em><em>strings</em>) – Name(s) of component(s)</p>
+</dd>
+<dt class="field-even">Returns<span class="colon">:</span></dt>
+<dd class="field-even"><p>Index of the component</p>
+</dd>
+<dt class="field-odd">Return type<span class="colon">:</span></dt>
+<dd class="field-odd"><p>int</p>
+</dd>
+</dl>
+</dd></dl>
+
+<dl class="py method">
+<dt class="sig sig-object py" id="helpers.spectral_helper.SpectralHelper.itransform">
+<span class="sig-name descname"><span class="pre">itransform</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">u</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">axes</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">padding</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../_modules/helpers/spectral_helper.html#SpectralHelper.itransform"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#helpers.spectral_helper.SpectralHelper.itransform" title="Link to this definition">¶</a></dt>
+<dd><p>Inverse transform over all components of the solution</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><ul class="simple">
+<li><p><strong>transform</strong> (<em>u data to</em>)</p></li>
+<li><p><strong>axes</strong> (<em>tuple</em>) – Axes over which to transform</p></li>
+<li><p><strong>padding</strong> (<em>list</em>) – Padding factor for transform. E.g. a padding factor of 2 will add as many zeros as there were modes before before transforming</p></li>
+</ul>
+</dd>
+<dt class="field-even">Returns<span class="colon">:</span></dt>
+<dd class="field-even"><p>Transformed data</p>
+</dd>
+</dl>
+</dd></dl>
+
+<dl class="py method">
+<dt class="sig sig-object py" id="helpers.spectral_helper.SpectralHelper.itransform_single_component">
+<span class="sig-name descname"><span class="pre">itransform_single_component</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">u</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">axes</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">padding</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../_modules/helpers/spectral_helper.html#SpectralHelper.itransform_single_component"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#helpers.spectral_helper.SpectralHelper.itransform_single_component" title="Link to this definition">¶</a></dt>
+<dd><p>Inverse transform over single component of the solution</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><ul class="simple">
+<li><p><strong>transform</strong> (<em>u data to</em>)</p></li>
+<li><p><strong>axes</strong> (<em>tuple</em>) – Axes over which to transform</p></li>
+<li><p><strong>padding</strong> (<em>list</em>) – Padding factor for transform. E.g. a padding factor of 2 will add as many zeros as there were modes before before transforming</p></li>
+</ul>
+</dd>
+<dt class="field-even">Returns<span class="colon">:</span></dt>
+<dd class="field-even"><p>Transformed data</p>
+</dd>
+</dl>
+</dd></dl>
+
+<dl class="py attribute">
+<dt class="sig sig-object py" id="helpers.spectral_helper.SpectralHelper.linalg">
+<span class="sig-name descname"><span class="pre">linalg</span></span><em class="property"><span class="w"> </span><span class="p"><span class="pre">=</span></span><span class="w"> </span><span class="pre">&lt;module</span> <span class="pre">'scipy.sparse.linalg'</span> <span class="pre">from</span> <span class="pre">'/home/runner/micromamba/envs/pySDC/lib/python3.12/site-packages/scipy/sparse/linalg/__init__.py'&gt;</span></em><a class="headerlink" href="#helpers.spectral_helper.SpectralHelper.linalg" title="Link to this definition">¶</a></dt>
+<dd></dd></dl>
+
+<dl class="py property">
+<dt class="sig sig-object py" id="helpers.spectral_helper.SpectralHelper.ncomponents">
+<em class="property"><span class="pre">property</span><span class="w"> </span></em><span class="sig-name descname"><span class="pre">ncomponents</span></span><a class="headerlink" href="#helpers.spectral_helper.SpectralHelper.ncomponents" title="Link to this definition">¶</a></dt>
+<dd></dd></dl>
+
+<dl class="py property">
+<dt class="sig sig-object py" id="helpers.spectral_helper.SpectralHelper.ndim">
+<em class="property"><span class="pre">property</span><span class="w"> </span></em><span class="sig-name descname"><span class="pre">ndim</span></span><a class="headerlink" href="#helpers.spectral_helper.SpectralHelper.ndim" title="Link to this definition">¶</a></dt>
+<dd></dd></dl>
+
+<dl class="py method">
+<dt class="sig sig-object py" id="helpers.spectral_helper.SpectralHelper.put_BCs_in_matrix">
+<span class="sig-name descname"><span class="pre">put_BCs_in_matrix</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">A</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../_modules/helpers/spectral_helper.html#SpectralHelper.put_BCs_in_matrix"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#helpers.spectral_helper.SpectralHelper.put_BCs_in_matrix" title="Link to this definition">¶</a></dt>
+<dd><p>Put the boundary conditions in a matrix by replacing rows with BCs.</p>
+</dd></dl>
+
+<dl class="py method">
+<dt class="sig sig-object py" id="helpers.spectral_helper.SpectralHelper.put_BCs_in_rhs">
+<span class="sig-name descname"><span class="pre">put_BCs_in_rhs</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">rhs</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../_modules/helpers/spectral_helper.html#SpectralHelper.put_BCs_in_rhs"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#helpers.spectral_helper.SpectralHelper.put_BCs_in_rhs" title="Link to this definition">¶</a></dt>
+<dd><p>Put the BCs in the right hand side for solving.
+This function will transform along each axis individually and add all BCs in that axis.
+Consider <cite>put_BCs_in_rhs_hat</cite> to add BCs with no extra transforms needed.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><p><strong>rhs</strong> – Right hand side in physical space</p>
+</dd>
+<dt class="field-even">Returns<span class="colon">:</span></dt>
+<dd class="field-even"><p>rhs in physical space with BCs</p>
+</dd>
+</dl>
+</dd></dl>
+
+<dl class="py method">
+<dt class="sig sig-object py" id="helpers.spectral_helper.SpectralHelper.put_BCs_in_rhs_hat">
+<span class="sig-name descname"><span class="pre">put_BCs_in_rhs_hat</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">rhs_hat</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../_modules/helpers/spectral_helper.html#SpectralHelper.put_BCs_in_rhs_hat"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#helpers.spectral_helper.SpectralHelper.put_BCs_in_rhs_hat" title="Link to this definition">¶</a></dt>
+<dd><p>Put the BCs in the right hand side in spectral space for solving.
+This function needs no transforms.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><p><strong>rhs_hat</strong> – Right hand side in spectral space</p>
+</dd>
+<dt class="field-even">Returns<span class="colon">:</span></dt>
+<dd class="field-even"><p>rhs in spectral space with BCs</p>
+</dd>
+</dl>
+</dd></dl>
+
+<dl class="py method">
+<dt class="sig sig-object py" id="helpers.spectral_helper.SpectralHelper.remove_BC">
+<span class="sig-name descname"><span class="pre">remove_BC</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">component</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">equation</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">axis</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">kind</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">line</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">-1</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">scalar</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">False</span></span></em>, <em class="sig-param"><span class="o"><span class="pre">**</span></span><span class="n"><span class="pre">kwargs</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../_modules/helpers/spectral_helper.html#SpectralHelper.remove_BC"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#helpers.spectral_helper.SpectralHelper.remove_BC" title="Link to this definition">¶</a></dt>
+<dd><p>Remove a BC from the matrix. This is useful e.g. when you add a non-scalar BC and then need to selectively
+remove single BCs again, as in incompressible Navier-Stokes, for instance.
+Forward arguments for the boundary conditions using <cite>kwargs</cite>. Refer to documentation of 1D bases for details.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><ul class="simple">
+<li><p><strong>component</strong> (<em>str</em>) – Name of the component the BC should act on</p></li>
+<li><p><strong>equation</strong> (<em>str</em>) – Name of the equation for the component you want to put the BC in</p></li>
+<li><p><strong>axis</strong> (<em>int</em>) – Axis you want to add the BC to</p></li>
+<li><p><strong>kind</strong> (<em>str</em>) – kind of BC, e.g. Dirichlet</p></li>
+<li><p><strong>v</strong> – Value of the BC</p></li>
+<li><p><strong>line</strong> (<em>int</em>) – Line you want the BC to go in</p></li>
+<li><p><strong>scalar</strong> (<em>bool</em>) – Put the BC in all space positions in the other direction</p></li>
+</ul>
+</dd>
+</dl>
+</dd></dl>
+
+<dl class="py method">
+<dt class="sig sig-object py" id="helpers.spectral_helper.SpectralHelper.setup_BCs">
+<span class="sig-name descname"><span class="pre">setup_BCs</span></span><span class="sig-paren">(</span><span class="sig-paren">)</span><a class="reference internal" href="../_modules/helpers/spectral_helper.html#SpectralHelper.setup_BCs"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#helpers.spectral_helper.SpectralHelper.setup_BCs" title="Link to this definition">¶</a></dt>
+<dd><p>Convert the list of lists of BCs to the boundary condition operator.
+Also, boundary bordering requires to zero out all other entries in the matrix in rows containing a boundary
+condition. This method sets up a suitable sparse matrix to do this.</p>
+</dd></dl>
+
+<dl class="py method">
+<dt class="sig sig-object py" id="helpers.spectral_helper.SpectralHelper.setup_GPU">
+<em class="property"><span class="pre">classmethod</span><span class="w"> </span></em><span class="sig-name descname"><span class="pre">setup_GPU</span></span><span class="sig-paren">(</span><span class="sig-paren">)</span><a class="reference internal" href="../_modules/helpers/spectral_helper.html#SpectralHelper.setup_GPU"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#helpers.spectral_helper.SpectralHelper.setup_GPU" title="Link to this definition">¶</a></dt>
+<dd><p>switch to GPU modules</p>
+</dd></dl>
+
+<dl class="py method">
+<dt class="sig sig-object py" id="helpers.spectral_helper.SpectralHelper.setup_fft">
+<span class="sig-name descname"><span class="pre">setup_fft</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">real_spectral_coefficients</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">False</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../_modules/helpers/spectral_helper.html#SpectralHelper.setup_fft"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#helpers.spectral_helper.SpectralHelper.setup_fft" title="Link to this definition">¶</a></dt>
+<dd><p>This function must be called after all axes have been setup in order to prepare the local shapes of the data.
+This must also be called before setting up any BCs.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><p><strong>real_spectral_coefficients</strong> (<em>bool</em>) – Allow only real coefficients in spectral space</p>
+</dd>
+</dl>
+</dd></dl>
+
+<dl class="py property">
+<dt class="sig sig-object py" id="helpers.spectral_helper.SpectralHelper.shape">
+<em class="property"><span class="pre">property</span><span class="w"> </span></em><span class="sig-name descname"><span class="pre">shape</span></span><a class="headerlink" href="#helpers.spectral_helper.SpectralHelper.shape" title="Link to this definition">¶</a></dt>
+<dd><p>Get shape of individual solution component</p>
+</dd></dl>
+
+<dl class="py attribute">
+<dt class="sig sig-object py" id="helpers.spectral_helper.SpectralHelper.sparse_lib">
+<span class="sig-name descname"><span class="pre">sparse_lib</span></span><em class="property"><span class="w"> </span><span class="p"><span class="pre">=</span></span><span class="w"> </span><span class="pre">&lt;module</span> <span class="pre">'scipy.sparse'</span> <span class="pre">from</span> <span class="pre">'/home/runner/micromamba/envs/pySDC/lib/python3.12/site-packages/scipy/sparse/__init__.py'&gt;</span></em><a class="headerlink" href="#helpers.spectral_helper.SpectralHelper.sparse_lib" title="Link to this definition">¶</a></dt>
+<dd></dd></dl>
+
+<dl class="py method">
+<dt class="sig sig-object py" id="helpers.spectral_helper.SpectralHelper.transform">
+<span class="sig-name descname"><span class="pre">transform</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">u</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">axes</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">padding</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../_modules/helpers/spectral_helper.html#SpectralHelper.transform"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#helpers.spectral_helper.SpectralHelper.transform" title="Link to this definition">¶</a></dt>
+<dd><p>Transform all components from physical space to spectral space</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><ul class="simple">
+<li><p><strong>transform</strong> (<em>u data to</em>)</p></li>
+<li><p><strong>axes</strong> (<em>tuple</em>) – Axes over which to transform</p></li>
+<li><p><strong>padding</strong> (<em>list</em>) – Padding factor for transform. E.g. a padding factor of 2 will discard the upper half of modes after transforming</p></li>
+</ul>
+</dd>
+<dt class="field-even">Returns<span class="colon">:</span></dt>
+<dd class="field-even"><p>Transformed data</p>
+</dd>
+</dl>
+</dd></dl>
+
+<dl class="py method">
+<dt class="sig sig-object py" id="helpers.spectral_helper.SpectralHelper.transform_single_component">
+<span class="sig-name descname"><span class="pre">transform_single_component</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">u</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">axes</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">padding</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../_modules/helpers/spectral_helper.html#SpectralHelper.transform_single_component"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#helpers.spectral_helper.SpectralHelper.transform_single_component" title="Link to this definition">¶</a></dt>
+<dd><p>Transform a single component of the solution</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><ul class="simple">
+<li><p><strong>transform</strong> (<em>u data to</em>)</p></li>
+<li><p><strong>axes</strong> (<em>tuple</em>) – Axes over which to transform</p></li>
+<li><p><strong>padding</strong> (<em>list</em>) – Padding factor for transform. E.g. a padding factor of 2 will discard the upper half of modes after transforming</p></li>
+</ul>
+</dd>
+<dt class="field-even">Returns<span class="colon">:</span></dt>
+<dd class="field-even"><p>Transformed data</p>
+</dd>
+</dl>
+</dd></dl>
+
+<dl class="py property">
+<dt class="sig sig-object py" id="helpers.spectral_helper.SpectralHelper.u_init">
+<em class="property"><span class="pre">property</span><span class="w"> </span></em><span class="sig-name descname"><span class="pre">u_init</span></span><a class="headerlink" href="#helpers.spectral_helper.SpectralHelper.u_init" title="Link to this definition">¶</a></dt>
+<dd><p>Get empty data container in physical space</p>
+</dd></dl>
+
+<dl class="py property">
+<dt class="sig sig-object py" id="helpers.spectral_helper.SpectralHelper.u_init_forward">
+<em class="property"><span class="pre">property</span><span class="w"> </span></em><span class="sig-name descname"><span class="pre">u_init_forward</span></span><a class="headerlink" href="#helpers.spectral_helper.SpectralHelper.u_init_forward" title="Link to this definition">¶</a></dt>
+<dd><p>Get empty data container in spectral space</p>
+</dd></dl>
+
+<dl class="py attribute">
+<dt class="sig sig-object py" id="helpers.spectral_helper.SpectralHelper.xp">
+<span class="sig-name descname"><span class="pre">xp</span></span><em class="property"><span class="w"> </span><span class="p"><span class="pre">=</span></span><span class="w"> </span><span class="pre">&lt;module</span> <span class="pre">'numpy'</span> <span class="pre">from</span> <span class="pre">'/home/runner/micromamba/envs/pySDC/lib/python3.12/site-packages/numpy/__init__.py'&gt;</span></em><a class="headerlink" href="#helpers.spectral_helper.SpectralHelper.xp" title="Link to this definition">¶</a></dt>
+<dd></dd></dl>
+
+</dd></dl>
+
+<dl class="py class">
+<dt class="sig sig-object py" id="helpers.spectral_helper.SpectralHelper1D">
+<em class="property"><span class="pre">class</span><span class="w"> </span></em><span class="sig-name descname"><span class="pre">SpectralHelper1D</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">N</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">x0</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">x1</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">useGPU</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">False</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../_modules/helpers/spectral_helper.html#SpectralHelper1D"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#helpers.spectral_helper.SpectralHelper1D" title="Link to this definition">¶</a></dt>
+<dd><p>Bases: <code class="xref py py-class docutils literal notranslate"><span class="pre">object</span></code></p>
+<p>Abstract base class for 1D spectral discretizations. Defines a common interface with parameters and functions that
+all bases need to have.</p>
+<p>When implementing new bases, please take care to use the modules that are supplied as class attributes to enable
+the code for GPUs.</p>
+<dl class="py attribute">
+<dt class="sig sig-object py" id="helpers.spectral_helper.SpectralHelper1D.N">
+<span class="sig-name descname"><span class="pre">N</span></span><a class="headerlink" href="#helpers.spectral_helper.SpectralHelper1D.N" title="Link to this definition">¶</a></dt>
+<dd><p>Resolution</p>
+<dl class="field-list simple">
+<dt class="field-odd">Type<span class="colon">:</span></dt>
+<dd class="field-odd"><p>int</p>
+</dd>
+</dl>
+</dd></dl>
+
+<dl class="py attribute">
+<dt class="sig sig-object py" id="helpers.spectral_helper.SpectralHelper1D.x0">
+<span class="sig-name descname"><span class="pre">x0</span></span><a class="headerlink" href="#helpers.spectral_helper.SpectralHelper1D.x0" title="Link to this definition">¶</a></dt>
+<dd><p>Coordinate of left boundary</p>
+<dl class="field-list simple">
+<dt class="field-odd">Type<span class="colon">:</span></dt>
+<dd class="field-odd"><p>float</p>
+</dd>
+</dl>
+</dd></dl>
+
+<dl class="py attribute">
+<dt class="sig sig-object py" id="helpers.spectral_helper.SpectralHelper1D.x1">
+<span class="sig-name descname"><span class="pre">x1</span></span><a class="headerlink" href="#helpers.spectral_helper.SpectralHelper1D.x1" title="Link to this definition">¶</a></dt>
+<dd><p>Coordinate of right boundary</p>
+<dl class="field-list simple">
+<dt class="field-odd">Type<span class="colon">:</span></dt>
+<dd class="field-odd"><p>float</p>
+</dd>
+</dl>
+</dd></dl>
+
+<dl class="py attribute">
+<dt class="sig sig-object py" id="helpers.spectral_helper.SpectralHelper1D.L">
+<span class="sig-name descname"><span class="pre">L</span></span><a class="headerlink" href="#helpers.spectral_helper.SpectralHelper1D.L" title="Link to this definition">¶</a></dt>
+<dd><p>Length of the domain</p>
+<dl class="field-list simple">
+<dt class="field-odd">Type<span class="colon">:</span></dt>
+<dd class="field-odd"><p>float</p>
+</dd>
+</dl>
+</dd></dl>
+
+<dl class="py attribute">
+<dt class="sig sig-object py" id="helpers.spectral_helper.SpectralHelper1D.useGPU">
+<span class="sig-name descname"><span class="pre">useGPU</span></span><a class="headerlink" href="#helpers.spectral_helper.SpectralHelper1D.useGPU" title="Link to this definition">¶</a></dt>
+<dd><p>Whether to use GPUs</p>
+<dl class="field-list simple">
+<dt class="field-odd">Type<span class="colon">:</span></dt>
+<dd class="field-odd"><p>bool</p>
+</dd>
+</dl>
+</dd></dl>
+
+<dl class="py attribute">
+<dt class="sig sig-object py" id="helpers.spectral_helper.SpectralHelper1D.fft_lib">
+<span class="sig-name descname"><span class="pre">fft_lib</span></span><em class="property"><span class="w"> </span><span class="p"><span class="pre">=</span></span><span class="w"> </span><span class="pre">&lt;module</span> <span class="pre">'scipy.fft'</span> <span class="pre">from</span> <span class="pre">'/home/runner/micromamba/envs/pySDC/lib/python3.12/site-packages/scipy/fft/__init__.py'&gt;</span></em><a class="headerlink" href="#helpers.spectral_helper.SpectralHelper1D.fft_lib" title="Link to this definition">¶</a></dt>
+<dd></dd></dl>
+
+<dl class="py method">
+<dt class="sig sig-object py" id="helpers.spectral_helper.SpectralHelper1D.get_1dgrid">
+<span class="sig-name descname"><span class="pre">get_1dgrid</span></span><span class="sig-paren">(</span><span class="sig-paren">)</span><a class="reference internal" href="../_modules/helpers/spectral_helper.html#SpectralHelper1D.get_1dgrid"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#helpers.spectral_helper.SpectralHelper1D.get_1dgrid" title="Link to this definition">¶</a></dt>
+<dd><p>Get the grid in physical space</p>
+<dl class="field-list simple">
+<dt class="field-odd">Returns<span class="colon">:</span></dt>
+<dd class="field-odd"><p>Grid</p>
+</dd>
+<dt class="field-even">Return type<span class="colon">:</span></dt>
+<dd class="field-even"><p>self.xp.array</p>
+</dd>
+</dl>
+</dd></dl>
+
+<dl class="py method">
+<dt class="sig sig-object py" id="helpers.spectral_helper.SpectralHelper1D.get_BC">
+<span class="sig-name descname"><span class="pre">get_BC</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">kind</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../_modules/helpers/spectral_helper.html#SpectralHelper1D.get_BC"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#helpers.spectral_helper.SpectralHelper1D.get_BC" title="Link to this definition">¶</a></dt>
+<dd><p>To facilitate boundary conditions (BCs) we use either a basis where all functions satisfy the BCs automatically,
+as is the case in FFT basis for periodic BCs, or boundary bordering. In boundary bordering, specific lines in
+the matrix are replaced by the boundary conditions as obtained by this method.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><ul class="simple">
+<li><p><strong>kind</strong> (<em>str</em>) – The type of BC you want to implement please refer to the implementations of this method in the</p></li>
+<li><p><strong>implemented</strong> (<em>individual 1D bases for what is</em>)</p></li>
+</ul>
+</dd>
+<dt class="field-even">Returns<span class="colon">:</span></dt>
+<dd class="field-even"><p>Boundary condition</p>
+</dd>
+<dt class="field-odd">Return type<span class="colon">:</span></dt>
+<dd class="field-odd"><p>self.xp.array</p>
+</dd>
+</dl>
+</dd></dl>
+
+<dl class="py method">
+<dt class="sig sig-object py" id="helpers.spectral_helper.SpectralHelper1D.get_Id">
+<span class="sig-name descname"><span class="pre">get_Id</span></span><span class="sig-paren">(</span><span class="sig-paren">)</span><a class="reference internal" href="../_modules/helpers/spectral_helper.html#SpectralHelper1D.get_Id"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#helpers.spectral_helper.SpectralHelper1D.get_Id" title="Link to this definition">¶</a></dt>
+<dd><p>Get identity matrix</p>
+<dl class="field-list simple">
+<dt class="field-odd">Returns<span class="colon">:</span></dt>
+<dd class="field-odd"><p>sparse diagonal identity matrix</p>
+</dd>
+</dl>
+</dd></dl>
+
+<dl class="py method">
+<dt class="sig sig-object py" id="helpers.spectral_helper.SpectralHelper1D.get_basis_change_matrix">
+<span class="sig-name descname"><span class="pre">get_basis_change_matrix</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="o"><span class="pre">*</span></span><span class="n"><span class="pre">args</span></span></em>, <em class="sig-param"><span class="o"><span class="pre">**</span></span><span class="n"><span class="pre">kwargs</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../_modules/helpers/spectral_helper.html#SpectralHelper1D.get_basis_change_matrix"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#helpers.spectral_helper.SpectralHelper1D.get_basis_change_matrix" title="Link to this definition">¶</a></dt>
+<dd><p>Some spectral discretization change the basis during differentiation. This method can be used to transfer
+between the various bases.</p>
+<p>This method accepts arbitrary arguments that may not be used in order to provide an easy interface for multi-
+dimensional bases. For instance, you may combine an FFT discretization with an ultraspherical discretization.
+The FFT discretization will always be in the same base, but the ultraspherical discretization uses a different
+base for every derivative. You can then ask all bases for transfer matrices from one ultraspherical derivative
+base to the next. The FFT discretization will ignore this and return an identity while the ultraspherical
+discretization will return the desired matrix. After a Kronecker product, you get the 2D version of the matrix
+you want. This is what the <cite>SpectralHelper</cite> does when you call the method of the same name on it.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Returns<span class="colon">:</span></dt>
+<dd class="field-odd"><p>sparse bases change matrix</p>
+</dd>
+</dl>
+</dd></dl>
+
+<dl class="py method">
+<dt class="sig sig-object py" id="helpers.spectral_helper.SpectralHelper1D.get_differentiation_matrix">
+<span class="sig-name descname"><span class="pre">get_differentiation_matrix</span></span><span class="sig-paren">(</span><span class="sig-paren">)</span><a class="reference internal" href="../_modules/helpers/spectral_helper.html#SpectralHelper1D.get_differentiation_matrix"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#helpers.spectral_helper.SpectralHelper1D.get_differentiation_matrix" title="Link to this definition">¶</a></dt>
+<dd></dd></dl>
+
+<dl class="py method">
+<dt class="sig sig-object py" id="helpers.spectral_helper.SpectralHelper1D.get_empty_operator_matrix">
+<span class="sig-name descname"><span class="pre">get_empty_operator_matrix</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">S</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">O</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../_modules/helpers/spectral_helper.html#SpectralHelper1D.get_empty_operator_matrix"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#helpers.spectral_helper.SpectralHelper1D.get_empty_operator_matrix" title="Link to this definition">¶</a></dt>
+<dd><p>Return a matrix of operators to be filled with the connections between the solution components.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><ul class="simple">
+<li><p><strong>S</strong> (<em>int</em>) – Number of components in the solution</p></li>
+<li><p><strong>O</strong> (<em>sparse matrix</em>) – Zero matrix used for initialization</p></li>
+</ul>
+</dd>
+<dt class="field-even">Returns<span class="colon">:</span></dt>
+<dd class="field-even"><p>list of lists containing sparse zeros</p>
+</dd>
+</dl>
+</dd></dl>
+
+<dl class="py method">
+<dt class="sig sig-object py" id="helpers.spectral_helper.SpectralHelper1D.get_filter_matrix">
+<span class="sig-name descname"><span class="pre">get_filter_matrix</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">kmin</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">0</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">kmax</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../_modules/helpers/spectral_helper.html#SpectralHelper1D.get_filter_matrix"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#helpers.spectral_helper.SpectralHelper1D.get_filter_matrix" title="Link to this definition">¶</a></dt>
+<dd><p>Get a bandpass filter.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><ul class="simple">
+<li><p><strong>kmin</strong> (<em>int</em>) – Lower limit of the bandpass filter</p></li>
+<li><p><strong>kmax</strong> (<em>int</em>) – Upper limit of the bandpass filter</p></li>
+</ul>
+</dd>
+<dt class="field-even">Returns<span class="colon">:</span></dt>
+<dd class="field-even"><p>sparse matrix</p>
+</dd>
+</dl>
+</dd></dl>
+
+<dl class="py method">
+<dt class="sig sig-object py" id="helpers.spectral_helper.SpectralHelper1D.get_integration_matrix">
+<span class="sig-name descname"><span class="pre">get_integration_matrix</span></span><span class="sig-paren">(</span><span class="sig-paren">)</span><a class="reference internal" href="../_modules/helpers/spectral_helper.html#SpectralHelper1D.get_integration_matrix"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#helpers.spectral_helper.SpectralHelper1D.get_integration_matrix" title="Link to this definition">¶</a></dt>
+<dd></dd></dl>
+
+<dl class="py method">
+<dt class="sig sig-object py" id="helpers.spectral_helper.SpectralHelper1D.get_wavenumbers">
+<span class="sig-name descname"><span class="pre">get_wavenumbers</span></span><span class="sig-paren">(</span><span class="sig-paren">)</span><a class="reference internal" href="../_modules/helpers/spectral_helper.html#SpectralHelper1D.get_wavenumbers"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#helpers.spectral_helper.SpectralHelper1D.get_wavenumbers" title="Link to this definition">¶</a></dt>
+<dd><p>Get the grid in spectral space</p>
+</dd></dl>
+
+<dl class="py method">
+<dt class="sig sig-object py" id="helpers.spectral_helper.SpectralHelper1D.get_zero">
+<span class="sig-name descname"><span class="pre">get_zero</span></span><span class="sig-paren">(</span><span class="sig-paren">)</span><a class="reference internal" href="../_modules/helpers/spectral_helper.html#SpectralHelper1D.get_zero"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#helpers.spectral_helper.SpectralHelper1D.get_zero" title="Link to this definition">¶</a></dt>
+<dd><p>Get a matrix with all zeros of the correct size.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Returns<span class="colon">:</span></dt>
+<dd class="field-odd"><p>sparse matrix with zeros everywhere</p>
+</dd>
+</dl>
+</dd></dl>
+
+<dl class="py attribute">
+<dt class="sig sig-object py" id="helpers.spectral_helper.SpectralHelper1D.linalg">
+<span class="sig-name descname"><span class="pre">linalg</span></span><em class="property"><span class="w"> </span><span class="p"><span class="pre">=</span></span><span class="w"> </span><span class="pre">&lt;module</span> <span class="pre">'scipy.sparse.linalg'</span> <span class="pre">from</span> <span class="pre">'/home/runner/micromamba/envs/pySDC/lib/python3.12/site-packages/scipy/sparse/linalg/__init__.py'&gt;</span></em><a class="headerlink" href="#helpers.spectral_helper.SpectralHelper1D.linalg" title="Link to this definition">¶</a></dt>
+<dd></dd></dl>
+
+<dl class="py method">
+<dt class="sig sig-object py" id="helpers.spectral_helper.SpectralHelper1D.setup_GPU">
+<em class="property"><span class="pre">classmethod</span><span class="w"> </span></em><span class="sig-name descname"><span class="pre">setup_GPU</span></span><span class="sig-paren">(</span><span class="sig-paren">)</span><a class="reference internal" href="../_modules/helpers/spectral_helper.html#SpectralHelper1D.setup_GPU"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#helpers.spectral_helper.SpectralHelper1D.setup_GPU" title="Link to this definition">¶</a></dt>
+<dd><p>switch to GPU modules</p>
+</dd></dl>
+
+<dl class="py attribute">
+<dt class="sig sig-object py" id="helpers.spectral_helper.SpectralHelper1D.sparse_lib">
+<span class="sig-name descname"><span class="pre">sparse_lib</span></span><em class="property"><span class="w"> </span><span class="p"><span class="pre">=</span></span><span class="w"> </span><span class="pre">&lt;module</span> <span class="pre">'scipy.sparse'</span> <span class="pre">from</span> <span class="pre">'/home/runner/micromamba/envs/pySDC/lib/python3.12/site-packages/scipy/sparse/__init__.py'&gt;</span></em><a class="headerlink" href="#helpers.spectral_helper.SpectralHelper1D.sparse_lib" title="Link to this definition">¶</a></dt>
+<dd></dd></dl>
+
+<dl class="py attribute">
+<dt class="sig sig-object py" id="helpers.spectral_helper.SpectralHelper1D.xp">
+<span class="sig-name descname"><span class="pre">xp</span></span><em class="property"><span class="w"> </span><span class="p"><span class="pre">=</span></span><span class="w"> </span><span class="pre">&lt;module</span> <span class="pre">'numpy'</span> <span class="pre">from</span> <span class="pre">'/home/runner/micromamba/envs/pySDC/lib/python3.12/site-packages/numpy/__init__.py'&gt;</span></em><a class="headerlink" href="#helpers.spectral_helper.SpectralHelper1D.xp" title="Link to this definition">¶</a></dt>
+<dd></dd></dl>
+
+</dd></dl>
+
+<dl class="py class">
+<dt class="sig sig-object py" id="helpers.spectral_helper.UltrasphericalHelper">
+<em class="property"><span class="pre">class</span><span class="w"> </span></em><span class="sig-name descname"><span class="pre">UltrasphericalHelper</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="o"><span class="pre">*</span></span><span class="n"><span class="pre">args</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">transform_type</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">'fft'</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">x0</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">-1</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">x1</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">1</span></span></em>, <em class="sig-param"><span class="o"><span class="pre">**</span></span><span class="n"><span class="pre">kwargs</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../_modules/helpers/spectral_helper.html#UltrasphericalHelper"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#helpers.spectral_helper.UltrasphericalHelper" title="Link to this definition">¶</a></dt>
+<dd><p>Bases: <a class="reference internal" href="#helpers.spectral_helper.ChebychevHelper" title="helpers.spectral_helper.ChebychevHelper"><code class="xref py py-class docutils literal notranslate"><span class="pre">ChebychevHelper</span></code></a></p>
+<p>This implementation follows <a class="reference external" href="https://doi.org/10.1137/120865458">https://doi.org/10.1137/120865458</a>.
+The ultraspherical method works in Chebychev polynomials as well, but also uses various Gegenbauer polynomials.
+The idea is that for every derivative of Chebychev T polynomials, there is a basis of Gegenbauer polynomials where the differentiation matrix is a single off-diagonal.
+There are also conversion operators from one derivative basis to the next that are sparse.</p>
+<p>This basis is used like this: For every equation that you have, look for the highest derivative and bump all matrices to the correct basis. If your highest derivative is 2 and you have an identity, it needs to get bumped from 0 to 1 and from 1 to 2. If you have a first derivative as well, it needs to be bumped from 1 to 2.
+You don’t need the same resulting basis in all equations. You just need to take care that you translate the right hand side to the correct basis as well.</p>
+<dl class="py method">
+<dt class="sig sig-object py" id="helpers.spectral_helper.UltrasphericalHelper.get_S">
+<span class="sig-name descname"><span class="pre">get_S</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">lmbda</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../_modules/helpers/spectral_helper.html#UltrasphericalHelper.get_S"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#helpers.spectral_helper.UltrasphericalHelper.get_S" title="Link to this definition">¶</a></dt>
+<dd><p>Get matrix for bumping the derivative base by one from lmbda to lmbda + 1. This is the same language as in
+<a class="reference external" href="https://doi.org/10.1137/120865458">https://doi.org/10.1137/120865458</a>.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><p><strong>lmbda</strong> (<em>int</em>) – Ingoing derivative base</p>
+</dd>
+<dt class="field-even">Returns<span class="colon">:</span></dt>
+<dd class="field-even"><p>Conversion from derivative base lmbda to lmbda + 1</p>
+</dd>
+<dt class="field-odd">Return type<span class="colon">:</span></dt>
+<dd class="field-odd"><p>sparse matrix</p>
+</dd>
+</dl>
+</dd></dl>
+
+<dl class="py method">
+<dt class="sig sig-object py" id="helpers.spectral_helper.UltrasphericalHelper.get_basis_change_matrix">
+<span class="sig-name descname"><span class="pre">get_basis_change_matrix</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">p_in</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">0</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">p_out</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">0</span></span></em>, <em class="sig-param"><span class="o"><span class="pre">**</span></span><span class="n"><span class="pre">kwargs</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../_modules/helpers/spectral_helper.html#UltrasphericalHelper.get_basis_change_matrix"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#helpers.spectral_helper.UltrasphericalHelper.get_basis_change_matrix" title="Link to this definition">¶</a></dt>
+<dd><p>Get a conversion matrix from derivative base <cite>p_in</cite> to <cite>p_out</cite>.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><ul class="simple">
+<li><p><strong>p_out</strong> (<em>int</em>) – Resulting derivative base</p></li>
+<li><p><strong>p_in</strong> (<em>int</em>) – Ingoing derivative base</p></li>
+</ul>
+</dd>
+</dl>
+</dd></dl>
+
+<dl class="py method">
+<dt class="sig sig-object py" id="helpers.spectral_helper.UltrasphericalHelper.get_differentiation_matrix">
+<span class="sig-name descname"><span class="pre">get_differentiation_matrix</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">p</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">1</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../_modules/helpers/spectral_helper.html#UltrasphericalHelper.get_differentiation_matrix"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#helpers.spectral_helper.UltrasphericalHelper.get_differentiation_matrix" title="Link to this definition">¶</a></dt>
+<dd><p>Notice that while sparse, this matrix is not diagonal, which means the inversion cannot be parallelized easily.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><p><strong>p</strong> (<em>int</em>) – Order of the derivative</p>
+</dd>
+<dt class="field-even">Returns<span class="colon">:</span></dt>
+<dd class="field-even"><p>sparse differentiation matrix</p>
+</dd>
+</dl>
+</dd></dl>
+
+<dl class="py method">
+<dt class="sig sig-object py" id="helpers.spectral_helper.UltrasphericalHelper.get_integration_constant">
+<span class="sig-name descname"><span class="pre">get_integration_constant</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">u_hat</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">axis</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../_modules/helpers/spectral_helper.html#UltrasphericalHelper.get_integration_constant"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#helpers.spectral_helper.UltrasphericalHelper.get_integration_constant" title="Link to this definition">¶</a></dt>
+<dd><p>Get integration constant for lower bound of -1. See documentation of <cite>UltrasphericalHelper.get_integration_matrix</cite> for details.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><ul class="simple">
+<li><p><strong>u_hat</strong> – Solution in spectral space</p></li>
+<li><p><strong>axis</strong> – Axis you want to integrate over</p></li>
+</ul>
+</dd>
+<dt class="field-even">Returns<span class="colon">:</span></dt>
+<dd class="field-even"><p>Integration constant, has one less dimension than <cite>u_hat</cite></p>
+</dd>
+</dl>
+</dd></dl>
+
+<dl class="py method">
+<dt class="sig sig-object py" id="helpers.spectral_helper.UltrasphericalHelper.get_integration_matrix">
+<span class="sig-name descname"><span class="pre">get_integration_matrix</span></span><span class="sig-paren">(</span><span class="sig-paren">)</span><a class="reference internal" href="../_modules/helpers/spectral_helper.html#UltrasphericalHelper.get_integration_matrix"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#helpers.spectral_helper.UltrasphericalHelper.get_integration_matrix" title="Link to this definition">¶</a></dt>
+<dd><p>Get an integration matrix. Please use <cite>UltrasphericalHelper.get_integration_constant</cite> afterwards to compute the
+integration constant such that integration starts from x=-1.</p>
+<p>Example:</p>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
+<span class="kn">from</span> <span class="nn">pySDC.helpers.spectral_helper</span> <span class="kn">import</span> <span class="n">UltrasphericalHelper</span>
+
+<span class="n">N</span> <span class="o">=</span> <span class="mi">4</span>
+<span class="n">helper</span> <span class="o">=</span> <span class="n">UltrasphericalHelper</span><span class="p">(</span><span class="n">N</span><span class="p">)</span>
+<span class="n">coeffs</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">random</span><span class="p">(</span><span class="n">N</span><span class="p">)</span>
+<span class="n">coeffs</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="mi">0</span>
+
+<span class="n">poly</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">polynomial</span><span class="o">.</span><span class="n">Chebyshev</span><span class="p">(</span><span class="n">coeffs</span><span class="p">)</span>
+
+<span class="n">S</span> <span class="o">=</span> <span class="n">helper</span><span class="o">.</span><span class="n">get_integration_matrix</span><span class="p">()</span>
+<span class="n">U_hat</span> <span class="o">=</span> <span class="n">S</span> <span class="o">@</span> <span class="n">coeffs</span>
+<span class="n">U_hat</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="n">helper</span><span class="o">.</span><span class="n">get_integration_constant</span><span class="p">(</span><span class="n">U_hat</span><span class="p">,</span> <span class="n">axis</span><span class="o">=-</span><span class="mi">1</span><span class="p">)</span>
+
+<span class="k">assert</span> <span class="n">np</span><span class="o">.</span><span class="n">allclose</span><span class="p">(</span><span class="n">poly</span><span class="o">.</span><span class="n">integ</span><span class="p">(</span><span class="n">lbnd</span><span class="o">=-</span><span class="mi">1</span><span class="p">)</span><span class="o">.</span><span class="n">coef</span><span class="p">[:</span><span class="o">-</span><span class="mi">1</span><span class="p">],</span> <span class="n">U_hat</span><span class="p">)</span>
+</pre></div>
+</div>
+<dl class="field-list simple">
+<dt class="field-odd">Returns<span class="colon">:</span></dt>
+<dd class="field-odd"><p>sparse integration matrix</p>
+</dd>
+</dl>
+</dd></dl>
+
+</dd></dl>
+
+</section>
+
+
+            <div class="clearer"></div>
+          </div>
+        </div>
+      </div>
+      <div class="sphinxsidebar" role="navigation" aria-label="Main">
+        <div class="sphinxsidebarwrapper">
+  <div>
+    <h3><a href="../index.html">Table of Contents</a></h3>
+    <ul>
+<li><a class="reference internal" href="#">helpers.spectral_helper module</a><ul>
+<li><a class="reference internal" href="#helpers.spectral_helper.ChebychevHelper"><code class="docutils literal notranslate"><span class="pre">ChebychevHelper</span></code></a><ul>
+<li><a class="reference internal" href="#helpers.spectral_helper.ChebychevHelper.get_1dgrid"><code class="docutils literal notranslate"><span class="pre">get_1dgrid()</span></code></a></li>
+<li><a class="reference internal" href="#helpers.spectral_helper.ChebychevHelper.get_BC"><code class="docutils literal notranslate"><span class="pre">get_BC()</span></code></a></li>
+<li><a class="reference internal" href="#helpers.spectral_helper.ChebychevHelper.get_Dirichlet_BC_row"><code class="docutils literal notranslate"><span class="pre">get_Dirichlet_BC_row()</span></code></a></li>
+<li><a class="reference internal" href="#helpers.spectral_helper.ChebychevHelper.get_Dirichlet_recombination_matrix"><code class="docutils literal notranslate"><span class="pre">get_Dirichlet_recombination_matrix()</span></code></a></li>
+<li><a class="reference internal" href="#helpers.spectral_helper.ChebychevHelper.get_basis_change_matrix"><code class="docutils literal notranslate"><span class="pre">get_basis_change_matrix()</span></code></a></li>
+<li><a class="reference internal" href="#helpers.spectral_helper.ChebychevHelper.get_conv"><code class="docutils literal notranslate"><span class="pre">get_conv()</span></code></a></li>
+<li><a class="reference internal" href="#helpers.spectral_helper.ChebychevHelper.get_differentiation_matrix"><code class="docutils literal notranslate"><span class="pre">get_differentiation_matrix()</span></code></a></li>
+<li><a class="reference internal" href="#helpers.spectral_helper.ChebychevHelper.get_fft_shift"><code class="docutils literal notranslate"><span class="pre">get_fft_shift()</span></code></a></li>
+<li><a class="reference internal" href="#helpers.spectral_helper.ChebychevHelper.get_fft_shuffle"><code class="docutils literal notranslate"><span class="pre">get_fft_shuffle()</span></code></a></li>
+<li><a class="reference internal" href="#helpers.spectral_helper.ChebychevHelper.get_fft_utils"><code class="docutils literal notranslate"><span class="pre">get_fft_utils()</span></code></a></li>
+<li><a class="reference internal" href="#helpers.spectral_helper.ChebychevHelper.get_integ_BC_row"><code class="docutils literal notranslate"><span class="pre">get_integ_BC_row()</span></code></a></li>
+<li><a class="reference internal" href="#helpers.spectral_helper.ChebychevHelper.get_integration_matrix"><code class="docutils literal notranslate"><span class="pre">get_integration_matrix()</span></code></a></li>
+<li><a class="reference internal" href="#helpers.spectral_helper.ChebychevHelper.get_norm"><code class="docutils literal notranslate"><span class="pre">get_norm()</span></code></a></li>
+<li><a class="reference internal" href="#helpers.spectral_helper.ChebychevHelper.get_wavenumbers"><code class="docutils literal notranslate"><span class="pre">get_wavenumbers()</span></code></a></li>
+<li><a class="reference internal" href="#helpers.spectral_helper.ChebychevHelper.itransform"><code class="docutils literal notranslate"><span class="pre">itransform()</span></code></a></li>
+<li><a class="reference internal" href="#helpers.spectral_helper.ChebychevHelper.transform"><code class="docutils literal notranslate"><span class="pre">transform()</span></code></a></li>
+</ul>
+</li>
+<li><a class="reference internal" href="#helpers.spectral_helper.FFTHelper"><code class="docutils literal notranslate"><span class="pre">FFTHelper</span></code></a><ul>
+<li><a class="reference internal" href="#helpers.spectral_helper.FFTHelper.get_1dgrid"><code class="docutils literal notranslate"><span class="pre">get_1dgrid()</span></code></a></li>
+<li><a class="reference internal" href="#helpers.spectral_helper.FFTHelper.get_BC"><code class="docutils literal notranslate"><span class="pre">get_BC()</span></code></a></li>
+<li><a class="reference internal" href="#helpers.spectral_helper.FFTHelper.get_Nyquist_mode_index"><code class="docutils literal notranslate"><span class="pre">get_Nyquist_mode_index()</span></code></a></li>
+<li><a class="reference internal" href="#helpers.spectral_helper.FFTHelper.get_differentiation_matrix"><code class="docutils literal notranslate"><span class="pre">get_differentiation_matrix()</span></code></a></li>
+<li><a class="reference internal" href="#helpers.spectral_helper.FFTHelper.get_integ_BC_row"><code class="docutils literal notranslate"><span class="pre">get_integ_BC_row()</span></code></a></li>
+<li><a class="reference internal" href="#helpers.spectral_helper.FFTHelper.get_integration_matrix"><code class="docutils literal notranslate"><span class="pre">get_integration_matrix()</span></code></a></li>
+<li><a class="reference internal" href="#helpers.spectral_helper.FFTHelper.get_wavenumbers"><code class="docutils literal notranslate"><span class="pre">get_wavenumbers()</span></code></a></li>
+<li><a class="reference internal" href="#helpers.spectral_helper.FFTHelper.itransform"><code class="docutils literal notranslate"><span class="pre">itransform()</span></code></a></li>
+<li><a class="reference internal" href="#helpers.spectral_helper.FFTHelper.transform"><code class="docutils literal notranslate"><span class="pre">transform()</span></code></a></li>
+</ul>
+</li>
+<li><a class="reference internal" href="#helpers.spectral_helper.SpectralHelper"><code class="docutils literal notranslate"><span class="pre">SpectralHelper</span></code></a><ul>
+<li><a class="reference internal" href="#helpers.spectral_helper.SpectralHelper.comm"><code class="docutils literal notranslate"><span class="pre">comm</span></code></a></li>
+<li><a class="reference internal" href="#helpers.spectral_helper.SpectralHelper.debug"><code class="docutils literal notranslate"><span class="pre">debug</span></code></a></li>
+<li><a class="reference internal" href="#helpers.spectral_helper.SpectralHelper.useGPU"><code class="docutils literal notranslate"><span class="pre">useGPU</span></code></a></li>
+<li><a class="reference internal" href="#helpers.spectral_helper.SpectralHelper.axes"><code class="docutils literal notranslate"><span class="pre">axes</span></code></a></li>
+<li><a class="reference internal" href="#helpers.spectral_helper.SpectralHelper.components"><code class="docutils literal notranslate"><span class="pre">components</span></code></a></li>
+<li><a class="reference internal" href="#helpers.spectral_helper.SpectralHelper.full_BCs"><code class="docutils literal notranslate"><span class="pre">full_BCs</span></code></a></li>
+<li><a class="reference internal" href="#helpers.spectral_helper.SpectralHelper.BC_mat"><code class="docutils literal notranslate"><span class="pre">BC_mat</span></code></a></li>
+<li><a class="reference internal" href="#helpers.spectral_helper.SpectralHelper.BCs"><code class="docutils literal notranslate"><span class="pre">BCs</span></code></a></li>
+<li><a class="reference internal" href="#helpers.spectral_helper.SpectralHelper.fft_cache"><code class="docutils literal notranslate"><span class="pre">fft_cache</span></code></a></li>
+<li><a class="reference internal" href="#helpers.spectral_helper.SpectralHelper.BC_rhs_mask"><code class="docutils literal notranslate"><span class="pre">BC_rhs_mask</span></code></a></li>
+<li><a class="reference internal" href="#helpers.spectral_helper.SpectralHelper.BC_zero_index"><code class="docutils literal notranslate"><span class="pre">BC_zero_index</span></code></a></li>
+<li><a class="reference internal" href="#helpers.spectral_helper.SpectralHelper.BC_line_zero_matrix"><code class="docutils literal notranslate"><span class="pre">BC_line_zero_matrix</span></code></a></li>
+<li><a class="reference internal" href="#helpers.spectral_helper.SpectralHelper.rhs_BCs_hat"><code class="docutils literal notranslate"><span class="pre">rhs_BCs_hat</span></code></a></li>
+<li><a class="reference internal" href="#helpers.spectral_helper.SpectralHelper.global_shape"><code class="docutils literal notranslate"><span class="pre">global_shape</span></code></a></li>
+<li><a class="reference internal" href="#helpers.spectral_helper.SpectralHelper.local_slice"><code class="docutils literal notranslate"><span class="pre">local_slice</span></code></a></li>
+<li><a class="reference internal" href="#helpers.spectral_helper.SpectralHelper.fft_obj"><code class="docutils literal notranslate"><span class="pre">fft_obj</span></code></a></li>
+<li><a class="reference internal" href="#helpers.spectral_helper.SpectralHelper.init"><code class="docutils literal notranslate"><span class="pre">init</span></code></a></li>
+<li><a class="reference internal" href="#helpers.spectral_helper.SpectralHelper.init_forward"><code class="docutils literal notranslate"><span class="pre">init_forward</span></code></a></li>
+<li><a class="reference internal" href="#helpers.spectral_helper.SpectralHelper.V"><code class="docutils literal notranslate"><span class="pre">V</span></code></a></li>
+<li><a class="reference internal" href="#helpers.spectral_helper.SpectralHelper.add_BC"><code class="docutils literal notranslate"><span class="pre">add_BC()</span></code></a></li>
+<li><a class="reference internal" href="#helpers.spectral_helper.SpectralHelper.add_axis"><code class="docutils literal notranslate"><span class="pre">add_axis()</span></code></a></li>
+<li><a class="reference internal" href="#helpers.spectral_helper.SpectralHelper.add_component"><code class="docutils literal notranslate"><span class="pre">add_component()</span></code></a></li>
+<li><a class="reference internal" href="#helpers.spectral_helper.SpectralHelper.add_equation_lhs"><code class="docutils literal notranslate"><span class="pre">add_equation_lhs()</span></code></a></li>
+<li><a class="reference internal" href="#helpers.spectral_helper.SpectralHelper.check_BCs"><code class="docutils literal notranslate"><span class="pre">check_BCs()</span></code></a></li>
+<li><a class="reference internal" href="#helpers.spectral_helper.SpectralHelper.convert_operator_matrix_to_operator"><code class="docutils literal notranslate"><span class="pre">convert_operator_matrix_to_operator()</span></code></a></li>
+<li><a class="reference internal" href="#helpers.spectral_helper.SpectralHelper.dtype"><code class="docutils literal notranslate"><span class="pre">dtype</span></code></a></li>
+<li><a class="reference internal" href="#helpers.spectral_helper.SpectralHelper.fft_backend"><code class="docutils literal notranslate"><span class="pre">fft_backend</span></code></a></li>
+<li><a class="reference internal" href="#helpers.spectral_helper.SpectralHelper.fft_comm_backend"><code class="docutils literal notranslate"><span class="pre">fft_comm_backend</span></code></a></li>
+<li><a class="reference internal" href="#helpers.spectral_helper.SpectralHelper.fft_lib"><code class="docutils literal notranslate"><span class="pre">fft_lib</span></code></a></li>
+<li><a class="reference internal" href="#helpers.spectral_helper.SpectralHelper.get_BC"><code class="docutils literal notranslate"><span class="pre">get_BC()</span></code></a></li>
+<li><a class="reference internal" href="#helpers.spectral_helper.SpectralHelper.get_Dirichlet_recombination_matrix"><code class="docutils literal notranslate"><span class="pre">get_Dirichlet_recombination_matrix()</span></code></a></li>
+<li><a class="reference internal" href="#helpers.spectral_helper.SpectralHelper.get_Id"><code class="docutils literal notranslate"><span class="pre">get_Id()</span></code></a></li>
+<li><a class="reference internal" href="#helpers.spectral_helper.SpectralHelper.get_aligned"><code class="docutils literal notranslate"><span class="pre">get_aligned()</span></code></a></li>
+<li><a class="reference internal" href="#helpers.spectral_helper.SpectralHelper.get_basis_change_matrix"><code class="docutils literal notranslate"><span class="pre">get_basis_change_matrix()</span></code></a></li>
+<li><a class="reference internal" href="#helpers.spectral_helper.SpectralHelper.get_differentiation_matrix"><code class="docutils literal notranslate"><span class="pre">get_differentiation_matrix()</span></code></a></li>
+<li><a class="reference internal" href="#helpers.spectral_helper.SpectralHelper.get_empty_operator_matrix"><code class="docutils literal notranslate"><span class="pre">get_empty_operator_matrix()</span></code></a></li>
+<li><a class="reference internal" href="#helpers.spectral_helper.SpectralHelper.get_fft"><code class="docutils literal notranslate"><span class="pre">get_fft()</span></code></a></li>
+<li><a class="reference internal" href="#helpers.spectral_helper.SpectralHelper.get_filter_matrix"><code class="docutils literal notranslate"><span class="pre">get_filter_matrix()</span></code></a></li>
+<li><a class="reference internal" href="#helpers.spectral_helper.SpectralHelper.get_grid"><code class="docutils literal notranslate"><span class="pre">get_grid()</span></code></a></li>
+<li><a class="reference internal" href="#helpers.spectral_helper.SpectralHelper.get_integration_matrix"><code class="docutils literal notranslate"><span class="pre">get_integration_matrix()</span></code></a></li>
+<li><a class="reference internal" href="#helpers.spectral_helper.SpectralHelper.get_local_slice_of_1D_matrix"><code class="docutils literal notranslate"><span class="pre">get_local_slice_of_1D_matrix()</span></code></a></li>
+<li><a class="reference internal" href="#helpers.spectral_helper.SpectralHelper.get_wavenumbers"><code class="docutils literal notranslate"><span class="pre">get_wavenumbers()</span></code></a></li>
+<li><a class="reference internal" href="#helpers.spectral_helper.SpectralHelper.index"><code class="docutils literal notranslate"><span class="pre">index()</span></code></a></li>
+<li><a class="reference internal" href="#helpers.spectral_helper.SpectralHelper.itransform"><code class="docutils literal notranslate"><span class="pre">itransform()</span></code></a></li>
+<li><a class="reference internal" href="#helpers.spectral_helper.SpectralHelper.itransform_single_component"><code class="docutils literal notranslate"><span class="pre">itransform_single_component()</span></code></a></li>
+<li><a class="reference internal" href="#helpers.spectral_helper.SpectralHelper.linalg"><code class="docutils literal notranslate"><span class="pre">linalg</span></code></a></li>
+<li><a class="reference internal" href="#helpers.spectral_helper.SpectralHelper.ncomponents"><code class="docutils literal notranslate"><span class="pre">ncomponents</span></code></a></li>
+<li><a class="reference internal" href="#helpers.spectral_helper.SpectralHelper.ndim"><code class="docutils literal notranslate"><span class="pre">ndim</span></code></a></li>
+<li><a class="reference internal" href="#helpers.spectral_helper.SpectralHelper.put_BCs_in_matrix"><code class="docutils literal notranslate"><span class="pre">put_BCs_in_matrix()</span></code></a></li>
+<li><a class="reference internal" href="#helpers.spectral_helper.SpectralHelper.put_BCs_in_rhs"><code class="docutils literal notranslate"><span class="pre">put_BCs_in_rhs()</span></code></a></li>
+<li><a class="reference internal" href="#helpers.spectral_helper.SpectralHelper.put_BCs_in_rhs_hat"><code class="docutils literal notranslate"><span class="pre">put_BCs_in_rhs_hat()</span></code></a></li>
+<li><a class="reference internal" href="#helpers.spectral_helper.SpectralHelper.remove_BC"><code class="docutils literal notranslate"><span class="pre">remove_BC()</span></code></a></li>
+<li><a class="reference internal" href="#helpers.spectral_helper.SpectralHelper.setup_BCs"><code class="docutils literal notranslate"><span class="pre">setup_BCs()</span></code></a></li>
+<li><a class="reference internal" href="#helpers.spectral_helper.SpectralHelper.setup_GPU"><code class="docutils literal notranslate"><span class="pre">setup_GPU()</span></code></a></li>
+<li><a class="reference internal" href="#helpers.spectral_helper.SpectralHelper.setup_fft"><code class="docutils literal notranslate"><span class="pre">setup_fft()</span></code></a></li>
+<li><a class="reference internal" href="#helpers.spectral_helper.SpectralHelper.shape"><code class="docutils literal notranslate"><span class="pre">shape</span></code></a></li>
+<li><a class="reference internal" href="#helpers.spectral_helper.SpectralHelper.sparse_lib"><code class="docutils literal notranslate"><span class="pre">sparse_lib</span></code></a></li>
+<li><a class="reference internal" href="#helpers.spectral_helper.SpectralHelper.transform"><code class="docutils literal notranslate"><span class="pre">transform()</span></code></a></li>
+<li><a class="reference internal" href="#helpers.spectral_helper.SpectralHelper.transform_single_component"><code class="docutils literal notranslate"><span class="pre">transform_single_component()</span></code></a></li>
+<li><a class="reference internal" href="#helpers.spectral_helper.SpectralHelper.u_init"><code class="docutils literal notranslate"><span class="pre">u_init</span></code></a></li>
+<li><a class="reference internal" href="#helpers.spectral_helper.SpectralHelper.u_init_forward"><code class="docutils literal notranslate"><span class="pre">u_init_forward</span></code></a></li>
+<li><a class="reference internal" href="#helpers.spectral_helper.SpectralHelper.xp"><code class="docutils literal notranslate"><span class="pre">xp</span></code></a></li>
+</ul>
+</li>
+<li><a class="reference internal" href="#helpers.spectral_helper.SpectralHelper1D"><code class="docutils literal notranslate"><span class="pre">SpectralHelper1D</span></code></a><ul>
+<li><a class="reference internal" href="#helpers.spectral_helper.SpectralHelper1D.N"><code class="docutils literal notranslate"><span class="pre">N</span></code></a></li>
+<li><a class="reference internal" href="#helpers.spectral_helper.SpectralHelper1D.x0"><code class="docutils literal notranslate"><span class="pre">x0</span></code></a></li>
+<li><a class="reference internal" href="#helpers.spectral_helper.SpectralHelper1D.x1"><code class="docutils literal notranslate"><span class="pre">x1</span></code></a></li>
+<li><a class="reference internal" href="#helpers.spectral_helper.SpectralHelper1D.L"><code class="docutils literal notranslate"><span class="pre">L</span></code></a></li>
+<li><a class="reference internal" href="#helpers.spectral_helper.SpectralHelper1D.useGPU"><code class="docutils literal notranslate"><span class="pre">useGPU</span></code></a></li>
+<li><a class="reference internal" href="#helpers.spectral_helper.SpectralHelper1D.fft_lib"><code class="docutils literal notranslate"><span class="pre">fft_lib</span></code></a></li>
+<li><a class="reference internal" href="#helpers.spectral_helper.SpectralHelper1D.get_1dgrid"><code class="docutils literal notranslate"><span class="pre">get_1dgrid()</span></code></a></li>
+<li><a class="reference internal" href="#helpers.spectral_helper.SpectralHelper1D.get_BC"><code class="docutils literal notranslate"><span class="pre">get_BC()</span></code></a></li>
+<li><a class="reference internal" href="#helpers.spectral_helper.SpectralHelper1D.get_Id"><code class="docutils literal notranslate"><span class="pre">get_Id()</span></code></a></li>
+<li><a class="reference internal" href="#helpers.spectral_helper.SpectralHelper1D.get_basis_change_matrix"><code class="docutils literal notranslate"><span class="pre">get_basis_change_matrix()</span></code></a></li>
+<li><a class="reference internal" href="#helpers.spectral_helper.SpectralHelper1D.get_differentiation_matrix"><code class="docutils literal notranslate"><span class="pre">get_differentiation_matrix()</span></code></a></li>
+<li><a class="reference internal" href="#helpers.spectral_helper.SpectralHelper1D.get_empty_operator_matrix"><code class="docutils literal notranslate"><span class="pre">get_empty_operator_matrix()</span></code></a></li>
+<li><a class="reference internal" href="#helpers.spectral_helper.SpectralHelper1D.get_filter_matrix"><code class="docutils literal notranslate"><span class="pre">get_filter_matrix()</span></code></a></li>
+<li><a class="reference internal" href="#helpers.spectral_helper.SpectralHelper1D.get_integration_matrix"><code class="docutils literal notranslate"><span class="pre">get_integration_matrix()</span></code></a></li>
+<li><a class="reference internal" href="#helpers.spectral_helper.SpectralHelper1D.get_wavenumbers"><code class="docutils literal notranslate"><span class="pre">get_wavenumbers()</span></code></a></li>
+<li><a class="reference internal" href="#helpers.spectral_helper.SpectralHelper1D.get_zero"><code class="docutils literal notranslate"><span class="pre">get_zero()</span></code></a></li>
+<li><a class="reference internal" href="#helpers.spectral_helper.SpectralHelper1D.linalg"><code class="docutils literal notranslate"><span class="pre">linalg</span></code></a></li>
+<li><a class="reference internal" href="#helpers.spectral_helper.SpectralHelper1D.setup_GPU"><code class="docutils literal notranslate"><span class="pre">setup_GPU()</span></code></a></li>
+<li><a class="reference internal" href="#helpers.spectral_helper.SpectralHelper1D.sparse_lib"><code class="docutils literal notranslate"><span class="pre">sparse_lib</span></code></a></li>
+<li><a class="reference internal" href="#helpers.spectral_helper.SpectralHelper1D.xp"><code class="docutils literal notranslate"><span class="pre">xp</span></code></a></li>
+</ul>
+</li>
+<li><a class="reference internal" href="#helpers.spectral_helper.UltrasphericalHelper"><code class="docutils literal notranslate"><span class="pre">UltrasphericalHelper</span></code></a><ul>
+<li><a class="reference internal" href="#helpers.spectral_helper.UltrasphericalHelper.get_S"><code class="docutils literal notranslate"><span class="pre">get_S()</span></code></a></li>
+<li><a class="reference internal" href="#helpers.spectral_helper.UltrasphericalHelper.get_basis_change_matrix"><code class="docutils literal notranslate"><span class="pre">get_basis_change_matrix()</span></code></a></li>
+<li><a class="reference internal" href="#helpers.spectral_helper.UltrasphericalHelper.get_differentiation_matrix"><code class="docutils literal notranslate"><span class="pre">get_differentiation_matrix()</span></code></a></li>
+<li><a class="reference internal" href="#helpers.spectral_helper.UltrasphericalHelper.get_integration_constant"><code class="docutils literal notranslate"><span class="pre">get_integration_constant()</span></code></a></li>
+<li><a class="reference internal" href="#helpers.spectral_helper.UltrasphericalHelper.get_integration_matrix"><code class="docutils literal notranslate"><span class="pre">get_integration_matrix()</span></code></a></li>
+</ul>
+</li>
+</ul>
+</li>
+</ul>
+
+  </div>
+  <div>
+    <h4>Previous topic</h4>
+    <p class="topless"><a href="helpers.setup_helper.html"
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+    <p class="topless"><a href="helpers.spectral_helper.html"
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+  <section id="module-implementations.problem_classes.Burgers">
+<span id="implementations-problem-classes-burgers-module"></span><h1>implementations.problem_classes.Burgers module<a class="headerlink" href="#module-implementations.problem_classes.Burgers" title="Link to this heading">¶</a></h1>
+<dl class="py class">
+<dt class="sig sig-object py" id="implementations.problem_classes.Burgers.Burgers1D">
+<em class="property"><span class="pre">class</span><span class="w"> </span></em><span class="sig-name descname"><span class="pre">Burgers1D</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">N</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">64</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">epsilon</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">0.1</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">BCl</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">1</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">BCr</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">-1</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">f</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">0</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">mode</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">'T2U'</span></span></em>, <em class="sig-param"><span class="o"><span class="pre">**</span></span><span class="n"><span class="pre">kwargs</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../_modules/implementations/problem_classes/Burgers.html#Burgers1D"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#implementations.problem_classes.Burgers.Burgers1D" title="Link to this definition">¶</a></dt>
+<dd><p>Bases: <code class="xref py py-class docutils literal notranslate"><span class="pre">GenericSpectralLinear</span></code></p>
+<p>See <a class="reference external" href="https://en.wikipedia.org/wiki/Burgers'_equation">https://en.wikipedia.org/wiki/Burgers’_equation</a> for the equation that is solved.
+Discretization is done with a Chebychev method, which requires a first order derivative formulation.
+Feel free to do a more efficient implementation using an ultraspherical method to avoid the first order business.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><ul class="simple">
+<li><p><strong>N</strong> (<em>int</em>) – Spatial resolution</p></li>
+<li><p><strong>epsilon</strong> (<em>float</em>) – viscosity</p></li>
+<li><p><strong>BCl</strong> (<em>float</em>) – Value at left boundary</p></li>
+<li><p><strong>BCr</strong> (<em>float</em>) – Value at right boundary</p></li>
+<li><p><strong>f</strong> (<em>int</em>) – Frequency of the initial conditions</p></li>
+<li><p><strong>mode</strong> (<em>str</em>) – ‘T2U’ or ‘T2T’. Use ‘T2U’ to get sparse differentiation matrices</p></li>
+</ul>
+</dd>
+</dl>
+<dl class="py attribute">
+<dt class="sig sig-object py" id="implementations.problem_classes.Burgers.Burgers1D.dtype_f">
+<span class="sig-name descname"><span class="pre">dtype_f</span></span><a class="headerlink" href="#implementations.problem_classes.Burgers.Burgers1D.dtype_f" title="Link to this definition">¶</a></dt>
+<dd><p>alias of <code class="xref py py-class docutils literal notranslate"><span class="pre">imex_mesh</span></code></p>
+</dd></dl>
+
+<dl class="py attribute">
+<dt class="sig sig-object py" id="implementations.problem_classes.Burgers.Burgers1D.dtype_u">
+<span class="sig-name descname"><span class="pre">dtype_u</span></span><a class="headerlink" href="#implementations.problem_classes.Burgers.Burgers1D.dtype_u" title="Link to this definition">¶</a></dt>
+<dd><p>alias of <code class="xref py py-class docutils literal notranslate"><span class="pre">mesh</span></code></p>
+</dd></dl>
+
+<dl class="py method">
+<dt class="sig sig-object py" id="implementations.problem_classes.Burgers.Burgers1D.eval_f">
+<span class="sig-name descname"><span class="pre">eval_f</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">u</span></span></em>, <em class="sig-param"><span class="o"><span class="pre">*</span></span><span class="n"><span class="pre">args</span></span></em>, <em class="sig-param"><span class="o"><span class="pre">**</span></span><span class="n"><span class="pre">kwargs</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../_modules/implementations/problem_classes/Burgers.html#Burgers1D.eval_f"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#implementations.problem_classes.Burgers.Burgers1D.eval_f" title="Link to this definition">¶</a></dt>
+<dd><p>Abstract interface to RHS computation of the ODE</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><ul class="simple">
+<li><p><strong>u</strong> (<em>dtype_u</em>) – Current values.</p></li>
+<li><p><strong>t</strong> (<em>float</em>) – Current time.</p></li>
+</ul>
+</dd>
+<dt class="field-even">Returns<span class="colon">:</span></dt>
+<dd class="field-even"><p><strong>f</strong> – The RHS values.</p>
+</dd>
+<dt class="field-odd">Return type<span class="colon">:</span></dt>
+<dd class="field-odd"><p>dtype_f</p>
+</dd>
+</dl>
+</dd></dl>
+
+<dl class="py method">
+<dt class="sig sig-object py" id="implementations.problem_classes.Burgers.Burgers1D.get_fig">
+<span class="sig-name descname"><span class="pre">get_fig</span></span><span class="sig-paren">(</span><span class="sig-paren">)</span><a class="reference internal" href="../_modules/implementations/problem_classes/Burgers.html#Burgers1D.get_fig"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#implementations.problem_classes.Burgers.Burgers1D.get_fig" title="Link to this definition">¶</a></dt>
+<dd><p>Get a figure suitable to plot the solution of this problem.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Returns<span class="colon">:</span></dt>
+<dd class="field-odd"><p><strong>self.fig</strong></p>
+</dd>
+<dt class="field-even">Return type<span class="colon">:</span></dt>
+<dd class="field-even"><p>matplotlib.pyplot.figure.Figure</p>
+</dd>
+</dl>
+</dd></dl>
+
+<dl class="py method">
+<dt class="sig sig-object py" id="implementations.problem_classes.Burgers.Burgers1D.plot">
+<span class="sig-name descname"><span class="pre">plot</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">u</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">t</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">fig</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">comp</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">'u'</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../_modules/implementations/problem_classes/Burgers.html#Burgers1D.plot"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#implementations.problem_classes.Burgers.Burgers1D.plot" title="Link to this definition">¶</a></dt>
+<dd><p>Plot the solution.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><ul class="simple">
+<li><p><strong>u</strong> (<em>dtype_u</em>) – Solution to be plotted</p></li>
+<li><p><strong>t</strong> (<em>float</em>) – Time to display at the top of the figure</p></li>
+<li><p><strong>fig</strong> (<em>matplotlib.pyplot.figure.Figure</em><em>, </em><em>optional</em>) – Figure with the same structure as a figure generated by <cite>self.get_fig</cite>. If none is supplied, a new figure will be generated.</p></li>
+</ul>
+</dd>
+<dt class="field-even">Return type<span class="colon">:</span></dt>
+<dd class="field-even"><p>None</p>
+</dd>
+</dl>
+</dd></dl>
+
+<dl class="py method">
+<dt class="sig sig-object py" id="implementations.problem_classes.Burgers.Burgers1D.u_exact">
+<span class="sig-name descname"><span class="pre">u_exact</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">t</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">0</span></span></em>, <em class="sig-param"><span class="o"><span class="pre">*</span></span><span class="n"><span class="pre">args</span></span></em>, <em class="sig-param"><span class="o"><span class="pre">**</span></span><span class="n"><span class="pre">kwargs</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../_modules/implementations/problem_classes/Burgers.html#Burgers1D.u_exact"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#implementations.problem_classes.Burgers.Burgers1D.u_exact" title="Link to this definition">¶</a></dt>
+<dd></dd></dl>
+
+</dd></dl>
+
+<dl class="py class">
+<dt class="sig sig-object py" id="implementations.problem_classes.Burgers.Burgers2D">
+<em class="property"><span class="pre">class</span><span class="w"> </span></em><span class="sig-name descname"><span class="pre">Burgers2D</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">nx</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">64</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">nz</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">64</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">epsilon</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">0.1</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">fux</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">2</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">fuz</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">1</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">mode</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">'T2U'</span></span></em>, <em class="sig-param"><span class="o"><span class="pre">**</span></span><span class="n"><span class="pre">kwargs</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../_modules/implementations/problem_classes/Burgers.html#Burgers2D"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#implementations.problem_classes.Burgers.Burgers2D" title="Link to this definition">¶</a></dt>
+<dd><p>Bases: <code class="xref py py-class docutils literal notranslate"><span class="pre">GenericSpectralLinear</span></code></p>
+<p>See <a class="reference external" href="https://en.wikipedia.org/wiki/Burgers'_equation">https://en.wikipedia.org/wiki/Burgers’_equation</a> for the equation that is solved.
+This implementation is discretized with FFTs in x and Chebychev in z.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><ul class="simple">
+<li><p><strong>nx</strong> (<em>int</em>) – Spatial resolution in x direction</p></li>
+<li><p><strong>nz</strong> (<em>int</em>) – Spatial resolution in z direction</p></li>
+<li><p><strong>epsilon</strong> (<em>float</em>) – viscosity</p></li>
+<li><p><strong>BCl</strong> (<em>float</em>) – Value at left boundary</p></li>
+<li><p><strong>BCr</strong> (<em>float</em>) – Value at right boundary</p></li>
+<li><p><strong>fux</strong> (<em>int</em>) – Frequency of the initial conditions in x-direction</p></li>
+<li><p><strong>fuz</strong> (<em>int</em>) – Frequency of the initial conditions in z-direction</p></li>
+<li><p><strong>mode</strong> (<em>str</em>) – ‘T2U’ or ‘T2T’. Use ‘T2U’ to get sparse differentiation matrices</p></li>
+</ul>
+</dd>
+</dl>
+<dl class="py method">
+<dt class="sig sig-object py" id="implementations.problem_classes.Burgers.Burgers2D.compute_vorticity">
+<span class="sig-name descname"><span class="pre">compute_vorticity</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">u</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../_modules/implementations/problem_classes/Burgers.html#Burgers2D.compute_vorticity"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#implementations.problem_classes.Burgers.Burgers2D.compute_vorticity" title="Link to this definition">¶</a></dt>
+<dd></dd></dl>
+
+<dl class="py attribute">
+<dt class="sig sig-object py" id="implementations.problem_classes.Burgers.Burgers2D.dtype_f">
+<span class="sig-name descname"><span class="pre">dtype_f</span></span><a class="headerlink" href="#implementations.problem_classes.Burgers.Burgers2D.dtype_f" title="Link to this definition">¶</a></dt>
+<dd><p>alias of <code class="xref py py-class docutils literal notranslate"><span class="pre">imex_mesh</span></code></p>
+</dd></dl>
+
+<dl class="py attribute">
+<dt class="sig sig-object py" id="implementations.problem_classes.Burgers.Burgers2D.dtype_u">
+<span class="sig-name descname"><span class="pre">dtype_u</span></span><a class="headerlink" href="#implementations.problem_classes.Burgers.Burgers2D.dtype_u" title="Link to this definition">¶</a></dt>
+<dd><p>alias of <code class="xref py py-class docutils literal notranslate"><span class="pre">mesh</span></code></p>
+</dd></dl>
+
+<dl class="py method">
+<dt class="sig sig-object py" id="implementations.problem_classes.Burgers.Burgers2D.eval_f">
+<span class="sig-name descname"><span class="pre">eval_f</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">u</span></span></em>, <em class="sig-param"><span class="o"><span class="pre">*</span></span><span class="n"><span class="pre">args</span></span></em>, <em class="sig-param"><span class="o"><span class="pre">**</span></span><span class="n"><span class="pre">kwargs</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../_modules/implementations/problem_classes/Burgers.html#Burgers2D.eval_f"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#implementations.problem_classes.Burgers.Burgers2D.eval_f" title="Link to this definition">¶</a></dt>
+<dd><p>Abstract interface to RHS computation of the ODE</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><ul class="simple">
+<li><p><strong>u</strong> (<em>dtype_u</em>) – Current values.</p></li>
+<li><p><strong>t</strong> (<em>float</em>) – Current time.</p></li>
+</ul>
+</dd>
+<dt class="field-even">Returns<span class="colon">:</span></dt>
+<dd class="field-even"><p><strong>f</strong> – The RHS values.</p>
+</dd>
+<dt class="field-odd">Return type<span class="colon">:</span></dt>
+<dd class="field-odd"><p>dtype_f</p>
+</dd>
+</dl>
+</dd></dl>
+
+<dl class="py method">
+<dt class="sig sig-object py" id="implementations.problem_classes.Burgers.Burgers2D.get_fig">
+<span class="sig-name descname"><span class="pre">get_fig</span></span><span class="sig-paren">(</span><span class="sig-paren">)</span><a class="reference internal" href="../_modules/implementations/problem_classes/Burgers.html#Burgers2D.get_fig"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#implementations.problem_classes.Burgers.Burgers2D.get_fig" title="Link to this definition">¶</a></dt>
+<dd><p>Get a figure suitable to plot the solution of this problem</p>
+<dl class="field-list simple">
+<dt class="field-odd">Returns<span class="colon">:</span></dt>
+<dd class="field-odd"><p><strong>self.fig</strong></p>
+</dd>
+<dt class="field-even">Return type<span class="colon">:</span></dt>
+<dd class="field-even"><p>matplotlib.pyplot.figure.Figure</p>
+</dd>
+</dl>
+</dd></dl>
+
+<dl class="py method">
+<dt class="sig sig-object py" id="implementations.problem_classes.Burgers.Burgers2D.plot">
+<span class="sig-name descname"><span class="pre">plot</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">u</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">t</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">fig</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">vmin</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">vmax</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../_modules/implementations/problem_classes/Burgers.html#Burgers2D.plot"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#implementations.problem_classes.Burgers.Burgers2D.plot" title="Link to this definition">¶</a></dt>
+<dd><p>Plot the solution. Please supply a figure with the same structure as returned by <code class="docutils literal notranslate"><span class="pre">self.get_fig</span></code>.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><ul class="simple">
+<li><p><strong>u</strong> (<em>dtype_u</em>) – Solution to be plotted</p></li>
+<li><p><strong>t</strong> (<em>float</em>) – Time to display at the top of the figure</p></li>
+<li><p><strong>fig</strong> (<em>matplotlib.pyplot.figure.Figure</em>) – Figure with the correct structure</p></li>
+</ul>
+</dd>
+<dt class="field-even">Return type<span class="colon">:</span></dt>
+<dd class="field-even"><p>None</p>
+</dd>
+</dl>
+</dd></dl>
+
+<dl class="py method">
+<dt class="sig sig-object py" id="implementations.problem_classes.Burgers.Burgers2D.u_exact">
+<span class="sig-name descname"><span class="pre">u_exact</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">t</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">0</span></span></em>, <em class="sig-param"><span class="o"><span class="pre">*</span></span><span class="n"><span class="pre">args</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">noise_level</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">0</span></span></em>, <em class="sig-param"><span class="o"><span class="pre">**</span></span><span class="n"><span class="pre">kwargs</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../_modules/implementations/problem_classes/Burgers.html#Burgers2D.u_exact"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#implementations.problem_classes.Burgers.Burgers2D.u_exact" title="Link to this definition">¶</a></dt>
+<dd></dd></dl>
+
+</dd></dl>
+
+</section>
+
+
+            <div class="clearer"></div>
+          </div>
+        </div>
+      </div>
+      <div class="sphinxsidebar" role="navigation" aria-label="Main">
+        <div class="sphinxsidebarwrapper">
+  <div>
+    <h3><a href="../index.html">Table of Contents</a></h3>
+    <ul>
+<li><a class="reference internal" href="#">implementations.problem_classes.Burgers module</a><ul>
+<li><a class="reference internal" href="#implementations.problem_classes.Burgers.Burgers1D"><code class="docutils literal notranslate"><span class="pre">Burgers1D</span></code></a><ul>
+<li><a class="reference internal" href="#implementations.problem_classes.Burgers.Burgers1D.dtype_f"><code class="docutils literal notranslate"><span class="pre">dtype_f</span></code></a></li>
+<li><a class="reference internal" href="#implementations.problem_classes.Burgers.Burgers1D.dtype_u"><code class="docutils literal notranslate"><span class="pre">dtype_u</span></code></a></li>
+<li><a class="reference internal" href="#implementations.problem_classes.Burgers.Burgers1D.eval_f"><code class="docutils literal notranslate"><span class="pre">eval_f()</span></code></a></li>
+<li><a class="reference internal" href="#implementations.problem_classes.Burgers.Burgers1D.get_fig"><code class="docutils literal notranslate"><span class="pre">get_fig()</span></code></a></li>
+<li><a class="reference internal" href="#implementations.problem_classes.Burgers.Burgers1D.plot"><code class="docutils literal notranslate"><span class="pre">plot()</span></code></a></li>
+<li><a class="reference internal" href="#implementations.problem_classes.Burgers.Burgers1D.u_exact"><code class="docutils literal notranslate"><span class="pre">u_exact()</span></code></a></li>
+</ul>
+</li>
+<li><a class="reference internal" href="#implementations.problem_classes.Burgers.Burgers2D"><code class="docutils literal notranslate"><span class="pre">Burgers2D</span></code></a><ul>
+<li><a class="reference internal" href="#implementations.problem_classes.Burgers.Burgers2D.compute_vorticity"><code class="docutils literal notranslate"><span class="pre">compute_vorticity()</span></code></a></li>
+<li><a class="reference internal" href="#implementations.problem_classes.Burgers.Burgers2D.dtype_f"><code class="docutils literal notranslate"><span class="pre">dtype_f</span></code></a></li>
+<li><a class="reference internal" href="#implementations.problem_classes.Burgers.Burgers2D.dtype_u"><code class="docutils literal notranslate"><span class="pre">dtype_u</span></code></a></li>
+<li><a class="reference internal" href="#implementations.problem_classes.Burgers.Burgers2D.eval_f"><code class="docutils literal notranslate"><span class="pre">eval_f()</span></code></a></li>
+<li><a class="reference internal" href="#implementations.problem_classes.Burgers.Burgers2D.get_fig"><code class="docutils literal notranslate"><span class="pre">get_fig()</span></code></a></li>
+<li><a class="reference internal" href="#implementations.problem_classes.Burgers.Burgers2D.plot"><code class="docutils literal notranslate"><span class="pre">plot()</span></code></a></li>
+<li><a class="reference internal" href="#implementations.problem_classes.Burgers.Burgers2D.u_exact"><code class="docutils literal notranslate"><span class="pre">u_exact()</span></code></a></li>
+</ul>
+</li>
+</ul>
+</li>
+</ul>
+
+  </div>
+  <div>
+    <h4>Previous topic</h4>
+    <p class="topless"><a href="implementations.problem_classes.BuckConverter.html"
+                          title="previous chapter">implementations.problem_classes.BuckConverter module</a></p>
+  </div>
+  <div>
+    <h4>Next topic</h4>
+    <p class="topless"><a href="implementations.problem_classes.DiscontinuousTestODE.html"
+                          title="next chapter">implementations.problem_classes.DiscontinuousTestODE module</a></p>
+  </div>
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+    <h3>This Page</h3>
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diff --git a/pySDC/implementations.problem_classes.DiscontinuousTestODE.html b/pySDC/implementations.problem_classes.DiscontinuousTestODE.html
index 4841ea9279..2c70d296b1 100644
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+    <link rel="prev" title="implementations.problem_classes.Burgers module" href="implementations.problem_classes.Burgers.html" /> 
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-                          title="previous chapter">implementations.problem_classes.BuckConverter module</a></p>
+    <p class="topless"><a href="implementations.problem_classes.Burgers.html"
+                          title="previous chapter">implementations.problem_classes.Burgers module</a></p>
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+    <p class="topless"><a href="implementations.problem_classes.HeatEquation_Chebychev.html"
+                          title="next chapter">implementations.problem_classes.HeatEquation_Chebychev module</a></p>
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new file mode 100644
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+          <li class="nav-item nav-item-2"><a href="implementations.problem_classes.html" accesskey="U">implementations.problem_classes package</a> &#187;</li>
+        <li class="nav-item nav-item-this"><a href="">implementations.problem_classes.HeatEquation_Chebychev module</a></li> 
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+            
+  <section id="module-implementations.problem_classes.HeatEquation_Chebychev">
+<span id="implementations-problem-classes-heatequation-chebychev-module"></span><h1>implementations.problem_classes.HeatEquation_Chebychev module<a class="headerlink" href="#module-implementations.problem_classes.HeatEquation_Chebychev" title="Link to this heading">¶</a></h1>
+<dl class="py class">
+<dt class="sig sig-object py" id="implementations.problem_classes.HeatEquation_Chebychev.Heat1DChebychev">
+<em class="property"><span class="pre">class</span><span class="w"> </span></em><span class="sig-name descname"><span class="pre">Heat1DChebychev</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">nvars</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">128</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">a</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">0</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">b</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">0</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">f</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">1</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">nu</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">1.0</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">mode</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">'T2U'</span></span></em>, <em class="sig-param"><span class="o"><span class="pre">**</span></span><span class="n"><span class="pre">kwargs</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../_modules/implementations/problem_classes/HeatEquation_Chebychev.html#Heat1DChebychev"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#implementations.problem_classes.HeatEquation_Chebychev.Heat1DChebychev" title="Link to this definition">¶</a></dt>
+<dd><p>Bases: <code class="xref py py-class docutils literal notranslate"><span class="pre">GenericSpectralLinear</span></code></p>
+<p>1D Heat equation with Dirichlet Boundary conditions discretized on (-1, 1) using a Chebychev spectral method.</p>
+<dl class="py attribute">
+<dt class="sig sig-object py" id="implementations.problem_classes.HeatEquation_Chebychev.Heat1DChebychev.dtype_f">
+<span class="sig-name descname"><span class="pre">dtype_f</span></span><a class="headerlink" href="#implementations.problem_classes.HeatEquation_Chebychev.Heat1DChebychev.dtype_f" title="Link to this definition">¶</a></dt>
+<dd><p>alias of <code class="xref py py-class docutils literal notranslate"><span class="pre">mesh</span></code></p>
+</dd></dl>
+
+<dl class="py attribute">
+<dt class="sig sig-object py" id="implementations.problem_classes.HeatEquation_Chebychev.Heat1DChebychev.dtype_u">
+<span class="sig-name descname"><span class="pre">dtype_u</span></span><a class="headerlink" href="#implementations.problem_classes.HeatEquation_Chebychev.Heat1DChebychev.dtype_u" title="Link to this definition">¶</a></dt>
+<dd><p>alias of <code class="xref py py-class docutils literal notranslate"><span class="pre">mesh</span></code></p>
+</dd></dl>
+
+<dl class="py method">
+<dt class="sig sig-object py" id="implementations.problem_classes.HeatEquation_Chebychev.Heat1DChebychev.eval_f">
+<span class="sig-name descname"><span class="pre">eval_f</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">u</span></span></em>, <em class="sig-param"><span class="o"><span class="pre">*</span></span><span class="n"><span class="pre">args</span></span></em>, <em class="sig-param"><span class="o"><span class="pre">**</span></span><span class="n"><span class="pre">kwargs</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../_modules/implementations/problem_classes/HeatEquation_Chebychev.html#Heat1DChebychev.eval_f"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#implementations.problem_classes.HeatEquation_Chebychev.Heat1DChebychev.eval_f" title="Link to this definition">¶</a></dt>
+<dd><p>Abstract interface to RHS computation of the ODE</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><ul class="simple">
+<li><p><strong>u</strong> (<em>dtype_u</em>) – Current values.</p></li>
+<li><p><strong>t</strong> (<em>float</em>) – Current time.</p></li>
+</ul>
+</dd>
+<dt class="field-even">Returns<span class="colon">:</span></dt>
+<dd class="field-even"><p><strong>f</strong> – The RHS values.</p>
+</dd>
+<dt class="field-odd">Return type<span class="colon">:</span></dt>
+<dd class="field-odd"><p>dtype_f</p>
+</dd>
+</dl>
+</dd></dl>
+
+<dl class="py method">
+<dt class="sig sig-object py" id="implementations.problem_classes.HeatEquation_Chebychev.Heat1DChebychev.u_exact">
+<span class="sig-name descname"><span class="pre">u_exact</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">t</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">noise</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">0</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">seed</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">666</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../_modules/implementations/problem_classes/HeatEquation_Chebychev.html#Heat1DChebychev.u_exact"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#implementations.problem_classes.HeatEquation_Chebychev.Heat1DChebychev.u_exact" title="Link to this definition">¶</a></dt>
+<dd><p>Get exact solution at time <cite>t</cite></p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><ul class="simple">
+<li><p><strong>t</strong> (<em>float</em>) – When you want the exact solution</p></li>
+<li><p><strong>noise</strong> (<em>float</em>) – Add noise of this level</p></li>
+<li><p><strong>seed</strong> (<em>int</em>) – Random seed for the noise</p></li>
+</ul>
+</dd>
+<dt class="field-even">Returns<span class="colon">:</span></dt>
+<dd class="field-even"><p>Exact solution</p>
+</dd>
+<dt class="field-odd">Return type<span class="colon">:</span></dt>
+<dd class="field-odd"><p>Heat1DChebychev.dtype_u</p>
+</dd>
+</dl>
+</dd></dl>
+
+</dd></dl>
+
+<dl class="py class">
+<dt class="sig sig-object py" id="implementations.problem_classes.HeatEquation_Chebychev.Heat1DUltraspherical">
+<em class="property"><span class="pre">class</span><span class="w"> </span></em><span class="sig-name descname"><span class="pre">Heat1DUltraspherical</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">nvars</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">128</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">a</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">0</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">b</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">0</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">f</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">1</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">nu</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">1.0</span></span></em>, <em class="sig-param"><span class="o"><span class="pre">**</span></span><span class="n"><span class="pre">kwargs</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../_modules/implementations/problem_classes/HeatEquation_Chebychev.html#Heat1DUltraspherical"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#implementations.problem_classes.HeatEquation_Chebychev.Heat1DUltraspherical" title="Link to this definition">¶</a></dt>
+<dd><p>Bases: <code class="xref py py-class docutils literal notranslate"><span class="pre">GenericSpectralLinear</span></code></p>
+<p>1D Heat equation with Dirichlet Boundary conditions discretized on (-1, 1) using an ultraspherical spectral method.</p>
+<dl class="py attribute">
+<dt class="sig sig-object py" id="implementations.problem_classes.HeatEquation_Chebychev.Heat1DUltraspherical.dtype_f">
+<span class="sig-name descname"><span class="pre">dtype_f</span></span><a class="headerlink" href="#implementations.problem_classes.HeatEquation_Chebychev.Heat1DUltraspherical.dtype_f" title="Link to this definition">¶</a></dt>
+<dd><p>alias of <code class="xref py py-class docutils literal notranslate"><span class="pre">mesh</span></code></p>
+</dd></dl>
+
+<dl class="py attribute">
+<dt class="sig sig-object py" id="implementations.problem_classes.HeatEquation_Chebychev.Heat1DUltraspherical.dtype_u">
+<span class="sig-name descname"><span class="pre">dtype_u</span></span><a class="headerlink" href="#implementations.problem_classes.HeatEquation_Chebychev.Heat1DUltraspherical.dtype_u" title="Link to this definition">¶</a></dt>
+<dd><p>alias of <code class="xref py py-class docutils literal notranslate"><span class="pre">mesh</span></code></p>
+</dd></dl>
+
+<dl class="py method">
+<dt class="sig sig-object py" id="implementations.problem_classes.HeatEquation_Chebychev.Heat1DUltraspherical.eval_f">
+<span class="sig-name descname"><span class="pre">eval_f</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">u</span></span></em>, <em class="sig-param"><span class="o"><span class="pre">*</span></span><span class="n"><span class="pre">args</span></span></em>, <em class="sig-param"><span class="o"><span class="pre">**</span></span><span class="n"><span class="pre">kwargs</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../_modules/implementations/problem_classes/HeatEquation_Chebychev.html#Heat1DUltraspherical.eval_f"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#implementations.problem_classes.HeatEquation_Chebychev.Heat1DUltraspherical.eval_f" title="Link to this definition">¶</a></dt>
+<dd><p>Abstract interface to RHS computation of the ODE</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><ul class="simple">
+<li><p><strong>u</strong> (<em>dtype_u</em>) – Current values.</p></li>
+<li><p><strong>t</strong> (<em>float</em>) – Current time.</p></li>
+</ul>
+</dd>
+<dt class="field-even">Returns<span class="colon">:</span></dt>
+<dd class="field-even"><p><strong>f</strong> – The RHS values.</p>
+</dd>
+<dt class="field-odd">Return type<span class="colon">:</span></dt>
+<dd class="field-odd"><p>dtype_f</p>
+</dd>
+</dl>
+</dd></dl>
+
+<dl class="py method">
+<dt class="sig sig-object py" id="implementations.problem_classes.HeatEquation_Chebychev.Heat1DUltraspherical.u_exact">
+<span class="sig-name descname"><span class="pre">u_exact</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">t</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">noise</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">0</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">seed</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">666</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../_modules/implementations/problem_classes/HeatEquation_Chebychev.html#Heat1DUltraspherical.u_exact"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#implementations.problem_classes.HeatEquation_Chebychev.Heat1DUltraspherical.u_exact" title="Link to this definition">¶</a></dt>
+<dd><p>Get exact solution at time <cite>t</cite></p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><ul class="simple">
+<li><p><strong>t</strong> (<em>float</em>) – When you want the exact solution</p></li>
+<li><p><strong>noise</strong> (<em>float</em>) – Add noise of this level</p></li>
+<li><p><strong>seed</strong> (<em>int</em>) – Random seed for the noise</p></li>
+</ul>
+</dd>
+<dt class="field-even">Returns<span class="colon">:</span></dt>
+<dd class="field-even"><p>Exact solution</p>
+</dd>
+<dt class="field-odd">Return type<span class="colon">:</span></dt>
+<dd class="field-odd"><p>Heat1DUltraspherical.dtype_u</p>
+</dd>
+</dl>
+</dd></dl>
+
+</dd></dl>
+
+<dl class="py class">
+<dt class="sig sig-object py" id="implementations.problem_classes.HeatEquation_Chebychev.Heat2DChebychev">
+<em class="property"><span class="pre">class</span><span class="w"> </span></em><span class="sig-name descname"><span class="pre">Heat2DChebychev</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">nx</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">128</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">ny</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">128</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">base_x</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">'fft'</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">base_y</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">'chebychev'</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">a</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">0</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">b</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">0</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">c</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">0</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">fx</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">1</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">fy</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">1</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">nu</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">1.0</span></span></em>, <em class="sig-param"><span class="o"><span class="pre">**</span></span><span class="n"><span class="pre">kwargs</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../_modules/implementations/problem_classes/HeatEquation_Chebychev.html#Heat2DChebychev"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#implementations.problem_classes.HeatEquation_Chebychev.Heat2DChebychev" title="Link to this definition">¶</a></dt>
+<dd><p>Bases: <code class="xref py py-class docutils literal notranslate"><span class="pre">GenericSpectralLinear</span></code></p>
+<p>2D Heat equation with Dirichlet Boundary conditions discretized on (-1, 1)x(-1,1) using spectral methods based on FFT and Chebychev.</p>
+<dl class="py attribute">
+<dt class="sig sig-object py" id="implementations.problem_classes.HeatEquation_Chebychev.Heat2DChebychev.dtype_f">
+<span class="sig-name descname"><span class="pre">dtype_f</span></span><a class="headerlink" href="#implementations.problem_classes.HeatEquation_Chebychev.Heat2DChebychev.dtype_f" title="Link to this definition">¶</a></dt>
+<dd><p>alias of <code class="xref py py-class docutils literal notranslate"><span class="pre">mesh</span></code></p>
+</dd></dl>
+
+<dl class="py attribute">
+<dt class="sig sig-object py" id="implementations.problem_classes.HeatEquation_Chebychev.Heat2DChebychev.dtype_u">
+<span class="sig-name descname"><span class="pre">dtype_u</span></span><a class="headerlink" href="#implementations.problem_classes.HeatEquation_Chebychev.Heat2DChebychev.dtype_u" title="Link to this definition">¶</a></dt>
+<dd><p>alias of <code class="xref py py-class docutils literal notranslate"><span class="pre">mesh</span></code></p>
+</dd></dl>
+
+<dl class="py method">
+<dt class="sig sig-object py" id="implementations.problem_classes.HeatEquation_Chebychev.Heat2DChebychev.eval_f">
+<span class="sig-name descname"><span class="pre">eval_f</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">u</span></span></em>, <em class="sig-param"><span class="o"><span class="pre">*</span></span><span class="n"><span class="pre">args</span></span></em>, <em class="sig-param"><span class="o"><span class="pre">**</span></span><span class="n"><span class="pre">kwargs</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../_modules/implementations/problem_classes/HeatEquation_Chebychev.html#Heat2DChebychev.eval_f"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#implementations.problem_classes.HeatEquation_Chebychev.Heat2DChebychev.eval_f" title="Link to this definition">¶</a></dt>
+<dd><p>Abstract interface to RHS computation of the ODE</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><ul class="simple">
+<li><p><strong>u</strong> (<em>dtype_u</em>) – Current values.</p></li>
+<li><p><strong>t</strong> (<em>float</em>) – Current time.</p></li>
+</ul>
+</dd>
+<dt class="field-even">Returns<span class="colon">:</span></dt>
+<dd class="field-even"><p><strong>f</strong> – The RHS values.</p>
+</dd>
+<dt class="field-odd">Return type<span class="colon">:</span></dt>
+<dd class="field-odd"><p>dtype_f</p>
+</dd>
+</dl>
+</dd></dl>
+
+<dl class="py method">
+<dt class="sig sig-object py" id="implementations.problem_classes.HeatEquation_Chebychev.Heat2DChebychev.u_exact">
+<span class="sig-name descname"><span class="pre">u_exact</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">t</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../_modules/implementations/problem_classes/HeatEquation_Chebychev.html#Heat2DChebychev.u_exact"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#implementations.problem_classes.HeatEquation_Chebychev.Heat2DChebychev.u_exact" title="Link to this definition">¶</a></dt>
+<dd></dd></dl>
+
+</dd></dl>
+
+<dl class="py class">
+<dt class="sig sig-object py" id="implementations.problem_classes.HeatEquation_Chebychev.Heat2DUltraspherical">
+<em class="property"><span class="pre">class</span><span class="w"> </span></em><span class="sig-name descname"><span class="pre">Heat2DUltraspherical</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">nx</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">128</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">ny</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">128</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">base_x</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">'fft'</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">base_y</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">'ultraspherical'</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">a</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">0</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">b</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">0</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">c</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">0</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">fx</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">1</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">fy</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">1</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">nu</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">1.0</span></span></em>, <em class="sig-param"><span class="o"><span class="pre">**</span></span><span class="n"><span class="pre">kwargs</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../_modules/implementations/problem_classes/HeatEquation_Chebychev.html#Heat2DUltraspherical"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#implementations.problem_classes.HeatEquation_Chebychev.Heat2DUltraspherical" title="Link to this definition">¶</a></dt>
+<dd><p>Bases: <code class="xref py py-class docutils literal notranslate"><span class="pre">GenericSpectralLinear</span></code></p>
+<p>2D Heat equation with Dirichlet Boundary conditions discretized on (-1, 1)x(-1,1) using spectral methods based on FFT and Gegenbauer.</p>
+<dl class="py attribute">
+<dt class="sig sig-object py" id="implementations.problem_classes.HeatEquation_Chebychev.Heat2DUltraspherical.dtype_f">
+<span class="sig-name descname"><span class="pre">dtype_f</span></span><a class="headerlink" href="#implementations.problem_classes.HeatEquation_Chebychev.Heat2DUltraspherical.dtype_f" title="Link to this definition">¶</a></dt>
+<dd><p>alias of <code class="xref py py-class docutils literal notranslate"><span class="pre">mesh</span></code></p>
+</dd></dl>
+
+<dl class="py attribute">
+<dt class="sig sig-object py" id="implementations.problem_classes.HeatEquation_Chebychev.Heat2DUltraspherical.dtype_u">
+<span class="sig-name descname"><span class="pre">dtype_u</span></span><a class="headerlink" href="#implementations.problem_classes.HeatEquation_Chebychev.Heat2DUltraspherical.dtype_u" title="Link to this definition">¶</a></dt>
+<dd><p>alias of <code class="xref py py-class docutils literal notranslate"><span class="pre">mesh</span></code></p>
+</dd></dl>
+
+<dl class="py method">
+<dt class="sig sig-object py" id="implementations.problem_classes.HeatEquation_Chebychev.Heat2DUltraspherical.eval_f">
+<span class="sig-name descname"><span class="pre">eval_f</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">u</span></span></em>, <em class="sig-param"><span class="o"><span class="pre">*</span></span><span class="n"><span class="pre">args</span></span></em>, <em class="sig-param"><span class="o"><span class="pre">**</span></span><span class="n"><span class="pre">kwargs</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../_modules/implementations/problem_classes/HeatEquation_Chebychev.html#Heat2DUltraspherical.eval_f"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#implementations.problem_classes.HeatEquation_Chebychev.Heat2DUltraspherical.eval_f" title="Link to this definition">¶</a></dt>
+<dd><p>Abstract interface to RHS computation of the ODE</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><ul class="simple">
+<li><p><strong>u</strong> (<em>dtype_u</em>) – Current values.</p></li>
+<li><p><strong>t</strong> (<em>float</em>) – Current time.</p></li>
+</ul>
+</dd>
+<dt class="field-even">Returns<span class="colon">:</span></dt>
+<dd class="field-even"><p><strong>f</strong> – The RHS values.</p>
+</dd>
+<dt class="field-odd">Return type<span class="colon">:</span></dt>
+<dd class="field-odd"><p>dtype_f</p>
+</dd>
+</dl>
+</dd></dl>
+
+<dl class="py method">
+<dt class="sig sig-object py" id="implementations.problem_classes.HeatEquation_Chebychev.Heat2DUltraspherical.u_exact">
+<span class="sig-name descname"><span class="pre">u_exact</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">t</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../_modules/implementations/problem_classes/HeatEquation_Chebychev.html#Heat2DUltraspherical.u_exact"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#implementations.problem_classes.HeatEquation_Chebychev.Heat2DUltraspherical.u_exact" title="Link to this definition">¶</a></dt>
+<dd></dd></dl>
+
+</dd></dl>
+
+</section>
+
+
+            <div class="clearer"></div>
+          </div>
+        </div>
+      </div>
+      <div class="sphinxsidebar" role="navigation" aria-label="Main">
+        <div class="sphinxsidebarwrapper">
+  <div>
+    <h3><a href="../index.html">Table of Contents</a></h3>
+    <ul>
+<li><a class="reference internal" href="#">implementations.problem_classes.HeatEquation_Chebychev module</a><ul>
+<li><a class="reference internal" href="#implementations.problem_classes.HeatEquation_Chebychev.Heat1DChebychev"><code class="docutils literal notranslate"><span class="pre">Heat1DChebychev</span></code></a><ul>
+<li><a class="reference internal" href="#implementations.problem_classes.HeatEquation_Chebychev.Heat1DChebychev.dtype_f"><code class="docutils literal notranslate"><span class="pre">dtype_f</span></code></a></li>
+<li><a class="reference internal" href="#implementations.problem_classes.HeatEquation_Chebychev.Heat1DChebychev.dtype_u"><code class="docutils literal notranslate"><span class="pre">dtype_u</span></code></a></li>
+<li><a class="reference internal" href="#implementations.problem_classes.HeatEquation_Chebychev.Heat1DChebychev.eval_f"><code class="docutils literal notranslate"><span class="pre">eval_f()</span></code></a></li>
+<li><a class="reference internal" href="#implementations.problem_classes.HeatEquation_Chebychev.Heat1DChebychev.u_exact"><code class="docutils literal notranslate"><span class="pre">u_exact()</span></code></a></li>
+</ul>
+</li>
+<li><a class="reference internal" href="#implementations.problem_classes.HeatEquation_Chebychev.Heat1DUltraspherical"><code class="docutils literal notranslate"><span class="pre">Heat1DUltraspherical</span></code></a><ul>
+<li><a class="reference internal" href="#implementations.problem_classes.HeatEquation_Chebychev.Heat1DUltraspherical.dtype_f"><code class="docutils literal notranslate"><span class="pre">dtype_f</span></code></a></li>
+<li><a class="reference internal" href="#implementations.problem_classes.HeatEquation_Chebychev.Heat1DUltraspherical.dtype_u"><code class="docutils literal notranslate"><span class="pre">dtype_u</span></code></a></li>
+<li><a class="reference internal" href="#implementations.problem_classes.HeatEquation_Chebychev.Heat1DUltraspherical.eval_f"><code class="docutils literal notranslate"><span class="pre">eval_f()</span></code></a></li>
+<li><a class="reference internal" href="#implementations.problem_classes.HeatEquation_Chebychev.Heat1DUltraspherical.u_exact"><code class="docutils literal notranslate"><span class="pre">u_exact()</span></code></a></li>
+</ul>
+</li>
+<li><a class="reference internal" href="#implementations.problem_classes.HeatEquation_Chebychev.Heat2DChebychev"><code class="docutils literal notranslate"><span class="pre">Heat2DChebychev</span></code></a><ul>
+<li><a class="reference internal" href="#implementations.problem_classes.HeatEquation_Chebychev.Heat2DChebychev.dtype_f"><code class="docutils literal notranslate"><span class="pre">dtype_f</span></code></a></li>
+<li><a class="reference internal" href="#implementations.problem_classes.HeatEquation_Chebychev.Heat2DChebychev.dtype_u"><code class="docutils literal notranslate"><span class="pre">dtype_u</span></code></a></li>
+<li><a class="reference internal" href="#implementations.problem_classes.HeatEquation_Chebychev.Heat2DChebychev.eval_f"><code class="docutils literal notranslate"><span class="pre">eval_f()</span></code></a></li>
+<li><a class="reference internal" href="#implementations.problem_classes.HeatEquation_Chebychev.Heat2DChebychev.u_exact"><code class="docutils literal notranslate"><span class="pre">u_exact()</span></code></a></li>
+</ul>
+</li>
+<li><a class="reference internal" href="#implementations.problem_classes.HeatEquation_Chebychev.Heat2DUltraspherical"><code class="docutils literal notranslate"><span class="pre">Heat2DUltraspherical</span></code></a><ul>
+<li><a class="reference internal" href="#implementations.problem_classes.HeatEquation_Chebychev.Heat2DUltraspherical.dtype_f"><code class="docutils literal notranslate"><span class="pre">dtype_f</span></code></a></li>
+<li><a class="reference internal" href="#implementations.problem_classes.HeatEquation_Chebychev.Heat2DUltraspherical.dtype_u"><code class="docutils literal notranslate"><span class="pre">dtype_u</span></code></a></li>
+<li><a class="reference internal" href="#implementations.problem_classes.HeatEquation_Chebychev.Heat2DUltraspherical.eval_f"><code class="docutils literal notranslate"><span class="pre">eval_f()</span></code></a></li>
+<li><a class="reference internal" href="#implementations.problem_classes.HeatEquation_Chebychev.Heat2DUltraspherical.u_exact"><code class="docutils literal notranslate"><span class="pre">u_exact()</span></code></a></li>
+</ul>
+</li>
+</ul>
+</li>
+</ul>
+
+  </div>
+  <div>
+    <h4>Previous topic</h4>
+    <p class="topless"><a href="implementations.problem_classes.HeatEquation_2D_PETSc_forced.html"
+                          title="previous chapter">implementations.problem_classes.HeatEquation_2D_PETSc_forced module</a></p>
+  </div>
+  <div>
+    <h4>Next topic</h4>
+    <p class="topless"><a href="implementations.problem_classes.HeatEquation_ND_FD.html"
+                          title="next chapter">implementations.problem_classes.HeatEquation_ND_FD module</a></p>
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+    <h3>This Page</h3>
+    <ul class="this-page-menu">
+      <li><a href="../_sources/pySDC/implementations.problem_classes.HeatEquation_Chebychev.rst.txt"
+            rel="nofollow">Show Source</a></li>
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+        <li class="nav-item nav-item-0"><a href="../index.html">pySDC 5.5.1 documentation</a> &#187;</li>
+          <li class="nav-item nav-item-1"><a href="implementations.html" >implementations package</a> &#187;</li>
+          <li class="nav-item nav-item-2"><a href="implementations.problem_classes.html" >implementations.problem_classes package</a> &#187;</li>
+        <li class="nav-item nav-item-this"><a href="">implementations.problem_classes.HeatEquation_Chebychev module</a></li> 
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diff --git a/pySDC/implementations.problem_classes.HeatEquation_ND_FD.html b/pySDC/implementations.problem_classes.HeatEquation_ND_FD.html
index 0cc21f2ef8..ae8e55eb73 100644
--- a/pySDC/implementations.problem_classes.HeatEquation_ND_FD.html
+++ b/pySDC/implementations.problem_classes.HeatEquation_ND_FD.html
@@ -18,7 +18,7 @@
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@@ -33,7 +33,7 @@ <h3>Navigation</h3>
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+          <a href="implementations.problem_classes.HeatEquation_Chebychev.html" title="implementations.problem_classes.HeatEquation_Chebychev module"
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@@ -225,8 +225,8 @@ <h3><a href="../index.html">Table of Contents</a></h3>
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-    <p class="topless"><a href="implementations.problem_classes.HeatEquation_2D_PETSc_forced.html"
-                          title="previous chapter">implementations.problem_classes.HeatEquation_2D_PETSc_forced module</a></p>
+    <p class="topless"><a href="implementations.problem_classes.HeatEquation_Chebychev.html"
+                          title="previous chapter">implementations.problem_classes.HeatEquation_Chebychev module</a></p>
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@@ -267,7 +267,7 @@ <h3>Navigation</h3>
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-          <a href="implementations.problem_classes.HeatEquation_2D_PETSc_forced.html" title="implementations.problem_classes.HeatEquation_2D_PETSc_forced module"
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+  <section id="module-implementations.problem_classes.RayleighBenard">
+<span id="implementations-problem-classes-rayleighbenard-module"></span><h1>implementations.problem_classes.RayleighBenard module<a class="headerlink" href="#module-implementations.problem_classes.RayleighBenard" title="Link to this heading">¶</a></h1>
+<dl class="py class">
+<dt class="sig sig-object py" id="implementations.problem_classes.RayleighBenard.CFLLimit">
+<em class="property"><span class="pre">class</span><span class="w"> </span></em><span class="sig-name descname"><span class="pre">CFLLimit</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">controller</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">params</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">description</span></span></em>, <em class="sig-param"><span class="o"><span class="pre">**</span></span><span class="n"><span class="pre">kwargs</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../_modules/implementations/problem_classes/RayleighBenard.html#CFLLimit"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#implementations.problem_classes.RayleighBenard.CFLLimit" title="Link to this definition">¶</a></dt>
+<dd><p>Bases: <code class="xref py py-class docutils literal notranslate"><span class="pre">ConvergenceController</span></code></p>
+<dl class="py method">
+<dt class="sig sig-object py" id="implementations.problem_classes.RayleighBenard.CFLLimit.compute_max_step_size">
+<em class="property"><span class="pre">static</span><span class="w"> </span></em><span class="sig-name descname"><span class="pre">compute_max_step_size</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">P</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">u</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../_modules/implementations/problem_classes/RayleighBenard.html#CFLLimit.compute_max_step_size"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#implementations.problem_classes.RayleighBenard.CFLLimit.compute_max_step_size" title="Link to this definition">¶</a></dt>
+<dd></dd></dl>
+
+<dl class="py method">
+<dt class="sig sig-object py" id="implementations.problem_classes.RayleighBenard.CFLLimit.dependencies">
+<span class="sig-name descname"><span class="pre">dependencies</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">controller</span></span></em>, <em class="sig-param"><span class="o"><span class="pre">*</span></span><span class="n"><span class="pre">args</span></span></em>, <em class="sig-param"><span class="o"><span class="pre">**</span></span><span class="n"><span class="pre">kwargs</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../_modules/implementations/problem_classes/RayleighBenard.html#CFLLimit.dependencies"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#implementations.problem_classes.RayleighBenard.CFLLimit.dependencies" title="Link to this definition">¶</a></dt>
+<dd><p>Load dependencies on other convergence controllers here.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><ul class="simple">
+<li><p><strong>controller</strong> (<em>pySDC.Controller</em>) – The controller</p></li>
+<li><p><strong>description</strong> (<em>dict</em>) – The description object used to instantiate the controller</p></li>
+</ul>
+</dd>
+<dt class="field-even">Returns<span class="colon">:</span></dt>
+<dd class="field-even"><p>None</p>
+</dd>
+</dl>
+</dd></dl>
+
+<dl class="py method">
+<dt class="sig sig-object py" id="implementations.problem_classes.RayleighBenard.CFLLimit.get_new_step_size">
+<span class="sig-name descname"><span class="pre">get_new_step_size</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">controller</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">step</span></span></em>, <em class="sig-param"><span class="o"><span class="pre">**</span></span><span class="n"><span class="pre">kwargs</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../_modules/implementations/problem_classes/RayleighBenard.html#CFLLimit.get_new_step_size"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#implementations.problem_classes.RayleighBenard.CFLLimit.get_new_step_size" title="Link to this definition">¶</a></dt>
+<dd><p>This function allows to set a step size with arbitrary criteria.
+Make sure to give an order to the convergence controller by setting the <cite>control_order</cite> variable in the params.
+This variable is an integer and you can see what the current order is by using
+<cite>controller.print_convergence_controllers()</cite>.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><ul class="simple">
+<li><p><strong>controller</strong> (<em>pySDC.Controller</em>) – The controller</p></li>
+<li><p><strong>S</strong> (<em>pySDC.Step</em>) – The current step</p></li>
+</ul>
+</dd>
+<dt class="field-even">Returns<span class="colon">:</span></dt>
+<dd class="field-even"><p>None</p>
+</dd>
+</dl>
+</dd></dl>
+
+<dl class="py method">
+<dt class="sig sig-object py" id="implementations.problem_classes.RayleighBenard.CFLLimit.setup">
+<span class="sig-name descname"><span class="pre">setup</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">controller</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">params</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">description</span></span></em>, <em class="sig-param"><span class="o"><span class="pre">**</span></span><span class="n"><span class="pre">kwargs</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../_modules/implementations/problem_classes/RayleighBenard.html#CFLLimit.setup"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#implementations.problem_classes.RayleighBenard.CFLLimit.setup" title="Link to this definition">¶</a></dt>
+<dd><p>Define default parameters here.</p>
+<dl class="simple">
+<dt>Default parameters are:</dt><dd><ul class="simple">
+<li><p>control_order (int): The order relative to other convergence controllers</p></li>
+<li><p>dt_max (float): maximal step size</p></li>
+<li><p>dt_min (float): minimal step size</p></li>
+</ul>
+</dd>
+</dl>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><ul class="simple">
+<li><p><strong>controller</strong> (<em>pySDC.Controller</em>) – The controller</p></li>
+<li><p><strong>params</strong> (<em>dict</em>) – The params passed for this specific convergence controller</p></li>
+<li><p><strong>description</strong> (<em>dict</em>) – The description object used to instantiate the controller</p></li>
+</ul>
+</dd>
+<dt class="field-even">Returns<span class="colon">:</span></dt>
+<dd class="field-even"><p>The updated params dictionary</p>
+</dd>
+<dt class="field-odd">Return type<span class="colon">:</span></dt>
+<dd class="field-odd"><p>(dict)</p>
+</dd>
+</dl>
+</dd></dl>
+
+<dl class="py method">
+<dt class="sig sig-object py" id="implementations.problem_classes.RayleighBenard.CFLLimit.setup_status_variables">
+<span class="sig-name descname"><span class="pre">setup_status_variables</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">controller</span></span></em>, <em class="sig-param"><span class="o"><span class="pre">**</span></span><span class="n"><span class="pre">kwargs</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../_modules/implementations/problem_classes/RayleighBenard.html#CFLLimit.setup_status_variables"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#implementations.problem_classes.RayleighBenard.CFLLimit.setup_status_variables" title="Link to this definition">¶</a></dt>
+<dd><p>Add the embedded error variable to the error function.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><p><strong>controller</strong> (<em>pySDC.Controller</em>) – The controller</p>
+</dd>
+</dl>
+</dd></dl>
+
+</dd></dl>
+
+<dl class="py class">
+<dt class="sig sig-object py" id="implementations.problem_classes.RayleighBenard.LogAnalysisVariables">
+<em class="property"><span class="pre">class</span><span class="w"> </span></em><span class="sig-name descname"><span class="pre">LogAnalysisVariables</span></span><a class="reference internal" href="../_modules/implementations/problem_classes/RayleighBenard.html#LogAnalysisVariables"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#implementations.problem_classes.RayleighBenard.LogAnalysisVariables" title="Link to this definition">¶</a></dt>
+<dd><p>Bases: <code class="xref py py-class docutils literal notranslate"><span class="pre">Hooks</span></code></p>
+<dl class="py method">
+<dt class="sig sig-object py" id="implementations.problem_classes.RayleighBenard.LogAnalysisVariables.post_step">
+<span class="sig-name descname"><span class="pre">post_step</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">step</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">level_number</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../_modules/implementations/problem_classes/RayleighBenard.html#LogAnalysisVariables.post_step"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#implementations.problem_classes.RayleighBenard.LogAnalysisVariables.post_step" title="Link to this definition">¶</a></dt>
+<dd><p>Record Nusselt numbers.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><ul class="simple">
+<li><p><strong>step</strong> (<em>pySDC.Step.step</em>) – the current step</p></li>
+<li><p><strong>level_number</strong> (<em>int</em>) – the current level number</p></li>
+</ul>
+</dd>
+<dt class="field-even">Returns<span class="colon">:</span></dt>
+<dd class="field-even"><p>None</p>
+</dd>
+</dl>
+</dd></dl>
+
+</dd></dl>
+
+<dl class="py class">
+<dt class="sig sig-object py" id="implementations.problem_classes.RayleighBenard.LogCFL">
+<em class="property"><span class="pre">class</span><span class="w"> </span></em><span class="sig-name descname"><span class="pre">LogCFL</span></span><a class="reference internal" href="../_modules/implementations/problem_classes/RayleighBenard.html#LogCFL"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#implementations.problem_classes.RayleighBenard.LogCFL" title="Link to this definition">¶</a></dt>
+<dd><p>Bases: <code class="xref py py-class docutils literal notranslate"><span class="pre">Hooks</span></code></p>
+<dl class="py method">
+<dt class="sig sig-object py" id="implementations.problem_classes.RayleighBenard.LogCFL.post_step">
+<span class="sig-name descname"><span class="pre">post_step</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">step</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">level_number</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../_modules/implementations/problem_classes/RayleighBenard.html#LogCFL.post_step"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#implementations.problem_classes.RayleighBenard.LogCFL.post_step" title="Link to this definition">¶</a></dt>
+<dd><p>Record CFL limit.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><ul class="simple">
+<li><p><strong>step</strong> (<em>pySDC.Step.step</em>) – the current step</p></li>
+<li><p><strong>level_number</strong> (<em>int</em>) – the current level number</p></li>
+</ul>
+</dd>
+<dt class="field-even">Returns<span class="colon">:</span></dt>
+<dd class="field-even"><p>None</p>
+</dd>
+</dl>
+</dd></dl>
+
+</dd></dl>
+
+<dl class="py class">
+<dt class="sig sig-object py" id="implementations.problem_classes.RayleighBenard.RayleighBenard">
+<em class="property"><span class="pre">class</span><span class="w"> </span></em><span class="sig-name descname"><span class="pre">RayleighBenard</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">Prandl</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">1</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">Rayleigh</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">2000000.0</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">nx</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">256</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">nz</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">64</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">BCs</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">dealiasing</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">1.5</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">comm</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="o"><span class="pre">**</span></span><span class="n"><span class="pre">kwargs</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../_modules/implementations/problem_classes/RayleighBenard.html#RayleighBenard"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#implementations.problem_classes.RayleighBenard.RayleighBenard" title="Link to this definition">¶</a></dt>
+<dd><p>Bases: <code class="xref py py-class docutils literal notranslate"><span class="pre">GenericSpectralLinear</span></code></p>
+<p>Rayleigh-Benard Convection is a variation of incompressible Navier-Stokes.</p>
+<p>The equations we solve are</p>
+<blockquote>
+<div><p>u_x + v_z = 0
+T_t - kappa (T_xx + T_zz) = -uT_x - vT_z
+u_t - nu (u_xx + u_zz) + p_x = -uu_x - vu_z
+v_t - nu (v_xx + v_zz) + p_z - T = -uv_x - vv_z</p>
+</div></blockquote>
+<p>with u the horizontal velocity, v the vertical velocity (in z-direction), T the temperature, p the pressure, indices
+denoting derivatives, kappa=(Rayleigh * Prandl)**(-1/2) and nu = (Rayleigh / Prandl)**(-1/2). Everything on the left
+hand side, that is the viscous part, the pressure gradient and the buoyancy due to temperature are treated
+implicitly, while the non-linear convection part on the right hand side is integrated explicitly.</p>
+<p>The domain, vertical boundary conditions and pressure gauge are</p>
+<blockquote>
+<div><p>Omega = [0, 8) x (-1, 1)
+T(z=+1) = 0
+T(z=-1) = 2
+u(z=+-1) = v(z=+-1) = 0
+integral over p = 0</p>
+</div></blockquote>
+<p>The spectral discretization uses FFT horizontally, implying periodic BCs, and an ultraspherical method vertically to
+facilitate the Dirichlet BCs.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><ul class="simple">
+<li><p><strong>Prandl</strong> (<em>float</em>) – Prandl number</p></li>
+<li><p><strong>Rayleigh</strong> (<em>float</em>) – Rayleigh number</p></li>
+<li><p><strong>nx</strong> (<em>int</em>) – Horizontal resolution</p></li>
+<li><p><strong>nz</strong> (<em>int</em>) – Vertical resolution</p></li>
+<li><p><strong>BCs</strong> (<em>dict</em>) – Can specify boundary conditions here</p></li>
+<li><p><strong>dealiasing</strong> (<em>float</em>) – Dealiasing factor for evaluating the non-linear part</p></li>
+<li><p><strong>comm</strong> (<em>mpi4py.Intracomm</em>) – Space communicator</p></li>
+</ul>
+</dd>
+</dl>
+<dl class="py method">
+<dt class="sig sig-object py" id="implementations.problem_classes.RayleighBenard.RayleighBenard.compute_Nusselt_numbers">
+<span class="sig-name descname"><span class="pre">compute_Nusselt_numbers</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">u</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../_modules/implementations/problem_classes/RayleighBenard.html#RayleighBenard.compute_Nusselt_numbers"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#implementations.problem_classes.RayleighBenard.RayleighBenard.compute_Nusselt_numbers" title="Link to this definition">¶</a></dt>
+<dd><p>Compute the various versions of the Nusselt number. This reflects the type of heat transport.
+If the Nusselt number is equal to one, it indicates heat transport due to conduction. If it is larger,
+advection is present.
+Computing the Nusselt number at various places can be used to check the code.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><p><strong>u</strong> – The solution you want to compute the Nusselt numbers of</p>
+</dd>
+<dt class="field-even">Returns<span class="colon">:</span></dt>
+<dd class="field-even"><p>Nusselt number averaged over the entire volume and horizontally averaged at the top and bottom.</p>
+</dd>
+<dt class="field-odd">Return type<span class="colon">:</span></dt>
+<dd class="field-odd"><p>dict</p>
+</dd>
+</dl>
+</dd></dl>
+
+<dl class="py method">
+<dt class="sig sig-object py" id="implementations.problem_classes.RayleighBenard.RayleighBenard.compute_buoyancy_generation">
+<span class="sig-name descname"><span class="pre">compute_buoyancy_generation</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">u</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../_modules/implementations/problem_classes/RayleighBenard.html#RayleighBenard.compute_buoyancy_generation"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#implementations.problem_classes.RayleighBenard.RayleighBenard.compute_buoyancy_generation" title="Link to this definition">¶</a></dt>
+<dd></dd></dl>
+
+<dl class="py method">
+<dt class="sig sig-object py" id="implementations.problem_classes.RayleighBenard.RayleighBenard.compute_viscous_dissipation">
+<span class="sig-name descname"><span class="pre">compute_viscous_dissipation</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">u</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../_modules/implementations/problem_classes/RayleighBenard.html#RayleighBenard.compute_viscous_dissipation"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#implementations.problem_classes.RayleighBenard.RayleighBenard.compute_viscous_dissipation" title="Link to this definition">¶</a></dt>
+<dd></dd></dl>
+
+<dl class="py method">
+<dt class="sig sig-object py" id="implementations.problem_classes.RayleighBenard.RayleighBenard.compute_vorticity">
+<span class="sig-name descname"><span class="pre">compute_vorticity</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">u</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../_modules/implementations/problem_classes/RayleighBenard.html#RayleighBenard.compute_vorticity"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#implementations.problem_classes.RayleighBenard.RayleighBenard.compute_vorticity" title="Link to this definition">¶</a></dt>
+<dd></dd></dl>
+
+<dl class="py attribute">
+<dt class="sig sig-object py" id="implementations.problem_classes.RayleighBenard.RayleighBenard.dtype_f">
+<span class="sig-name descname"><span class="pre">dtype_f</span></span><a class="headerlink" href="#implementations.problem_classes.RayleighBenard.RayleighBenard.dtype_f" title="Link to this definition">¶</a></dt>
+<dd><p>alias of <code class="xref py py-class docutils literal notranslate"><span class="pre">imex_mesh</span></code></p>
+</dd></dl>
+
+<dl class="py attribute">
+<dt class="sig sig-object py" id="implementations.problem_classes.RayleighBenard.RayleighBenard.dtype_u">
+<span class="sig-name descname"><span class="pre">dtype_u</span></span><a class="headerlink" href="#implementations.problem_classes.RayleighBenard.RayleighBenard.dtype_u" title="Link to this definition">¶</a></dt>
+<dd><p>alias of <code class="xref py py-class docutils literal notranslate"><span class="pre">mesh</span></code></p>
+</dd></dl>
+
+<dl class="py method">
+<dt class="sig sig-object py" id="implementations.problem_classes.RayleighBenard.RayleighBenard.eval_f">
+<span class="sig-name descname"><span class="pre">eval_f</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">u</span></span></em>, <em class="sig-param"><span class="o"><span class="pre">*</span></span><span class="n"><span class="pre">args</span></span></em>, <em class="sig-param"><span class="o"><span class="pre">**</span></span><span class="n"><span class="pre">kwargs</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../_modules/implementations/problem_classes/RayleighBenard.html#RayleighBenard.eval_f"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#implementations.problem_classes.RayleighBenard.RayleighBenard.eval_f" title="Link to this definition">¶</a></dt>
+<dd><p>Abstract interface to RHS computation of the ODE</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><ul class="simple">
+<li><p><strong>u</strong> (<em>dtype_u</em>) – Current values.</p></li>
+<li><p><strong>t</strong> (<em>float</em>) – Current time.</p></li>
+</ul>
+</dd>
+<dt class="field-even">Returns<span class="colon">:</span></dt>
+<dd class="field-even"><p><strong>f</strong> – The RHS values.</p>
+</dd>
+<dt class="field-odd">Return type<span class="colon">:</span></dt>
+<dd class="field-odd"><p>dtype_f</p>
+</dd>
+</dl>
+</dd></dl>
+
+<dl class="py method">
+<dt class="sig sig-object py" id="implementations.problem_classes.RayleighBenard.RayleighBenard.get_fig">
+<span class="sig-name descname"><span class="pre">get_fig</span></span><span class="sig-paren">(</span><span class="sig-paren">)</span><a class="reference internal" href="../_modules/implementations/problem_classes/RayleighBenard.html#RayleighBenard.get_fig"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#implementations.problem_classes.RayleighBenard.RayleighBenard.get_fig" title="Link to this definition">¶</a></dt>
+<dd><p>Get a figure suitable to plot the solution of this problem</p>
+<dl class="field-list simple">
+<dt class="field-odd">Returns<span class="colon">:</span></dt>
+<dd class="field-odd"><p><strong>self.fig</strong></p>
+</dd>
+<dt class="field-even">Return type<span class="colon">:</span></dt>
+<dd class="field-even"><p>matplotlib.pyplot.figure.Figure</p>
+</dd>
+</dl>
+</dd></dl>
+
+<dl class="py method">
+<dt class="sig sig-object py" id="implementations.problem_classes.RayleighBenard.RayleighBenard.plot">
+<span class="sig-name descname"><span class="pre">plot</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">u</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">t</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">fig</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">quantity</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">'T'</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../_modules/implementations/problem_classes/RayleighBenard.html#RayleighBenard.plot"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#implementations.problem_classes.RayleighBenard.RayleighBenard.plot" title="Link to this definition">¶</a></dt>
+<dd><p>Plot the solution.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><ul class="simple">
+<li><p><strong>u</strong> (<em>dtype_u</em>) – Solution to be plotted</p></li>
+<li><p><strong>t</strong> (<em>float</em>) – Time to display at the top of the figure</p></li>
+<li><p><strong>fig</strong> (<em>matplotlib.pyplot.figure.Figure</em>) – Figure with the same structure as a figure generated by <cite>self.get_fig</cite>. If none is supplied, a new figure will be generated.</p></li>
+<li><p><strong>quantity</strong> (<em>(</em><em>str</em><em>)</em>) – quantity you want to plot</p></li>
+</ul>
+</dd>
+<dt class="field-even">Return type<span class="colon">:</span></dt>
+<dd class="field-even"><p>None</p>
+</dd>
+</dl>
+</dd></dl>
+
+<dl class="py method">
+<dt class="sig sig-object py" id="implementations.problem_classes.RayleighBenard.RayleighBenard.u_exact">
+<span class="sig-name descname"><span class="pre">u_exact</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">t</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">0</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">noise_level</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">0.001</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">seed</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">99</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../_modules/implementations/problem_classes/RayleighBenard.html#RayleighBenard.u_exact"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#implementations.problem_classes.RayleighBenard.RayleighBenard.u_exact" title="Link to this definition">¶</a></dt>
+<dd></dd></dl>
+
+</dd></dl>
+
+</section>
+
+
+            <div class="clearer"></div>
+          </div>
+        </div>
+      </div>
+      <div class="sphinxsidebar" role="navigation" aria-label="Main">
+        <div class="sphinxsidebarwrapper">
+  <div>
+    <h3><a href="../index.html">Table of Contents</a></h3>
+    <ul>
+<li><a class="reference internal" href="#">implementations.problem_classes.RayleighBenard module</a><ul>
+<li><a class="reference internal" href="#implementations.problem_classes.RayleighBenard.CFLLimit"><code class="docutils literal notranslate"><span class="pre">CFLLimit</span></code></a><ul>
+<li><a class="reference internal" href="#implementations.problem_classes.RayleighBenard.CFLLimit.compute_max_step_size"><code class="docutils literal notranslate"><span class="pre">compute_max_step_size()</span></code></a></li>
+<li><a class="reference internal" href="#implementations.problem_classes.RayleighBenard.CFLLimit.dependencies"><code class="docutils literal notranslate"><span class="pre">dependencies()</span></code></a></li>
+<li><a class="reference internal" href="#implementations.problem_classes.RayleighBenard.CFLLimit.get_new_step_size"><code class="docutils literal notranslate"><span class="pre">get_new_step_size()</span></code></a></li>
+<li><a class="reference internal" href="#implementations.problem_classes.RayleighBenard.CFLLimit.setup"><code class="docutils literal notranslate"><span class="pre">setup()</span></code></a></li>
+<li><a class="reference internal" href="#implementations.problem_classes.RayleighBenard.CFLLimit.setup_status_variables"><code class="docutils literal notranslate"><span class="pre">setup_status_variables()</span></code></a></li>
+</ul>
+</li>
+<li><a class="reference internal" href="#implementations.problem_classes.RayleighBenard.LogAnalysisVariables"><code class="docutils literal notranslate"><span class="pre">LogAnalysisVariables</span></code></a><ul>
+<li><a class="reference internal" href="#implementations.problem_classes.RayleighBenard.LogAnalysisVariables.post_step"><code class="docutils literal notranslate"><span class="pre">post_step()</span></code></a></li>
+</ul>
+</li>
+<li><a class="reference internal" href="#implementations.problem_classes.RayleighBenard.LogCFL"><code class="docutils literal notranslate"><span class="pre">LogCFL</span></code></a><ul>
+<li><a class="reference internal" href="#implementations.problem_classes.RayleighBenard.LogCFL.post_step"><code class="docutils literal notranslate"><span class="pre">post_step()</span></code></a></li>
+</ul>
+</li>
+<li><a class="reference internal" href="#implementations.problem_classes.RayleighBenard.RayleighBenard"><code class="docutils literal notranslate"><span class="pre">RayleighBenard</span></code></a><ul>
+<li><a class="reference internal" href="#implementations.problem_classes.RayleighBenard.RayleighBenard.compute_Nusselt_numbers"><code class="docutils literal notranslate"><span class="pre">compute_Nusselt_numbers()</span></code></a></li>
+<li><a class="reference internal" href="#implementations.problem_classes.RayleighBenard.RayleighBenard.compute_buoyancy_generation"><code class="docutils literal notranslate"><span class="pre">compute_buoyancy_generation()</span></code></a></li>
+<li><a class="reference internal" href="#implementations.problem_classes.RayleighBenard.RayleighBenard.compute_viscous_dissipation"><code class="docutils literal notranslate"><span class="pre">compute_viscous_dissipation()</span></code></a></li>
+<li><a class="reference internal" href="#implementations.problem_classes.RayleighBenard.RayleighBenard.compute_vorticity"><code class="docutils literal notranslate"><span class="pre">compute_vorticity()</span></code></a></li>
+<li><a class="reference internal" href="#implementations.problem_classes.RayleighBenard.RayleighBenard.dtype_f"><code class="docutils literal notranslate"><span class="pre">dtype_f</span></code></a></li>
+<li><a class="reference internal" href="#implementations.problem_classes.RayleighBenard.RayleighBenard.dtype_u"><code class="docutils literal notranslate"><span class="pre">dtype_u</span></code></a></li>
+<li><a class="reference internal" href="#implementations.problem_classes.RayleighBenard.RayleighBenard.eval_f"><code class="docutils literal notranslate"><span class="pre">eval_f()</span></code></a></li>
+<li><a class="reference internal" href="#implementations.problem_classes.RayleighBenard.RayleighBenard.get_fig"><code class="docutils literal notranslate"><span class="pre">get_fig()</span></code></a></li>
+<li><a class="reference internal" href="#implementations.problem_classes.RayleighBenard.RayleighBenard.plot"><code class="docutils literal notranslate"><span class="pre">plot()</span></code></a></li>
+<li><a class="reference internal" href="#implementations.problem_classes.RayleighBenard.RayleighBenard.u_exact"><code class="docutils literal notranslate"><span class="pre">u_exact()</span></code></a></li>
+</ul>
+</li>
+</ul>
+</li>
+</ul>
+
+  </div>
+  <div>
+    <h4>Previous topic</h4>
+    <p class="topless"><a href="implementations.problem_classes.Quench.html"
+                          title="previous chapter">implementations.problem_classes.Quench module</a></p>
+  </div>
+  <div>
+    <h4>Next topic</h4>
+    <p class="topless"><a href="implementations.problem_classes.TestEquation_0D.html"
+                          title="next chapter">implementations.problem_classes.TestEquation_0D module</a></p>
+  </div>
+  <div role="note" aria-label="source link">
+    <h3>This Page</h3>
+    <ul class="this-page-menu">
+      <li><a href="../_sources/pySDC/implementations.problem_classes.RayleighBenard.rst.txt"
+            rel="nofollow">Show Source</a></li>
+    </ul>
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+<script>document.getElementById('searchbox').style.display = "block"</script>
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+             >modules</a> |</li>
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+             >previous</a> |</li>
+        <li class="nav-item nav-item-0"><a href="../index.html">pySDC 5.5.1 documentation</a> &#187;</li>
+          <li class="nav-item nav-item-1"><a href="implementations.html" >implementations package</a> &#187;</li>
+          <li class="nav-item nav-item-2"><a href="implementations.problem_classes.html" >implementations.problem_classes package</a> &#187;</li>
+        <li class="nav-item nav-item-this"><a href="">implementations.problem_classes.RayleighBenard module</a></li> 
+      </ul>
+    </div>
+    <div class="footer" role="contentinfo">
+    &#169; Copyright 2024, Robert Speck.
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diff --git a/pySDC/implementations.problem_classes.TestEquation_0D.html b/pySDC/implementations.problem_classes.TestEquation_0D.html
index fdfdf7dd55..0a69bbbb65 100644
--- a/pySDC/implementations.problem_classes.TestEquation_0D.html
+++ b/pySDC/implementations.problem_classes.TestEquation_0D.html
@@ -18,7 +18,7 @@
     <link rel="index" title="Index" href="../genindex.html" />
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     <link rel="next" title="implementations.problem_classes.Van_der_Pol_implicit module" href="implementations.problem_classes.Van_der_Pol_implicit.html" />
-    <link rel="prev" title="implementations.problem_classes.Quench module" href="implementations.problem_classes.Quench.html" /> 
+    <link rel="prev" title="implementations.problem_classes.RayleighBenard module" href="implementations.problem_classes.RayleighBenard.html" /> 
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@@ -33,7 +33,7 @@ <h3>Navigation</h3>
           <a href="implementations.problem_classes.Van_der_Pol_implicit.html" title="implementations.problem_classes.Van_der_Pol_implicit module"
              accesskey="N">next</a> |</li>
         <li class="right" >
-          <a href="implementations.problem_classes.Quench.html" title="implementations.problem_classes.Quench module"
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@@ -201,8 +201,8 @@ <h3><a href="../index.html">Table of Contents</a></h3>
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-    <p class="topless"><a href="implementations.problem_classes.Quench.html"
-                          title="previous chapter">implementations.problem_classes.Quench module</a></p>
+    <p class="topless"><a href="implementations.problem_classes.RayleighBenard.html"
+                          title="previous chapter">implementations.problem_classes.RayleighBenard module</a></p>
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@@ -243,7 +243,7 @@ <h3>Navigation</h3>
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              >previous</a> |</li>
         <li class="nav-item nav-item-0"><a href="../index.html">pySDC 5.5.1 documentation</a> &#187;</li>
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diff --git a/pySDC/implementations.problem_classes.generic_ND_FD.html b/pySDC/implementations.problem_classes.generic_ND_FD.html
index 8c6199b4d4..f42d6bd070 100644
--- a/pySDC/implementations.problem_classes.generic_ND_FD.html
+++ b/pySDC/implementations.problem_classes.generic_ND_FD.html
@@ -17,7 +17,7 @@
     
     <link rel="index" title="Index" href="../genindex.html" />
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-    <link rel="next" title="implementations.problem_classes.nonlinear_ODE_1 module" href="implementations.problem_classes.nonlinear_ODE_1.html" />
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         <li class="right" >
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+          <a href="implementations.problem_classes.generic_spectral.html" title="implementations.problem_classes.generic_spectral module"
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-    <p class="topless"><a href="implementations.problem_classes.nonlinear_ODE_1.html"
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+    <p class="topless"><a href="implementations.problem_classes.generic_spectral.html"
+                          title="next chapter">implementations.problem_classes.generic_spectral module</a></p>
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@@ -281,7 +281,7 @@ <h3>Navigation</h3>
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+          <a href="implementations.problem_classes.generic_spectral.html" title="implementations.problem_classes.generic_spectral module"
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         <li class="right" >
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diff --git a/pySDC/implementations.problem_classes.generic_spectral.html b/pySDC/implementations.problem_classes.generic_spectral.html
new file mode 100644
index 0000000000..8c57ccce5e
--- /dev/null
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+  <section id="module-implementations.problem_classes.generic_spectral">
+<span id="implementations-problem-classes-generic-spectral-module"></span><h1>implementations.problem_classes.generic_spectral module<a class="headerlink" href="#module-implementations.problem_classes.generic_spectral" title="Link to this heading">¶</a></h1>
+<dl class="py class">
+<dt class="sig sig-object py" id="implementations.problem_classes.generic_spectral.GenericSpectralLinear">
+<em class="property"><span class="pre">class</span><span class="w"> </span></em><span class="sig-name descname"><span class="pre">GenericSpectralLinear</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">bases</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">components</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">comm</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">Dirichlet_recombination</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">True</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">left_preconditioner</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">True</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">solver_type</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">'cached_direct'</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">solver_args</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">useGPU</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">False</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">max_cached_factorizations</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">12</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">spectral_space</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">True</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">real_spectral_coefficients</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">False</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">debug</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">False</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../_modules/implementations/problem_classes/generic_spectral.html#GenericSpectralLinear"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#implementations.problem_classes.generic_spectral.GenericSpectralLinear" title="Link to this definition">¶</a></dt>
+<dd><p>Bases: <code class="xref py py-class docutils literal notranslate"><span class="pre">Problem</span></code></p>
+<p>Generic class to solve problems of the form M u_t + L u = y, with mass matrix M, linear operator L and some right
+hand side y using spectral methods.
+L may contain algebraic conditions, as long as (M + dt L) is invertible.</p>
+<p>Note that the <cite>__getattr__</cite> method is overloaded to pass requests on to the spectral helper if they are not
+attributes of this class itself. For instance, you can add a BC by calling <cite>self.spectral.add_BC</cite> or equivalently
+<cite>self.add_BC</cite>.</p>
+<p>You can port problems derived from this more or less seamlessly to GPU by using the numerical libraries that are
+class attributes of the spectral helper. This class will automatically switch the datatype using the <cite>setup_GPU</cite> class method.</p>
+<dl class="py attribute">
+<dt class="sig sig-object py" id="implementations.problem_classes.generic_spectral.GenericSpectralLinear.spectral">
+<span class="sig-name descname"><span class="pre">spectral</span></span><a class="headerlink" href="#implementations.problem_classes.generic_spectral.GenericSpectralLinear.spectral" title="Link to this definition">¶</a></dt>
+<dd><p>Spectral helper</p>
+<dl class="field-list simple">
+<dt class="field-odd">Type<span class="colon">:</span></dt>
+<dd class="field-odd"><p>pySDC.helpers.spectral_helper.SpectralHelper</p>
+</dd>
+</dl>
+</dd></dl>
+
+<dl class="py attribute">
+<dt class="sig sig-object py" id="implementations.problem_classes.generic_spectral.GenericSpectralLinear.work_counters">
+<span class="sig-name descname"><span class="pre">work_counters</span></span><a class="headerlink" href="#implementations.problem_classes.generic_spectral.GenericSpectralLinear.work_counters" title="Link to this definition">¶</a></dt>
+<dd><p>Dictionary for counting work</p>
+<dl class="field-list simple">
+<dt class="field-odd">Type<span class="colon">:</span></dt>
+<dd class="field-odd"><p>dict</p>
+</dd>
+</dl>
+</dd></dl>
+
+<dl class="py attribute">
+<dt class="sig sig-object py" id="implementations.problem_classes.generic_spectral.GenericSpectralLinear.cached_factorizations">
+<span class="sig-name descname"><span class="pre">cached_factorizations</span></span><a class="headerlink" href="#implementations.problem_classes.generic_spectral.GenericSpectralLinear.cached_factorizations" title="Link to this definition">¶</a></dt>
+<dd><p>Dictionary of cached matrix factorizations for solving</p>
+<dl class="field-list simple">
+<dt class="field-odd">Type<span class="colon">:</span></dt>
+<dd class="field-odd"><p>dict</p>
+</dd>
+</dl>
+</dd></dl>
+
+<dl class="py attribute">
+<dt class="sig sig-object py" id="implementations.problem_classes.generic_spectral.GenericSpectralLinear.L">
+<span class="sig-name descname"><span class="pre">L</span></span><a class="headerlink" href="#implementations.problem_classes.generic_spectral.GenericSpectralLinear.L" title="Link to this definition">¶</a></dt>
+<dd><p>Linear operator</p>
+<dl class="field-list simple">
+<dt class="field-odd">Type<span class="colon">:</span></dt>
+<dd class="field-odd"><p>sparse matrix</p>
+</dd>
+</dl>
+</dd></dl>
+
+<dl class="py attribute">
+<dt class="sig sig-object py" id="implementations.problem_classes.generic_spectral.GenericSpectralLinear.M">
+<span class="sig-name descname"><span class="pre">M</span></span><a class="headerlink" href="#implementations.problem_classes.generic_spectral.GenericSpectralLinear.M" title="Link to this definition">¶</a></dt>
+<dd><p>Mass matrix</p>
+<dl class="field-list simple">
+<dt class="field-odd">Type<span class="colon">:</span></dt>
+<dd class="field-odd"><p>sparse matrix</p>
+</dd>
+</dl>
+</dd></dl>
+
+<dl class="py attribute">
+<dt class="sig sig-object py" id="implementations.problem_classes.generic_spectral.GenericSpectralLinear.diff_mask">
+<span class="sig-name descname"><span class="pre">diff_mask</span></span><a class="headerlink" href="#implementations.problem_classes.generic_spectral.GenericSpectralLinear.diff_mask" title="Link to this definition">¶</a></dt>
+<dd><p>Mask for separating differential and algebraic terms</p>
+<dl class="field-list simple">
+<dt class="field-odd">Type<span class="colon">:</span></dt>
+<dd class="field-odd"><p>list</p>
+</dd>
+</dl>
+</dd></dl>
+
+<dl class="py attribute">
+<dt class="sig sig-object py" id="implementations.problem_classes.generic_spectral.GenericSpectralLinear.Pl">
+<span class="sig-name descname"><span class="pre">Pl</span></span><a class="headerlink" href="#implementations.problem_classes.generic_spectral.GenericSpectralLinear.Pl" title="Link to this definition">¶</a></dt>
+<dd><p>Left preconditioner</p>
+<dl class="field-list simple">
+<dt class="field-odd">Type<span class="colon">:</span></dt>
+<dd class="field-odd"><p>sparse matrix</p>
+</dd>
+</dl>
+</dd></dl>
+
+<dl class="py attribute">
+<dt class="sig sig-object py" id="implementations.problem_classes.generic_spectral.GenericSpectralLinear.Pr">
+<span class="sig-name descname"><span class="pre">Pr</span></span><a class="headerlink" href="#implementations.problem_classes.generic_spectral.GenericSpectralLinear.Pr" title="Link to this definition">¶</a></dt>
+<dd><p>Right preconditioner</p>
+<dl class="field-list simple">
+<dt class="field-odd">Type<span class="colon">:</span></dt>
+<dd class="field-odd"><p>sparse matrix</p>
+</dd>
+</dl>
+</dd></dl>
+
+<dl class="py method">
+<dt class="sig sig-object py" id="implementations.problem_classes.generic_spectral.GenericSpectralLinear.setup_GPU">
+<em class="property"><span class="pre">classmethod</span><span class="w"> </span></em><span class="sig-name descname"><span class="pre">setup_GPU</span></span><span class="sig-paren">(</span><span class="sig-paren">)</span><a class="reference internal" href="../_modules/implementations/problem_classes/generic_spectral.html#GenericSpectralLinear.setup_GPU"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#implementations.problem_classes.generic_spectral.GenericSpectralLinear.setup_GPU" title="Link to this definition">¶</a></dt>
+<dd><p>switch to GPU modules</p>
+</dd></dl>
+
+<dl class="py method">
+<dt class="sig sig-object py" id="implementations.problem_classes.generic_spectral.GenericSpectralLinear.setup_L">
+<span class="sig-name descname"><span class="pre">setup_L</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">LHS</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../_modules/implementations/problem_classes/generic_spectral.html#GenericSpectralLinear.setup_L"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#implementations.problem_classes.generic_spectral.GenericSpectralLinear.setup_L" title="Link to this definition">¶</a></dt>
+<dd><p>Setup the left hand side of the linear operator L and store it in <code class="docutils literal notranslate"><span class="pre">self.L</span></code>.</p>
+<p>The argument is meant to be a dictionary with the line you want to write the equation in as the key and the relationship between components as another dictionary. For instance, you can add an algebraic condition capturing a first derivative relationship between u and ux as follows:</p>
+<p><code class="docutils literal notranslate"><span class="pre">`</span>
+<span class="pre">Dx</span> <span class="pre">=</span> <span class="pre">self.get_differentiation_matrix(axes=(0,))</span>
+<span class="pre">I</span> <span class="pre">=</span> <span class="pre">self.get_Id()</span>
+<span class="pre">LHS</span> <span class="pre">=</span> <span class="pre">{'ux':</span> <span class="pre">{'u':</span> <span class="pre">Dx,</span> <span class="pre">'ux':</span> <span class="pre">-I}}</span>
+<span class="pre">self.setup_L(LHS)</span>
+<span class="pre">`</span></code></p>
+<p>If you put zero as right hand side for the solver in the line for ux, ux will contain the x-derivative of u afterwards.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><p><strong>LHS</strong> (<em>dict</em>) – Dictionary containing the equations.</p>
+</dd>
+</dl>
+</dd></dl>
+
+<dl class="py method">
+<dt class="sig sig-object py" id="implementations.problem_classes.generic_spectral.GenericSpectralLinear.setup_M">
+<span class="sig-name descname"><span class="pre">setup_M</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">LHS</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../_modules/implementations/problem_classes/generic_spectral.html#GenericSpectralLinear.setup_M"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#implementations.problem_classes.generic_spectral.GenericSpectralLinear.setup_M" title="Link to this definition">¶</a></dt>
+<dd><p>Setup mass matrix, see documentation of <code class="docutils literal notranslate"><span class="pre">GenericSpectralLinear.setup_L</span></code>.</p>
+</dd></dl>
+
+<dl class="py method">
+<dt class="sig sig-object py" id="implementations.problem_classes.generic_spectral.GenericSpectralLinear.setup_preconditioner">
+<span class="sig-name descname"><span class="pre">setup_preconditioner</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">Dirichlet_recombination</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">True</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">left_preconditioner</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">True</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../_modules/implementations/problem_classes/generic_spectral.html#GenericSpectralLinear.setup_preconditioner"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#implementations.problem_classes.generic_spectral.GenericSpectralLinear.setup_preconditioner" title="Link to this definition">¶</a></dt>
+<dd><p>Get left and right preconditioners.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><ul class="simple">
+<li><p><strong>Dirichlet_recombination</strong> (<em>bool</em>) – Basis conversion for right preconditioner. Useful for Chebychev and Ultraspherical methods. 10/10 would recommend.</p></li>
+<li><p><strong>left_preconditioner</strong> (<em>bool</em>) – If True, it will interleave the variables and reverse the Kronecker product</p></li>
+</ul>
+</dd>
+</dl>
+</dd></dl>
+
+<dl class="py method">
+<dt class="sig sig-object py" id="implementations.problem_classes.generic_spectral.GenericSpectralLinear.solve_system">
+<span class="sig-name descname"><span class="pre">solve_system</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">rhs</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">dt</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">u0</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="o"><span class="pre">*</span></span><span class="n"><span class="pre">args</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">skip_itransform</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">False</span></span></em>, <em class="sig-param"><span class="o"><span class="pre">**</span></span><span class="n"><span class="pre">kwargs</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../_modules/implementations/problem_classes/generic_spectral.html#GenericSpectralLinear.solve_system"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#implementations.problem_classes.generic_spectral.GenericSpectralLinear.solve_system" title="Link to this definition">¶</a></dt>
+<dd><p>Do an implicit Euler step to solve M u_t + Lu = rhs, with M the mass matrix and L the linear operator as setup by
+<code class="docutils literal notranslate"><span class="pre">GenericSpectralLinear.setup_L</span></code> and <code class="docutils literal notranslate"><span class="pre">GenericSpectralLinear.setup_M</span></code>.</p>
+<p>The implicit Euler step is (M - dt L) u = M rhs. Note that M need not be invertible as long as (M + dt*L) is.
+This means solving with dt=0 to mimic explicit methods does not work for all problems, in particular simple DAEs.</p>
+<p>Note that by putting M rhs on the right hand side, this function can only solve algebraic conditions equal to
+zero. If you want something else, it should be easy to overload this function.</p>
+</dd></dl>
+
+</dd></dl>
+
+<dl class="py function">
+<dt class="sig sig-object py" id="implementations.problem_classes.generic_spectral.compute_residual_DAE">
+<span class="sig-name descname"><span class="pre">compute_residual_DAE</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">self</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">stage</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">''</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../_modules/implementations/problem_classes/generic_spectral.html#compute_residual_DAE"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#implementations.problem_classes.generic_spectral.compute_residual_DAE" title="Link to this definition">¶</a></dt>
+<dd><p>Computation of the residual that does not add u_0 - u_m in algebraic equations.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><p><strong>stage</strong> (<em>str</em>) – The current stage of the step the level belongs to</p>
+</dd>
+</dl>
+</dd></dl>
+
+<dl class="py function">
+<dt class="sig sig-object py" id="implementations.problem_classes.generic_spectral.compute_residual_DAE_MPI">
+<span class="sig-name descname"><span class="pre">compute_residual_DAE_MPI</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">self</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">stage</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../_modules/implementations/problem_classes/generic_spectral.html#compute_residual_DAE_MPI"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#implementations.problem_classes.generic_spectral.compute_residual_DAE_MPI" title="Link to this definition">¶</a></dt>
+<dd><p>Computation of the residual using the collocation matrix Q</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><p><strong>stage</strong> (<em>str</em>) – The current stage of the step the level belongs to</p>
+</dd>
+</dl>
+</dd></dl>
+
+</section>
+
+
+            <div class="clearer"></div>
+          </div>
+        </div>
+      </div>
+      <div class="sphinxsidebar" role="navigation" aria-label="Main">
+        <div class="sphinxsidebarwrapper">
+  <div>
+    <h3><a href="../index.html">Table of Contents</a></h3>
+    <ul>
+<li><a class="reference internal" href="#">implementations.problem_classes.generic_spectral module</a><ul>
+<li><a class="reference internal" href="#implementations.problem_classes.generic_spectral.GenericSpectralLinear"><code class="docutils literal notranslate"><span class="pre">GenericSpectralLinear</span></code></a><ul>
+<li><a class="reference internal" href="#implementations.problem_classes.generic_spectral.GenericSpectralLinear.spectral"><code class="docutils literal notranslate"><span class="pre">spectral</span></code></a></li>
+<li><a class="reference internal" href="#implementations.problem_classes.generic_spectral.GenericSpectralLinear.work_counters"><code class="docutils literal notranslate"><span class="pre">work_counters</span></code></a></li>
+<li><a class="reference internal" href="#implementations.problem_classes.generic_spectral.GenericSpectralLinear.cached_factorizations"><code class="docutils literal notranslate"><span class="pre">cached_factorizations</span></code></a></li>
+<li><a class="reference internal" href="#implementations.problem_classes.generic_spectral.GenericSpectralLinear.L"><code class="docutils literal notranslate"><span class="pre">L</span></code></a></li>
+<li><a class="reference internal" href="#implementations.problem_classes.generic_spectral.GenericSpectralLinear.M"><code class="docutils literal notranslate"><span class="pre">M</span></code></a></li>
+<li><a class="reference internal" href="#implementations.problem_classes.generic_spectral.GenericSpectralLinear.diff_mask"><code class="docutils literal notranslate"><span class="pre">diff_mask</span></code></a></li>
+<li><a class="reference internal" href="#implementations.problem_classes.generic_spectral.GenericSpectralLinear.Pl"><code class="docutils literal notranslate"><span class="pre">Pl</span></code></a></li>
+<li><a class="reference internal" href="#implementations.problem_classes.generic_spectral.GenericSpectralLinear.Pr"><code class="docutils literal notranslate"><span class="pre">Pr</span></code></a></li>
+<li><a class="reference internal" href="#implementations.problem_classes.generic_spectral.GenericSpectralLinear.setup_GPU"><code class="docutils literal notranslate"><span class="pre">setup_GPU()</span></code></a></li>
+<li><a class="reference internal" href="#implementations.problem_classes.generic_spectral.GenericSpectralLinear.setup_L"><code class="docutils literal notranslate"><span class="pre">setup_L()</span></code></a></li>
+<li><a class="reference internal" href="#implementations.problem_classes.generic_spectral.GenericSpectralLinear.setup_M"><code class="docutils literal notranslate"><span class="pre">setup_M()</span></code></a></li>
+<li><a class="reference internal" href="#implementations.problem_classes.generic_spectral.GenericSpectralLinear.setup_preconditioner"><code class="docutils literal notranslate"><span class="pre">setup_preconditioner()</span></code></a></li>
+<li><a class="reference internal" href="#implementations.problem_classes.generic_spectral.GenericSpectralLinear.solve_system"><code class="docutils literal notranslate"><span class="pre">solve_system()</span></code></a></li>
+</ul>
+</li>
+<li><a class="reference internal" href="#implementations.problem_classes.generic_spectral.compute_residual_DAE"><code class="docutils literal notranslate"><span class="pre">compute_residual_DAE()</span></code></a></li>
+<li><a class="reference internal" href="#implementations.problem_classes.generic_spectral.compute_residual_DAE_MPI"><code class="docutils literal notranslate"><span class="pre">compute_residual_DAE_MPI()</span></code></a></li>
+</ul>
+</li>
+</ul>
+
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+  <div>
+    <h4>Previous topic</h4>
+    <p class="topless"><a href="implementations.problem_classes.generic_ND_FD.html"
+                          title="previous chapter">implementations.problem_classes.generic_ND_FD module</a></p>
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+    <h4>Next topic</h4>
+    <p class="topless"><a href="implementations.problem_classes.nonlinear_ODE_1.html"
+                          title="next chapter">implementations.problem_classes.nonlinear_ODE_1 module</a></p>
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+    <h3>This Page</h3>
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+      <li><a href="../_sources/pySDC/implementations.problem_classes.generic_spectral.rst.txt"
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+          <li class="nav-item nav-item-1"><a href="implementations.html" >implementations package</a> &#187;</li>
+          <li class="nav-item nav-item-2"><a href="implementations.problem_classes.html" >implementations.problem_classes package</a> &#187;</li>
+        <li class="nav-item nav-item-this"><a href="">implementations.problem_classes.generic_spectral module</a></li> 
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diff --git a/pySDC/implementations.problem_classes.html b/pySDC/implementations.problem_classes.html
index 029669d9f2..2ef5c99bb1 100644
--- a/pySDC/implementations.problem_classes.html
+++ b/pySDC/implementations.problem_classes.html
@@ -148,6 +148,11 @@ <h2>Submodules<a class="headerlink" href="#submodules" title="Link to this headi
 <li class="toctree-l2"><a class="reference internal" href="implementations.problem_classes.BuckConverter.html#implementations.problem_classes.BuckConverter.buck_converter"><code class="docutils literal notranslate"><span class="pre">buck_converter</span></code></a></li>
 </ul>
 </li>
+<li class="toctree-l1"><a class="reference internal" href="implementations.problem_classes.Burgers.html">implementations.problem_classes.Burgers module</a><ul>
+<li class="toctree-l2"><a class="reference internal" href="implementations.problem_classes.Burgers.html#implementations.problem_classes.Burgers.Burgers1D"><code class="docutils literal notranslate"><span class="pre">Burgers1D</span></code></a></li>
+<li class="toctree-l2"><a class="reference internal" href="implementations.problem_classes.Burgers.html#implementations.problem_classes.Burgers.Burgers2D"><code class="docutils literal notranslate"><span class="pre">Burgers2D</span></code></a></li>
+</ul>
+</li>
 <li class="toctree-l1"><a class="reference internal" href="implementations.problem_classes.DiscontinuousTestODE.html">implementations.problem_classes.DiscontinuousTestODE module</a><ul>
 <li class="toctree-l2"><a class="reference internal" href="implementations.problem_classes.DiscontinuousTestODE.html#implementations.problem_classes.DiscontinuousTestODE.DiscontinuousTestODE"><code class="docutils literal notranslate"><span class="pre">DiscontinuousTestODE</span></code></a></li>
 <li class="toctree-l2"><a class="reference internal" href="implementations.problem_classes.DiscontinuousTestODE.html#implementations.problem_classes.DiscontinuousTestODE.ExactDiscontinuousTestODE"><code class="docutils literal notranslate"><span class="pre">ExactDiscontinuousTestODE</span></code></a></li>
@@ -210,6 +215,13 @@ <h2>Submodules<a class="headerlink" href="#submodules" title="Link to this headi
 <li class="toctree-l2"><a class="reference internal" href="implementations.problem_classes.HeatEquation_2D_PETSc_forced.html#implementations.problem_classes.HeatEquation_2D_PETSc_forced.heat2d_petsc_forced"><code class="docutils literal notranslate"><span class="pre">heat2d_petsc_forced</span></code></a></li>
 </ul>
 </li>
+<li class="toctree-l1"><a class="reference internal" href="implementations.problem_classes.HeatEquation_Chebychev.html">implementations.problem_classes.HeatEquation_Chebychev module</a><ul>
+<li class="toctree-l2"><a class="reference internal" href="implementations.problem_classes.HeatEquation_Chebychev.html#implementations.problem_classes.HeatEquation_Chebychev.Heat1DChebychev"><code class="docutils literal notranslate"><span class="pre">Heat1DChebychev</span></code></a></li>
+<li class="toctree-l2"><a class="reference internal" href="implementations.problem_classes.HeatEquation_Chebychev.html#implementations.problem_classes.HeatEquation_Chebychev.Heat1DUltraspherical"><code class="docutils literal notranslate"><span class="pre">Heat1DUltraspherical</span></code></a></li>
+<li class="toctree-l2"><a class="reference internal" href="implementations.problem_classes.HeatEquation_Chebychev.html#implementations.problem_classes.HeatEquation_Chebychev.Heat2DChebychev"><code class="docutils literal notranslate"><span class="pre">Heat2DChebychev</span></code></a></li>
+<li class="toctree-l2"><a class="reference internal" href="implementations.problem_classes.HeatEquation_Chebychev.html#implementations.problem_classes.HeatEquation_Chebychev.Heat2DUltraspherical"><code class="docutils literal notranslate"><span class="pre">Heat2DUltraspherical</span></code></a></li>
+</ul>
+</li>
 <li class="toctree-l1"><a class="reference internal" href="implementations.problem_classes.HeatEquation_ND_FD.html">implementations.problem_classes.HeatEquation_ND_FD module</a><ul>
 <li class="toctree-l2"><a class="reference internal" href="implementations.problem_classes.HeatEquation_ND_FD.html#implementations.problem_classes.HeatEquation_ND_FD.heatNd_forced"><code class="docutils literal notranslate"><span class="pre">heatNd_forced</span></code></a></li>
 <li class="toctree-l2"><a class="reference internal" href="implementations.problem_classes.HeatEquation_ND_FD.html#implementations.problem_classes.HeatEquation_ND_FD.heatNd_unforced"><code class="docutils literal notranslate"><span class="pre">heatNd_unforced</span></code></a></li>
@@ -254,6 +266,13 @@ <h2>Submodules<a class="headerlink" href="#submodules" title="Link to this headi
 <li class="toctree-l2"><a class="reference internal" href="implementations.problem_classes.Quench.html#implementations.problem_classes.Quench.QuenchIMEX"><code class="docutils literal notranslate"><span class="pre">QuenchIMEX</span></code></a></li>
 </ul>
 </li>
+<li class="toctree-l1"><a class="reference internal" href="implementations.problem_classes.RayleighBenard.html">implementations.problem_classes.RayleighBenard module</a><ul>
+<li class="toctree-l2"><a class="reference internal" href="implementations.problem_classes.RayleighBenard.html#implementations.problem_classes.RayleighBenard.CFLLimit"><code class="docutils literal notranslate"><span class="pre">CFLLimit</span></code></a></li>
+<li class="toctree-l2"><a class="reference internal" href="implementations.problem_classes.RayleighBenard.html#implementations.problem_classes.RayleighBenard.LogAnalysisVariables"><code class="docutils literal notranslate"><span class="pre">LogAnalysisVariables</span></code></a></li>
+<li class="toctree-l2"><a class="reference internal" href="implementations.problem_classes.RayleighBenard.html#implementations.problem_classes.RayleighBenard.LogCFL"><code class="docutils literal notranslate"><span class="pre">LogCFL</span></code></a></li>
+<li class="toctree-l2"><a class="reference internal" href="implementations.problem_classes.RayleighBenard.html#implementations.problem_classes.RayleighBenard.RayleighBenard"><code class="docutils literal notranslate"><span class="pre">RayleighBenard</span></code></a></li>
+</ul>
+</li>
 <li class="toctree-l1"><a class="reference internal" href="implementations.problem_classes.TestEquation_0D.html">implementations.problem_classes.TestEquation_0D module</a><ul>
 <li class="toctree-l2"><a class="reference internal" href="implementations.problem_classes.TestEquation_0D.html#implementations.problem_classes.TestEquation_0D.testequation0d"><code class="docutils literal notranslate"><span class="pre">testequation0d</span></code></a></li>
 </ul>
@@ -275,6 +294,12 @@ <h2>Submodules<a class="headerlink" href="#submodules" title="Link to this headi
 <li class="toctree-l2"><a class="reference internal" href="implementations.problem_classes.generic_ND_FD.html#implementations.problem_classes.generic_ND_FD.GenericNDimFinDiff"><code class="docutils literal notranslate"><span class="pre">GenericNDimFinDiff</span></code></a></li>
 </ul>
 </li>
+<li class="toctree-l1"><a class="reference internal" href="implementations.problem_classes.generic_spectral.html">implementations.problem_classes.generic_spectral module</a><ul>
+<li class="toctree-l2"><a class="reference internal" href="implementations.problem_classes.generic_spectral.html#implementations.problem_classes.generic_spectral.GenericSpectralLinear"><code class="docutils literal notranslate"><span class="pre">GenericSpectralLinear</span></code></a></li>
+<li class="toctree-l2"><a class="reference internal" href="implementations.problem_classes.generic_spectral.html#implementations.problem_classes.generic_spectral.compute_residual_DAE"><code class="docutils literal notranslate"><span class="pre">compute_residual_DAE()</span></code></a></li>
+<li class="toctree-l2"><a class="reference internal" href="implementations.problem_classes.generic_spectral.html#implementations.problem_classes.generic_spectral.compute_residual_DAE_MPI"><code class="docutils literal notranslate"><span class="pre">compute_residual_DAE_MPI()</span></code></a></li>
+</ul>
+</li>
 <li class="toctree-l1"><a class="reference internal" href="implementations.problem_classes.nonlinear_ODE_1.html">implementations.problem_classes.nonlinear_ODE_1 module</a><ul>
 <li class="toctree-l2"><a class="reference internal" href="implementations.problem_classes.nonlinear_ODE_1.html#implementations.problem_classes.nonlinear_ODE_1.nonlinear_ODE_1"><code class="docutils literal notranslate"><span class="pre">nonlinear_ODE_1</span></code></a></li>
 </ul>
diff --git a/pySDC/implementations.problem_classes.nonlinear_ODE_1.html b/pySDC/implementations.problem_classes.nonlinear_ODE_1.html
index 0afb51ce0c..d0a2237584 100644
--- a/pySDC/implementations.problem_classes.nonlinear_ODE_1.html
+++ b/pySDC/implementations.problem_classes.nonlinear_ODE_1.html
@@ -18,7 +18,7 @@
     <link rel="index" title="Index" href="../genindex.html" />
     <link rel="search" title="Search" href="../search.html" />
     <link rel="next" title="implementations.problem_classes.odeScalar module" href="implementations.problem_classes.odeScalar.html" />
-    <link rel="prev" title="implementations.problem_classes.generic_ND_FD module" href="implementations.problem_classes.generic_ND_FD.html" /> 
+    <link rel="prev" title="implementations.problem_classes.generic_spectral module" href="implementations.problem_classes.generic_spectral.html" /> 
   </head><body>
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       <h3>Navigation</h3>
@@ -33,7 +33,7 @@ <h3>Navigation</h3>
           <a href="implementations.problem_classes.odeScalar.html" title="implementations.problem_classes.odeScalar module"
              accesskey="N">next</a> |</li>
         <li class="right" >
-          <a href="implementations.problem_classes.generic_ND_FD.html" title="implementations.problem_classes.generic_ND_FD module"
+          <a href="implementations.problem_classes.generic_spectral.html" title="implementations.problem_classes.generic_spectral module"
              accesskey="P">previous</a> |</li>
         <li class="nav-item nav-item-0"><a href="../index.html">pySDC 5.5.1 documentation</a> &#187;</li>
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@@ -196,8 +196,8 @@ <h3><a href="../index.html">Table of Contents</a></h3>
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-    <p class="topless"><a href="implementations.problem_classes.generic_ND_FD.html"
-                          title="previous chapter">implementations.problem_classes.generic_ND_FD module</a></p>
+    <p class="topless"><a href="implementations.problem_classes.generic_spectral.html"
+                          title="previous chapter">implementations.problem_classes.generic_spectral module</a></p>
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@@ -238,7 +238,7 @@ <h3>Navigation</h3>
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         <li class="right" >
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index d6c66bc890..4c1175fbe0 100644
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"implementations.problem_classes.generic_MPIFFT_Laplacian.IMEX_Laplacian_MPIFFT.fft", false]], "fft (nonlinearschroedinger_imex attribute)": [[124, "implementations.problem_classes.NonlinearSchroedinger_MPIFFT.nonlinearschroedinger_imex.fft", false]], "fft_backend (imex_laplacian_mpifft attribute)": [[143, "implementations.problem_classes.generic_MPIFFT_Laplacian.IMEX_Laplacian_MPIFFT.fft_backend", false]], "fft_comm_backend (imex_laplacian_mpifft attribute)": [[143, "implementations.problem_classes.generic_MPIFFT_Laplacian.IMEX_Laplacian_MPIFFT.fft_comm_backend", false]], "fft_to_fft (class in implementations.transfer_classes.transfermesh_mpifft)": [[169, "implementations.transfer_classes.TransferMesh_MPIFFT.fft_to_fft", false]], "fields (class in implementations.datatype_classes.particles)": [[89, "implementations.datatype_classes.particles.fields", false]], "fields.electric (class in implementations.datatype_classes.particles)": [[89, "implementations.datatype_classes.particles.fields.electric", false]], "fields.magnetic (class in implementations.datatype_classes.particles)": [[89, "implementations.datatype_classes.particles.fields.magnetic", false]], "figsize() (in module helpers.plot_helper)": [[56, "helpers.plot_helper.figsize", false]], "figsize_by_journal() (in module helpers.plot_helper)": [[56, "helpers.plot_helper.figsize_by_journal", false]], "filter_stats() (in module helpers.stats_helper)": [[60, "helpers.stats_helper.filter_stats", false]], "fine (base_transfer_mass attribute)": [[164, "implementations.transfer_classes.BaseTransfer_mass.base_transfer_mass.fine", false]], "fine (base_transfer_mpi attribute)": [[163, "implementations.transfer_classes.BaseTransferMPI.base_transfer_MPI.fine", false]], "fine (basetransfer attribute)": [[42, "core.base_transfer.BaseTransfer.fine", false]], "fine_prob (spacetransfer attribute)": [[51, "core.space_transfer.SpaceTransfer.fine_prob", false]], "firsttimestep() (bdf2 method)": [[135, "implementations.problem_classes.acoustic_helpers.standard_integrators.bdf2.firsttimestep", false], [141, "implementations.problem_classes.boussinesq_helpers.standard_integrators.bdf2.firsttimestep", false]], "fisher_full (class in implementations.problem_classes.generalizedfisher_1d_petsc)": [[112, "implementations.problem_classes.GeneralizedFisher_1D_PETSc.Fisher_full", false]], "fisher_reaction (class in implementations.problem_classes.generalizedfisher_1d_petsc)": [[112, "implementations.problem_classes.GeneralizedFisher_1D_PETSc.Fisher_reaction", false]], "fix_bc_for_residual (fenics_heat_mass attribute)": [[117, "implementations.problem_classes.HeatEquation_1D_FEniCS_matrix_forced.fenics_heat_mass.fix_bc_for_residual", false]], "fix_bc_for_residual (fenics_heat_mass_timebc attribute)": [[117, "implementations.problem_classes.HeatEquation_1D_FEniCS_matrix_forced.fenics_heat_mass_timebc.fix_bc_for_residual", false]], "fix_residual() (fenics_heat_mass method)": [[117, "implementations.problem_classes.HeatEquation_1D_FEniCS_matrix_forced.fenics_heat_mass.fix_residual", false]], "fold (level attribute)": [[49, "core.level.Level.fold", false]], "formfunction() (fisher_full method)": [[112, "implementations.problem_classes.GeneralizedFisher_1D_PETSc.Fisher_full.formFunction", false]], "formfunction() (fisher_reaction method)": [[112, "implementations.problem_classes.GeneralizedFisher_1D_PETSc.Fisher_reaction.formFunction", false]], "formfunction() (gs_full method)": [[114, "implementations.problem_classes.GrayScott_2D_PETSc_periodic.GS_full.formFunction", false]], "formfunction() (gs_reaction method)": [[114, "implementations.problem_classes.GrayScott_2D_PETSc_periodic.GS_reaction.formFunction", false]], "formjacobian() (fisher_full method)": [[112, "implementations.problem_classes.GeneralizedFisher_1D_PETSc.Fisher_full.formJacobian", false]], "formjacobian() (fisher_reaction method)": [[112, "implementations.problem_classes.GeneralizedFisher_1D_PETSc.Fisher_reaction.formJacobian", false]], "formjacobian() (gs_full method)": [[114, "implementations.problem_classes.GrayScott_2D_PETSc_periodic.GS_full.formJacobian", false]], "formjacobian() (gs_reaction method)": [[114, "implementations.problem_classes.GrayScott_2D_PETSc_periodic.GS_reaction.formJacobian", false]], "forwardeuler (class in implementations.sweeper_classes.runge_kutta)": [[151, "implementations.sweeper_classes.Runge_Kutta.ForwardEuler", false]], "frozenclass (class in helpers.pysdc_helper)": [[58, "helpers.pysdc_helper.FrozenClass", false]], "full_solar_system (class in implementations.problem_classes.fullsolarsystem)": [[110, "implementations.problem_classes.FullSolarSystem.full_solar_system", false]], "g (fenics_heat attribute)": [[117, "implementations.problem_classes.HeatEquation_1D_FEniCS_matrix_forced.fenics_heat.g", false]], "g (fenics_heat_mass attribute)": [[117, "implementations.problem_classes.HeatEquation_1D_FEniCS_matrix_forced.fenics_heat_mass.g", false]], "g (fenics_heat_mass_timebc attribute)": [[117, "implementations.problem_classes.HeatEquation_1D_FEniCS_matrix_forced.fenics_heat_mass_timebc.g", false]], "g (full_solar_system attribute)": [[110, "implementations.problem_classes.FullSolarSystem.full_solar_system.G", false]], "g (outer_solar_system attribute)": [[125, "id0", false], [125, "implementations.problem_classes.OuterSolarSystem.outer_solar_system.G", false]], "g() (protherorobinson method)": [[146, "implementations.problem_classes.odeScalar.ProtheroRobinson.g", false]], "g() (protherorobinsonautonomous method)": [[147, "implementations.problem_classes.odeSystem.ProtheroRobinsonAutonomous.g", false]], "generalized_fisher (class in implementations.problem_classes.generalizedfisher_1d_fd_implicit)": [[111, "implementations.problem_classes.GeneralizedFisher_1D_FD_implicit.generalized_fisher", false]], "generate_description() (in module helpers.setup_helper)": [[59, "helpers.setup_helper.generate_description", false]], "generate_scipy_reference_solution() (problem method)": [[50, "core.problem.Problem.generate_scipy_reference_solution", false]], "generate_starting_values() (adamsmoultonimplicit2step method)": [[150, "implementations.sweeper_classes.Multistep.AdamsMoultonImplicit2Step.generate_starting_values", false]], "generate_starting_values() (multistep method)": [[150, "implementations.sweeper_classes.Multistep.MultiStep.generate_starting_values", false]], "generator (ark548l2saerk attribute)": [[151, "implementations.sweeper_classes.Runge_Kutta.ARK548L2SAERK.generator", false]], "generator (ark548l2saesdirk2 attribute)": [[151, "implementations.sweeper_classes.Runge_Kutta.ARK548L2SAESDIRK2.generator", false]], "generator (backwardeuler attribute)": [[151, "implementations.sweeper_classes.Runge_Kutta.BackwardEuler.generator", false]], "generator (cash_karp attribute)": [[151, "implementations.sweeper_classes.Runge_Kutta.Cash_Karp.generator", false]], "generator (cranknicolson attribute)": [[151, "implementations.sweeper_classes.Runge_Kutta.CrankNicolson.generator", false]], "generator (dirk43 attribute)": [[151, "implementations.sweeper_classes.Runge_Kutta.DIRK43.generator", false]], "generator (dirk43_2 attribute)": [[151, "implementations.sweeper_classes.Runge_Kutta.DIRK43_2.generator", false]], "generator (edirk4 attribute)": [[151, "implementations.sweeper_classes.Runge_Kutta.EDIRK4.generator", false]], "generator (esdirk43 attribute)": [[151, "implementations.sweeper_classes.Runge_Kutta.ESDIRK43.generator", false]], "generator (esdirk53 attribute)": [[151, "implementations.sweeper_classes.Runge_Kutta.ESDIRK53.generator", false]], "generator (explicitmidpointmethod attribute)": [[151, "implementations.sweeper_classes.Runge_Kutta.ExplicitMidpointMethod.generator", false]], "generator (forwardeuler attribute)": [[151, "implementations.sweeper_classes.Runge_Kutta.ForwardEuler.generator", false]], "generator (heun_euler attribute)": [[151, "implementations.sweeper_classes.Runge_Kutta.Heun_Euler.generator", false]], "generator (implicitmidpointmethod attribute)": [[151, "implementations.sweeper_classes.Runge_Kutta.ImplicitMidpointMethod.generator", false]], "generator (rk4 attribute)": [[151, "implementations.sweeper_classes.Runge_Kutta.RK4.generator", false]], "generator_exp (ark548l2saerk2 attribute)": [[151, "implementations.sweeper_classes.Runge_Kutta.ARK548L2SAERK2.generator_EXP", false]], "generator_imp (ark548l2saesdirk attribute)": [[151, "implementations.sweeper_classes.Runge_Kutta.ARK548L2SAESDIRK.generator_IMP", false]], "generic_implicit (class in implementations.sweeper_classes.generic_implicit)": [[155, "implementations.sweeper_classes.generic_implicit.generic_implicit", false]], "generic_implicit_mpi (class in implementations.sweeper_classes.generic_implicit_mpi)": [[156, "implementations.sweeper_classes.generic_implicit_MPI.generic_implicit_MPI", false]], "genericndimfindiff (class in implementations.problem_classes.generic_nd_fd)": [[144, "implementations.problem_classes.generic_ND_FD.GenericNDimFinDiff", false]], "get() (frozenclass method)": [[58, "helpers.pysdc_helper.FrozenClass.get", false]], "get2dmatrix() (in module implementations.problem_classes.boussinesq_helpers.build2dfdmatrix)": [[137, "implementations.problem_classes.boussinesq_helpers.build2DFDMatrix.get2DMatrix", false]], "get2dmesh() (in module implementations.problem_classes.boussinesq_helpers.build2dfdmatrix)": [[137, "implementations.problem_classes.boussinesq_helpers.build2DFDMatrix.get2DMesh", false]], "get2dupwindmatrix() (in module implementations.problem_classes.boussinesq_helpers.build2dfdmatrix)": [[137, "implementations.problem_classes.boussinesq_helpers.build2DFDMatrix.get2DUpwindMatrix", false]], "get_1d_grid() (in module helpers.problem_helper)": [[57, "helpers.problem_helper.get_1d_grid", false]], "get_butcher_tableau() (rungekutta class method)": [[151, "implementations.sweeper_classes.Runge_Kutta.RungeKutta.get_Butcher_tableau", false]], "get_butcher_tableau_explicit() (rungekuttaimex class method)": [[151, "implementations.sweeper_classes.Runge_Kutta.RungeKuttaIMEX.get_Butcher_tableau_explicit", false]], "get_convergence() (adaptivitycollocation method)": [[70, "implementations.convergence_controller_classes.adaptivity.AdaptivityCollocation.get_convergence", false]], "get_convergence() (adaptivityextrapolationwithinq method)": [[70, "implementations.convergence_controller_classes.adaptivity.AdaptivityExtrapolationWithinQ.get_convergence", false]], "get_convergence() (adaptivityforconvergedcollocationproblems method)": [[70, "implementations.convergence_controller_classes.adaptivity.AdaptivityForConvergedCollocationProblems.get_convergence", false]], "get_convergence() (adaptivitypolynomialerror method)": [[70, "implementations.convergence_controller_classes.adaptivity.AdaptivityPolynomialError.get_convergence", false]], "get_convergence_controllers_as_table() (controller method)": [[45, "core.controller.Controller.get_convergence_controllers_as_table", false]], "get_count() (ncclcomm static method)": [[55, "helpers.NCCL_communicator.NCCLComm.get_count", false]], "get_default_sweeper_class() (genericndimfindiff class method)": [[144, "implementations.problem_classes.generic_ND_FD.GenericNDimFinDiff.get_default_sweeper_class", false]], "get_default_sweeper_class() (problem class method)": [[50, "core.problem.Problem.get_default_sweeper_class", false]], "get_dtype() (ncclcomm static method)": [[55, "helpers.NCCL_communicator.NCCLComm.get_dtype", false]], "get_extrapolated_error() (estimateextrapolationerrornonmpi method)": [[77, "implementations.convergence_controller_classes.estimate_extrapolation_error.EstimateExtrapolationErrorNonMPI.get_extrapolated_error", false]], "get_extrapolated_solution() (estimateextrapolationerrornonmpi method)": [[77, "implementations.convergence_controller_classes.estimate_extrapolation_error.EstimateExtrapolationErrorNonMPI.get_extrapolated_solution", false]], "get_extrapolation_coefficients() (estimateextrapolationerrorbase method)": [[77, "implementations.convergence_controller_classes.estimate_extrapolation_error.EstimateExtrapolationErrorBase.get_extrapolation_coefficients", false]], "get_fig() (brusselator method)": [[105, "implementations.problem_classes.Brusselator.Brusselator.get_fig", false]], "get_fig() (problem method)": [[50, "core.problem.Problem.get_fig", false]], "get_finite_difference_matrix() (in module helpers.problem_helper)": [[57, "helpers.problem_helper.get_finite_difference_matrix", false]], "get_finite_difference_stencil() (in module helpers.problem_helper)": [[57, "helpers.problem_helper.get_finite_difference_stencil", false]], "get_full_f() (rungekutta method)": [[151, "implementations.sweeper_classes.Runge_Kutta.RungeKutta.get_full_f", false]], "get_full_f() (rungekuttanystrom method)": [[152, "implementations.sweeper_classes.Runge_Kutta_Nystrom.RungeKuttaNystrom.get_full_f", false]], "get_implementation() (basicrestarting class method)": [[71, "implementations.convergence_controller_classes.basic_restarting.BasicRestarting.get_implementation", false]], "get_implementation() (estimateembeddederror class method)": [[76, "implementations.convergence_controller_classes.estimate_embedded_error.EstimateEmbeddedError.get_implementation", false]], "get_implementation() (spreadstepsizesblockwise class method)": [[82, "implementations.convergence_controller_classes.spread_step_sizes.SpreadStepSizesBlockwise.get_implementation", false]], "get_interactions() (penningtrap method)": [[126, "implementations.problem_classes.PenningTrap_3D.penningtrap.get_interactions", false]], "get_list_of_types() (in module helpers.stats_helper)": [[60, "helpers.stats_helper.get_list_of_types", false]], "get_local_error_estimate() (adaptivity method)": [[70, "implementations.convergence_controller_classes.adaptivity.Adaptivity.get_local_error_estimate", false]], "get_local_error_estimate() (adaptivitybase method)": [[70, "implementations.convergence_controller_classes.adaptivity.AdaptivityBase.get_local_error_estimate", false]], "get_local_error_estimate() (adaptivitycollocation method)": [[70, "implementations.convergence_controller_classes.adaptivity.AdaptivityCollocation.get_local_error_estimate", false]], "get_local_error_estimate() (adaptivityextrapolationwithinq method)": [[70, "implementations.convergence_controller_classes.adaptivity.AdaptivityExtrapolationWithinQ.get_local_error_estimate", false]], "get_local_error_estimate() (adaptivitypolynomialerror method)": [[70, "implementations.convergence_controller_classes.adaptivity.AdaptivityPolynomialError.get_local_error_estimate", false]], "get_local_error_estimate() (adaptivityresidual method)": [[70, "implementations.convergence_controller_classes.adaptivity.AdaptivityResidual.get_local_error_estimate", false]], "get_new_step_size() (adaptivity method)": [[70, "implementations.convergence_controller_classes.adaptivity.Adaptivity.get_new_step_size", false]], "get_new_step_size() (adaptivitybase method)": [[70, "implementations.convergence_controller_classes.adaptivity.AdaptivityBase.get_new_step_size", false]], "get_new_step_size() (adaptivitycollocation method)": [[70, "implementations.convergence_controller_classes.adaptivity.AdaptivityCollocation.get_new_step_size", false]], "get_new_step_size() (adaptivityextrapolationwithinq method)": [[70, "implementations.convergence_controller_classes.adaptivity.AdaptivityExtrapolationWithinQ.get_new_step_size", false]], "get_new_step_size() (adaptivitypolynomialerror method)": [[70, "implementations.convergence_controller_classes.adaptivity.AdaptivityPolynomialError.get_new_step_size", false]], "get_new_step_size() (adaptivityresidual method)": [[70, "implementations.convergence_controller_classes.adaptivity.AdaptivityResidual.get_new_step_size", false]], "get_new_step_size() (adaptivityrk method)": [[70, "implementations.convergence_controller_classes.adaptivity.AdaptivityRK.get_new_step_size", false]], "get_new_step_size() (convergencecontroller method)": [[46, "core.convergence_controller.ConvergenceController.get_new_step_size", false]], "get_new_step_size() (stepsizelimiter method)": [[83, "implementations.convergence_controller_classes.step_size_limiter.StepSizeLimiter.get_new_step_size", false]], "get_new_step_size() (stepsizeslopelimiter method)": [[83, "implementations.convergence_controller_classes.step_size_limiter.StepSizeSlopeLimiter.get_new_step_size", false]], "get_non_linear_jacobian() (quench method)": [[128, "implementations.problem_classes.Quench.Quench.get_non_linear_Jacobian", false]], "get_op() (ncclcomm method)": [[55, "helpers.NCCL_communicator.NCCLComm.get_op", false]], "get_problem_dict() (battery_n_capacitors method)": [[103, "implementations.problem_classes.Battery.battery_n_capacitors.get_problem_dict", false]], "get_q_matrix() (rungekutta class method)": [[151, "implementations.sweeper_classes.Runge_Kutta.RungeKutta.get_Q_matrix", false]], "get_qdelta_explicit() (sweeper method)": [[53, "core.sweeper.Sweeper.get_Qdelta_explicit", false]], "get_qdelta_implicit() (sweeper method)": [[53, "core.sweeper.Sweeper.get_Qdelta_implicit", false]], "get_scalar_problems_manysweep_mat() (imex_1st_order method)": [[157, "implementations.sweeper_classes.imex_1st_order.imex_1st_order.get_scalar_problems_manysweep_mat", false]], "get_scalar_problems_manysweep_mats() (boris_2nd_order method)": [[153, "implementations.sweeper_classes.boris_2nd_order.boris_2nd_order.get_scalar_problems_manysweep_mats", false]], "get_scalar_problems_picardsweep_mats() (boris_2nd_order method)": [[153, "implementations.sweeper_classes.boris_2nd_order.boris_2nd_order.get_scalar_problems_picardsweep_mats", false]], "get_scalar_problems_sweeper_mats() (boris_2nd_order method)": [[153, "implementations.sweeper_classes.boris_2nd_order.boris_2nd_order.get_scalar_problems_sweeper_mats", false]], "get_scalar_problems_sweeper_mats() (imex_1st_order method)": [[157, "implementations.sweeper_classes.imex_1st_order.imex_1st_order.get_scalar_problems_sweeper_mats", false]], "get_sorted() (in module helpers.stats_helper)": [[60, "helpers.stats_helper.get_sorted", false]], "get_step_from_which_to_spread() (spreadstepsizesblockwise method)": [[82, "implementations.convergence_controller_classes.spread_step_sizes.SpreadStepSizesBlockwise.get_step_from_which_to_spread", false]], "get_step_from_which_to_spread() (spreadstepsizesblockwisempi method)": [[82, "implementations.convergence_controller_classes.spread_step_sizes.SpreadStepSizesBlockwiseMPI.get_step_from_which_to_spread", false]], "get_step_from_which_to_spread() (spreadstepsizesblockwisenonmpi method)": [[82, "implementations.convergence_controller_classes.spread_step_sizes.SpreadStepSizesBlockwiseNonMPI.get_step_from_which_to_spread", false]], "get_steps() (in module helpers.problem_helper)": [[57, "helpers.problem_helper.get_steps", false]], "get_sweeper_mats() (boris_2nd_order method)": [[153, "implementations.sweeper_classes.boris_2nd_order.boris_2nd_order.get_sweeper_mats", false]], "get_sweeper_mats() (imex_1st_order method)": [[157, "implementations.sweeper_classes.imex_1st_order.imex_1st_order.get_sweeper_mats", false]], "get_switching_info() (battery_n_capacitors method)": [[103, "implementations.problem_classes.Battery.battery_n_capacitors.get_switching_info", false]], "get_switching_info() (discontinuoustestode method)": [[107, "implementations.problem_classes.DiscontinuousTestODE.DiscontinuousTestODE.get_switching_info", false]], "get_sys_mat() (petsc_fisher_multiimplicit method)": [[112, "implementations.problem_classes.GeneralizedFisher_1D_PETSc.petsc_fisher_multiimplicit.get_sys_mat", false]], "get_transfer_matrix_q() (basetransfer static method)": [[42, "core.base_transfer.BaseTransfer.get_transfer_matrix_Q", false]], "get_update_order() (ark54 class method)": [[151, "implementations.sweeper_classes.Runge_Kutta.ARK54.get_update_order", false]], "get_update_order() (ark548l2sa class method)": [[151, "implementations.sweeper_classes.Runge_Kutta.ARK548L2SA.get_update_order", false]], "get_update_order() (ark548l2saerk class method)": [[151, "implementations.sweeper_classes.Runge_Kutta.ARK548L2SAERK.get_update_order", false]], "get_update_order() (ark548l2saesdirk2 class method)": [[151, "implementations.sweeper_classes.Runge_Kutta.ARK548L2SAESDIRK2.get_update_order", false]], "get_update_order() (cash_karp class method)": [[151, "implementations.sweeper_classes.Runge_Kutta.Cash_Karp.get_update_order", false]], "get_update_order() (dirk43 class method)": [[151, "implementations.sweeper_classes.Runge_Kutta.DIRK43.get_update_order", false]], "get_update_order() (esdirk43 class method)": [[151, "implementations.sweeper_classes.Runge_Kutta.ESDIRK43.get_update_order", false]], "get_update_order() (esdirk53 class method)": [[151, "implementations.sweeper_classes.Runge_Kutta.ESDIRK53.get_update_order", false]], "get_update_order() (heun_euler class method)": [[151, "implementations.sweeper_classes.Runge_Kutta.Heun_Euler.get_update_order", false]], "get_update_order() (rungekutta class method)": [[151, "implementations.sweeper_classes.Runge_Kutta.RungeKutta.get_update_order", false]], "getbchorizontal() (in module implementations.problem_classes.boussinesq_helpers.build2dfdmatrix)": [[137, "implementations.problem_classes.boussinesq_helpers.build2DFDMatrix.getBCHorizontal", false]], "getbcleft() (in module implementations.problem_classes.acoustic_helpers.buildfdmatrix)": [[133, "implementations.problem_classes.acoustic_helpers.buildFDMatrix.getBCLeft", false]], "getbcleft() (in module implementations.problem_classes.boussinesq_helpers.buildfdmatrix)": [[139, "implementations.problem_classes.boussinesq_helpers.buildFDMatrix.getBCLeft", false]], "getbcright() (in module implementations.problem_classes.acoustic_helpers.buildfdmatrix)": [[133, "implementations.problem_classes.acoustic_helpers.buildFDMatrix.getBCRight", false]], "getbcright() (in module implementations.problem_classes.boussinesq_helpers.buildfdmatrix)": [[139, "implementations.problem_classes.boussinesq_helpers.buildFDMatrix.getBCRight", false]], "getbcvertical() (in module implementations.problem_classes.boussinesq_helpers.build2dfdmatrix)": [[137, "implementations.problem_classes.boussinesq_helpers.build2DFDMatrix.getBCVertical", false]], "getboussinesq2dmatrix() (in module implementations.problem_classes.boussinesq_helpers.buildboussinesq2dmatrix)": [[138, "implementations.problem_classes.boussinesq_helpers.buildBoussinesq2DMatrix.getBoussinesq2DMatrix", false]], "getboussinesq2dupwindmatrix() (in module implementations.problem_classes.boussinesq_helpers.buildboussinesq2dmatrix)": [[138, "implementations.problem_classes.boussinesq_helpers.buildBoussinesq2DMatrix.getBoussinesq2DUpwindMatrix", false]], "getboussinesqbchorizontal() (in module implementations.problem_classes.boussinesq_helpers.buildboussinesq2dmatrix)": [[138, "implementations.problem_classes.boussinesq_helpers.buildBoussinesq2DMatrix.getBoussinesqBCHorizontal", false]], "getboussinesqbcvertical() (in module implementations.problem_classes.boussinesq_helpers.buildboussinesq2dmatrix)": [[138, "implementations.problem_classes.boussinesq_helpers.buildBoussinesq2DMatrix.getBoussinesqBCVertical", false]], "getcounter() (callback method)": [[140, "implementations.problem_classes.boussinesq_helpers.helper_classes.Callback.getcounter", false]], "gethorizontaldx() (in module implementations.problem_classes.acoustic_helpers.buildfdmatrix)": [[133, "implementations.problem_classes.acoustic_helpers.buildFDMatrix.getHorizontalDx", false]], "getmatrix() (in module implementations.problem_classes.acoustic_helpers.buildfdmatrix)": [[133, "implementations.problem_classes.acoustic_helpers.buildFDMatrix.getMatrix", false]], "getmatrix() (in module implementations.problem_classes.boussinesq_helpers.buildfdmatrix)": [[139, "implementations.problem_classes.boussinesq_helpers.buildFDMatrix.getMatrix", false]], "getresidual() (callback method)": [[140, "implementations.problem_classes.boussinesq_helpers.helper_classes.Callback.getresidual", false]], "getupwindmatrix() (in module implementations.problem_classes.boussinesq_helpers.buildfdmatrix)": [[139, "implementations.problem_classes.boussinesq_helpers.buildFDMatrix.getUpwindMatrix", false]], "getwave1dadvectionmatrix() (in module implementations.problem_classes.acoustic_helpers.buildwave1dmatrix)": [[134, "implementations.problem_classes.acoustic_helpers.buildWave1DMatrix.getWave1DAdvectionMatrix", false]], "getwave1dmatrix() (in module implementations.problem_classes.acoustic_helpers.buildwave1dmatrix)": [[134, "implementations.problem_classes.acoustic_helpers.buildWave1DMatrix.getWave1DMatrix", false]], "getwavebcleft() (in module implementations.problem_classes.acoustic_helpers.buildwave1dmatrix)": [[134, "implementations.problem_classes.acoustic_helpers.buildWave1DMatrix.getWaveBCLeft", false]], "getwavebcright() (in module implementations.problem_classes.acoustic_helpers.buildwave1dmatrix)": [[134, "implementations.problem_classes.acoustic_helpers.buildWave1DMatrix.getWaveBCRight", false]], "gmres_logger (boussinesq_2d_imex attribute)": [[104, "implementations.problem_classes.Boussinesq_2D_FD_imex.boussinesq_2d_imex.gmres_logger", false]], "grayscott_imex_diffusion (class in implementations.problem_classes.grayscott_mpifft)": [[115, "implementations.problem_classes.GrayScott_MPIFFT.grayscott_imex_diffusion", false]], "grayscott_imex_linear (class in implementations.problem_classes.grayscott_mpifft)": [[115, "implementations.problem_classes.GrayScott_MPIFFT.grayscott_imex_linear", false]], "grayscott_mi_diffusion (class in implementations.problem_classes.grayscott_mpifft)": [[115, "implementations.problem_classes.GrayScott_MPIFFT.grayscott_mi_diffusion", false]], "grayscott_mi_linear (class in implementations.problem_classes.grayscott_mpifft)": [[115, "implementations.problem_classes.GrayScott_MPIFFT.grayscott_mi_linear", false]], "grids (genericndimfindiff property)": [[144, "implementations.problem_classes.generic_ND_FD.GenericNDimFinDiff.grids", false]], "gs_full (class in implementations.problem_classes.grayscott_2d_petsc_periodic)": [[114, "implementations.problem_classes.GrayScott_2D_PETSc_periodic.GS_full", false]], "gs_reaction (class in implementations.problem_classes.grayscott_2d_petsc_periodic)": [[114, "implementations.problem_classes.GrayScott_2D_PETSc_periodic.GS_reaction", false]], "h (boussinesq_2d_imex attribute)": [[104, "implementations.problem_classes.Boussinesq_2D_FD_imex.boussinesq_2d_imex.h", false]], "harmonic_oscillator (class in implementations.problem_classes.harmonicoscillator)": [[116, "implementations.problem_classes.HarmonicOscillator.harmonic_oscillator", false]], "heat2d_petsc_forced (class in implementations.problem_classes.heatequation_2d_petsc_forced)": [[118, "implementations.problem_classes.HeatEquation_2D_PETSc_forced.heat2d_petsc_forced", false]], "heatnd_forced (class in implementations.problem_classes.heatequation_nd_fd)": [[119, "implementations.problem_classes.HeatEquation_ND_FD.heatNd_forced", false]], "heatnd_forced (class in implementations.problem_classes.heatequation_nd_fd_cupy)": [[120, "implementations.problem_classes.HeatEquation_ND_FD_CuPy.heatNd_forced", false]], "heatnd_unforced (class in implementations.problem_classes.heatequation_nd_fd)": [[119, "implementations.problem_classes.HeatEquation_ND_FD.heatNd_unforced", false]], "heatnd_unforced (class in implementations.problem_classes.heatequation_nd_fd_cupy)": [[120, "implementations.problem_classes.HeatEquation_ND_FD_CuPy.heatNd_unforced", false]], "helpers": [[54, "module-helpers", false]], "helpers.nccl_communicator": [[55, "module-helpers.NCCL_communicator", false]], "helpers.plot_helper": [[56, "module-helpers.plot_helper", false]], "helpers.problem_helper": [[57, "module-helpers.problem_helper", false]], "helpers.pysdc_helper": [[58, "module-helpers.pysdc_helper", false]], "helpers.setup_helper": [[59, "module-helpers.setup_helper", false]], "helpers.stats_helper": [[60, "module-helpers.stats_helper", false]], "helpers.testing": [[61, "module-helpers.testing", false]], "helpers.transfer_helper": [[62, "module-helpers.transfer_helper", false]], "helpers.visualization_tools": [[63, "module-helpers.visualization_tools", false]], "henon_heiles (class in implementations.problem_classes.henonheiles)": [[121, "implementations.problem_classes.HenonHeiles.henon_heiles", false]], "heun_euler (class in implementations.sweeper_classes.runge_kutta)": [[151, "implementations.sweeper_classes.Runge_Kutta.Heun_Euler", false]], "hooks (class in core.hooks)": [[48, "core.hooks.Hooks", false]], "hooks (controller property)": [[45, "core.controller.Controller.hooks", false]], "hotrod (class in implementations.convergence_controller_classes.hotrod)": [[79, "implementations.convergence_controller_classes.hotrod.HotRod", false]], "id (acoustic_1d_imex attribute)": [[92, "implementations.problem_classes.AcousticAdvection_1D_FD_imex.acoustic_1d_imex.Id", false]], "id (advectionnd attribute)": [[94, "implementations.problem_classes.AdvectionEquation_ND_FD.advectionNd.Id", false]], "id (boussinesq_2d_imex attribute)": [[104, "implementations.problem_classes.Boussinesq_2D_FD_imex.boussinesq_2d_imex.Id", false]], "id (genericndimfindiff attribute)": [[144, "implementations.problem_classes.generic_ND_FD.GenericNDimFinDiff.Id", false]], "id (heat2d_petsc_forced attribute)": [[118, "implementations.problem_classes.HeatEquation_2D_PETSc_forced.heat2d_petsc_forced.Id", false]], "id (heatnd_forced attribute)": [[120, "implementations.problem_classes.HeatEquation_ND_FD_CuPy.heatNd_forced.Id", false]], "id (heatnd_unforced attribute)": [[119, "implementations.problem_classes.HeatEquation_ND_FD.heatNd_unforced.Id", false]], "id (petsc_grayscott_multiimplicit attribute)": [[114, "implementations.problem_classes.GrayScott_2D_PETSc_periodic.petsc_grayscott_multiimplicit.Id", false]], "id (quench attribute)": [[128, "implementations.problem_classes.Quench.Quench.Id", false]], "imex_1st_order (class in implementations.sweeper_classes.imex_1st_order)": [[157, "implementations.sweeper_classes.imex_1st_order.imex_1st_order", false]], "imex_1st_order_mass (class in implementations.sweeper_classes.imex_1st_order_mass)": [[159, "implementations.sweeper_classes.imex_1st_order_mass.imex_1st_order_mass", false]], "imex_1st_order_mpi (class in implementations.sweeper_classes.imex_1st_order_mpi)": [[158, "implementations.sweeper_classes.imex_1st_order_MPI.imex_1st_order_MPI", false]], "imex_cupy_mesh (class in implementations.datatype_classes.cupy_mesh)": [[86, "implementations.datatype_classes.cupy_mesh.imex_cupy_mesh", false]], "imex_laplacian_mpifft (class in implementations.problem_classes.generic_mpifft_laplacian)": [[143, "implementations.problem_classes.generic_MPIFFT_Laplacian.IMEX_Laplacian_MPIFFT", false]], "imex_mesh (class in implementations.datatype_classes.mesh)": [[88, "implementations.datatype_classes.mesh.imex_mesh", false]], "impl (petsc_vec_comp2 attribute)": [[90, "implementations.datatype_classes.petsc_vec.petsc_vec_comp2.impl", false]], "impl (petsc_vec_imex attribute)": [[90, "implementations.datatype_classes.petsc_vec.petsc_vec_imex.impl", false]], "impl (rhs_fenics_mesh attribute)": [[87, "implementations.datatype_classes.fenics_mesh.rhs_fenics_mesh.impl", false]], "implementations": [[64, "module-implementations", false]], "implementations.controller_classes": [[65, "module-implementations.controller_classes", false]], "implementations.controller_classes.controller_mpi": [[66, "module-implementations.controller_classes.controller_MPI", false]], "implementations.controller_classes.controller_nonmpi": [[67, "module-implementations.controller_classes.controller_nonMPI", false]], "implementations.convergence_controller_classes": [[68, "module-implementations.convergence_controller_classes", false]], "implementations.convergence_controller_classes.adaptive_collocation": [[69, "module-implementations.convergence_controller_classes.adaptive_collocation", false]], "implementations.convergence_controller_classes.adaptivity": [[70, "module-implementations.convergence_controller_classes.adaptivity", false]], "implementations.convergence_controller_classes.basic_restarting": [[71, "module-implementations.convergence_controller_classes.basic_restarting", false]], "implementations.convergence_controller_classes.check_convergence": [[72, "module-implementations.convergence_controller_classes.check_convergence", false]], "implementations.convergence_controller_classes.check_iteration_estimator": [[73, "module-implementations.convergence_controller_classes.check_iteration_estimator", false]], "implementations.convergence_controller_classes.crash": [[74, "module-implementations.convergence_controller_classes.crash", false]], "implementations.convergence_controller_classes.estimate_contraction_factor": [[75, "module-implementations.convergence_controller_classes.estimate_contraction_factor", false]], "implementations.convergence_controller_classes.estimate_embedded_error": [[76, "module-implementations.convergence_controller_classes.estimate_embedded_error", false]], "implementations.convergence_controller_classes.estimate_extrapolation_error": [[77, "module-implementations.convergence_controller_classes.estimate_extrapolation_error", false]], "implementations.convergence_controller_classes.estimate_polynomial_error": [[78, "module-implementations.convergence_controller_classes.estimate_polynomial_error", false]], "implementations.convergence_controller_classes.hotrod": [[79, "module-implementations.convergence_controller_classes.hotrod", false]], "implementations.convergence_controller_classes.inexactness": [[80, "module-implementations.convergence_controller_classes.inexactness", false]], "implementations.convergence_controller_classes.interpolate_between_restarts": [[81, "module-implementations.convergence_controller_classes.interpolate_between_restarts", false]], "implementations.convergence_controller_classes.spread_step_sizes": [[82, "module-implementations.convergence_controller_classes.spread_step_sizes", false]], "implementations.convergence_controller_classes.step_size_limiter": [[83, "module-implementations.convergence_controller_classes.step_size_limiter", false]], "implementations.convergence_controller_classes.store_uold": [[84, "module-implementations.convergence_controller_classes.store_uold", false]], "implementations.datatype_classes": [[85, "module-implementations.datatype_classes", false]], "implementations.datatype_classes.cupy_mesh": [[86, "module-implementations.datatype_classes.cupy_mesh", false]], "implementations.datatype_classes.fenics_mesh": [[87, "module-implementations.datatype_classes.fenics_mesh", false]], "implementations.datatype_classes.mesh": [[88, "module-implementations.datatype_classes.mesh", false]], "implementations.datatype_classes.particles": [[89, "module-implementations.datatype_classes.particles", false]], "implementations.datatype_classes.petsc_vec": [[90, "module-implementations.datatype_classes.petsc_vec", false]], "implementations.problem_classes": [[91, "module-implementations.problem_classes", false]], "implementations.problem_classes.acoustic_helpers": [[132, "module-implementations.problem_classes.acoustic_helpers", false]], "implementations.problem_classes.acoustic_helpers.buildfdmatrix": [[133, "module-implementations.problem_classes.acoustic_helpers.buildFDMatrix", false]], "implementations.problem_classes.acoustic_helpers.buildwave1dmatrix": [[134, "module-implementations.problem_classes.acoustic_helpers.buildWave1DMatrix", false]], "implementations.problem_classes.acoustic_helpers.standard_integrators": [[135, "module-implementations.problem_classes.acoustic_helpers.standard_integrators", false]], "implementations.problem_classes.acousticadvection_1d_fd_imex": [[92, "module-implementations.problem_classes.AcousticAdvection_1D_FD_imex", false]], "implementations.problem_classes.advectiondiffusionequation_1d_fft": [[93, "module-implementations.problem_classes.AdvectionDiffusionEquation_1D_FFT", false]], "implementations.problem_classes.advectionequation_nd_fd": [[94, "module-implementations.problem_classes.AdvectionEquation_ND_FD", false]], "implementations.problem_classes.allencahn_1d_fd": [[95, "module-implementations.problem_classes.AllenCahn_1D_FD", false]], "implementations.problem_classes.allencahn_2d_fd": [[96, "module-implementations.problem_classes.AllenCahn_2D_FD", false]], "implementations.problem_classes.allencahn_2d_fd_gpu": [[97, "module-implementations.problem_classes.AllenCahn_2D_FD_gpu", false]], "implementations.problem_classes.allencahn_2d_fft": [[98, "module-implementations.problem_classes.AllenCahn_2D_FFT", false]], "implementations.problem_classes.allencahn_2d_fft_gpu": [[99, "module-implementations.problem_classes.AllenCahn_2D_FFT_gpu", false]], "implementations.problem_classes.allencahn_mpifft": [[100, "module-implementations.problem_classes.AllenCahn_MPIFFT", false]], "implementations.problem_classes.allencahn_temp_mpifft": [[101, "module-implementations.problem_classes.AllenCahn_Temp_MPIFFT", false]], "implementations.problem_classes.auzinger_implicit": [[102, "module-implementations.problem_classes.Auzinger_implicit", false]], "implementations.problem_classes.battery": [[103, "module-implementations.problem_classes.Battery", false]], "implementations.problem_classes.boussinesq_2d_fd_imex": [[104, "module-implementations.problem_classes.Boussinesq_2D_FD_imex", false]], "implementations.problem_classes.boussinesq_helpers": [[136, "module-implementations.problem_classes.boussinesq_helpers", false]], "implementations.problem_classes.boussinesq_helpers.build2dfdmatrix": [[137, "module-implementations.problem_classes.boussinesq_helpers.build2DFDMatrix", false]], "implementations.problem_classes.boussinesq_helpers.buildboussinesq2dmatrix": [[138, "module-implementations.problem_classes.boussinesq_helpers.buildBoussinesq2DMatrix", false]], "implementations.problem_classes.boussinesq_helpers.buildfdmatrix": [[139, "module-implementations.problem_classes.boussinesq_helpers.buildFDMatrix", false]], "implementations.problem_classes.boussinesq_helpers.helper_classes": [[140, "module-implementations.problem_classes.boussinesq_helpers.helper_classes", false]], "implementations.problem_classes.boussinesq_helpers.standard_integrators": [[141, "module-implementations.problem_classes.boussinesq_helpers.standard_integrators", false]], "implementations.problem_classes.boussinesq_helpers.unflatten": [[142, "module-implementations.problem_classes.boussinesq_helpers.unflatten", false]], "implementations.problem_classes.brusselator": [[105, "module-implementations.problem_classes.Brusselator", false]], "implementations.problem_classes.buckconverter": [[106, "module-implementations.problem_classes.BuckConverter", false]], "implementations.problem_classes.discontinuoustestode": [[107, "module-implementations.problem_classes.DiscontinuousTestODE", false]], "implementations.problem_classes.fastwaveslowwave_0d": [[108, "module-implementations.problem_classes.FastWaveSlowWave_0D", false]], "implementations.problem_classes.fermipastaulamtsingou": [[109, "module-implementations.problem_classes.FermiPastaUlamTsingou", false]], "implementations.problem_classes.fullsolarsystem": [[110, "module-implementations.problem_classes.FullSolarSystem", false]], "implementations.problem_classes.generalizedfisher_1d_fd_implicit": [[111, "module-implementations.problem_classes.GeneralizedFisher_1D_FD_implicit", false]], "implementations.problem_classes.generalizedfisher_1d_petsc": [[112, "module-implementations.problem_classes.GeneralizedFisher_1D_PETSc", false]], "implementations.problem_classes.generic_mpifft_laplacian": [[143, "module-implementations.problem_classes.generic_MPIFFT_Laplacian", false]], "implementations.problem_classes.generic_nd_fd": [[144, "module-implementations.problem_classes.generic_ND_FD", false]], "implementations.problem_classes.grayscott_1d_fenics_implicit": [[113, "module-implementations.problem_classes.GrayScott_1D_FEniCS_implicit", false]], "implementations.problem_classes.grayscott_2d_petsc_periodic": [[114, "module-implementations.problem_classes.GrayScott_2D_PETSc_periodic", false]], "implementations.problem_classes.grayscott_mpifft": [[115, "module-implementations.problem_classes.GrayScott_MPIFFT", false]], "implementations.problem_classes.harmonicoscillator": [[116, "module-implementations.problem_classes.HarmonicOscillator", false]], "implementations.problem_classes.heatequation_1d_fenics_matrix_forced": [[117, "module-implementations.problem_classes.HeatEquation_1D_FEniCS_matrix_forced", false]], "implementations.problem_classes.heatequation_2d_petsc_forced": [[118, "module-implementations.problem_classes.HeatEquation_2D_PETSc_forced", false]], "implementations.problem_classes.heatequation_nd_fd": [[119, "module-implementations.problem_classes.HeatEquation_ND_FD", false]], "implementations.problem_classes.heatequation_nd_fd_cupy": [[120, "module-implementations.problem_classes.HeatEquation_ND_FD_CuPy", false]], "implementations.problem_classes.henonheiles": [[121, "module-implementations.problem_classes.HenonHeiles", false]], "implementations.problem_classes.logisticequation": [[122, "module-implementations.problem_classes.LogisticEquation", false]], "implementations.problem_classes.lorenz": [[123, "module-implementations.problem_classes.Lorenz", false]], "implementations.problem_classes.nonlinear_ode_1": [[145, "module-implementations.problem_classes.nonlinear_ODE_1", false]], "implementations.problem_classes.nonlinearschroedinger_mpifft": [[124, "module-implementations.problem_classes.NonlinearSchroedinger_MPIFFT", false]], "implementations.problem_classes.odescalar": [[146, "module-implementations.problem_classes.odeScalar", false]], "implementations.problem_classes.odesystem": [[147, "module-implementations.problem_classes.odeSystem", false]], "implementations.problem_classes.outersolarsystem": [[125, "module-implementations.problem_classes.OuterSolarSystem", false]], "implementations.problem_classes.penningtrap_3d": [[126, "module-implementations.problem_classes.PenningTrap_3D", false]], "implementations.problem_classes.piline": [[127, "module-implementations.problem_classes.Piline", false]], "implementations.problem_classes.polynomial_test_problem": [[148, "module-implementations.problem_classes.polynomial_test_problem", false]], "implementations.problem_classes.quench": [[128, "module-implementations.problem_classes.Quench", false]], "implementations.problem_classes.testequation_0d": [[129, "module-implementations.problem_classes.TestEquation_0D", false]], "implementations.problem_classes.van_der_pol_implicit": [[130, "module-implementations.problem_classes.Van_der_Pol_implicit", false]], "implementations.problem_classes.vorticityvelocity_2d_fenics_periodic": [[131, "module-implementations.problem_classes.VorticityVelocity_2D_FEniCS_periodic", false]], "implementations.sweeper_classes": [[149, "module-implementations.sweeper_classes", false]], "implementations.sweeper_classes.boris_2nd_order": [[153, "module-implementations.sweeper_classes.boris_2nd_order", false]], "implementations.sweeper_classes.explicit": [[154, "module-implementations.sweeper_classes.explicit", false]], "implementations.sweeper_classes.generic_implicit": [[155, "module-implementations.sweeper_classes.generic_implicit", false]], "implementations.sweeper_classes.generic_implicit_mpi": [[156, "module-implementations.sweeper_classes.generic_implicit_MPI", false]], "implementations.sweeper_classes.imex_1st_order": [[157, "module-implementations.sweeper_classes.imex_1st_order", false]], "implementations.sweeper_classes.imex_1st_order_mass": [[159, "module-implementations.sweeper_classes.imex_1st_order_mass", false]], "implementations.sweeper_classes.imex_1st_order_mpi": [[158, "module-implementations.sweeper_classes.imex_1st_order_MPI", false]], "implementations.sweeper_classes.multi_implicit": [[160, "module-implementations.sweeper_classes.multi_implicit", 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106, 117, 118, 119, 120, 122, 127, 131, 173, 174, 175, 176, 177, 178, 179, 180, 183, 184, 185, 187, 188, 189, 190, 195, 198, 199, 200, 201, 202, 203, 204, 205, 206, 207], "coil": 14, "col": 112, "coll": [53, 175, 176, 184, 200, 203], "coll_err": [199, 207], "coll_error": [199, 207], "coll_error_after_step": [199, 207], "collabor": [0, 19], "collbas": [41, 43, 53, 175, 176, 179, 181, 200, 201, 202], "collect": [0, 5, 13, 22, 26, 30, 31, 33, 37, 109, 174, 176, 183, 200, 203], "colloc": [0, 2, 3, 10, 15, 19, 34, 35, 36, 37, 41, 47, 53, 69, 70, 76, 77, 78, 81, 82, 103, 107, 150, 151, 152, 153, 154, 155, 156, 157, 158, 159, 160, 161, 175, 176, 178, 179, 181, 182, 196, 199, 201, 203, 206, 207], "collocation_class": [151, 152, 200], "collocation_nod": [196, 206], "collocationerror": [41, 47], "collumn": 53, "color": [5, 19, 29, 30, 31, 37, 174, 176, 195, 200, 206], "color_list": [29, 30, 31, 37], "column": [8, 77, 112, 157, 175, 176, 200], "com": [2, 3, 10, 105, 152], "combin": [5, 12, 13, 36, 77, 192, 195, 203, 205, 206], "come": [0, 4, 5, 11, 16, 19, 29, 37, 46, 200, 205, 207], "comm": [31, 37, 46, 55, 60, 66, 71, 72, 74, 82, 86, 88, 89, 90, 100, 101, 112, 114, 115, 118, 124, 143, 156, 192, 194, 195, 205, 206], "comm_world": [31, 37, 112, 114, 115, 118, 124, 143, 192, 194, 195, 205, 206], "command": [5, 16, 36], "comment": [1, 4, 5, 38], "commerci": 37, "commit": [1, 2, 4, 5, 9, 16], "commit_msg_fil": 5, "common": [1, 12, 14, 41], "commonli": 12, "commun": [0, 1, 3, 10, 11, 19, 46, 47, 48, 55, 60, 66, 69, 71, 72, 73, 74, 76, 78, 82, 86, 88, 89, 90, 100, 101, 112, 114, 115, 118, 124, 143, 156, 192, 195, 205, 206], "communicate_converg": 72, "communicate_crash": 74, "communicate_tau_correction_for_full_interv": 156, "communicationerror": [41, 47], "comp1": [86, 88], "comp2": [86, 88], "comp2_cupy_mesh": [85, 86, 97], "comp2_mesh": [85, 88, 95, 96, 115], "compact": 36, "compactli": 35, "companion": 2, "compar": [5, 11, 14, 17, 20, 27, 28, 29, 34, 35, 36, 37, 61, 69, 70, 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201, 202, 203, 204, 205, 206, 207], "conditionali": 5, "conduct": [3, 5, 10, 128], "config": 36, "configur": [34, 38, 40, 58, 207], "conflict": [4, 5], "conform": 10, "confus": [0, 5, 12, 156], "conjug": [96, 97], "connect": 0, "connect_level": 52, "connoisseur": 151, "consecut": [76, 174, 176, 200, 207], "consequ": 1, "conserv": 202, "consid": [1, 5, 12, 14, 34, 36, 50, 94, 119, 120], "consider": 204, "consist": [0, 2, 4, 6, 12, 107, 125, 126, 127, 200], "const": [31, 37], "constant": [11, 12, 77, 96, 97, 98, 99, 110, 116, 117, 125, 126, 153, 204, 207], "constantli": [5, 128], "constrain": 12, "construct": [1, 57, 62, 202], "constructor": [0, 7, 12], "consumpt": 14, "contain": [0, 5, 8, 10, 11, 12, 13, 17, 19, 20, 34, 37, 39, 42, 43, 44, 45, 46, 48, 49, 50, 53, 57, 60, 62, 66, 67, 71, 74, 77, 81, 82, 87, 89, 93, 95, 96, 97, 102, 103, 104, 107, 109, 110, 112, 114, 115, 117, 122, 124, 126, 129, 130, 144, 150, 151, 153, 154, 155, 156, 157, 158, 160, 161, 163, 164, 174, 176, 177, 184, 190, 200, 201, 202, 203, 205], "content": [5, 10], "continu": [1, 2, 3, 4, 6, 7, 10, 16, 62, 70, 72, 81, 86, 88, 113, 114, 115, 117, 131], "continue_periodic_arrai": [54, 62], "contour": 38, "contract": [70, 75], "contrast": [12, 201, 202, 204], "contribut": [0, 1, 5, 6, 7, 8, 9], "contributor": 4, "control": [0, 10, 13, 14, 17, 19, 20, 22, 26, 27, 28, 29, 30, 31, 33, 35, 37, 39, 41, 46, 47, 48, 52, 60, 66, 67, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 106, 179, 180, 181, 182, 185, 186, 187, 188, 189, 190, 192, 193, 194, 195, 197, 198, 199, 201, 202, 203, 204, 206, 207], "control_ord": [46, 70, 71], "controller_class": [13, 17, 20, 22, 26, 27, 29, 30, 31, 33, 35, 37, 64, 179, 180, 181, 182, 185, 186, 187, 188, 189, 190, 192, 193, 194, 195, 197, 198, 199, 201, 202, 203, 204, 205, 206, 207], "controller_mat": [27, 35], "controller_matrix_nonmpi": [27, 28, 35], "controller_mlsdc": [185, 203], "controller_mpi": [0, 65, 192, 195, 205, 206, 207], "controller_mssdc_g": [198, 207], "controller_mssdc_jac": [198, 207], "controller_nomat": [27, 35], "controller_nonmpi": [0, 13, 17, 20, 22, 26, 27, 29, 30, 31, 33, 35, 37, 65, 179, 180, 181, 182, 185, 186, 187, 188, 189, 190, 193, 194, 197, 198, 199, 201, 202, 203, 204, 205, 206, 207], "controller_param": [13, 17, 20, 22, 26, 27, 28, 29, 30, 31, 33, 35, 37, 45, 66, 67, 179, 180, 181, 182, 185, 186, 187, 188, 189, 190, 192, 193, 194, 195, 197, 198, 199, 201, 202, 203, 204, 205, 206, 207], "controller_paramet": 201, "controller_params_g": [198, 207], "controller_params_jac": [198, 207], "controller_pfasst": [198, 207], "controller_sdc": [185, 203], "controllererror": [41, 47], "contruct": 62, "conv": 40, "conv_test_to0": 18, "conv_test_toinf": 18, "convect": 123, "conveni": [16, 49, 200], "convent": [2, 5, 7], "converg": [0, 10, 12, 13, 14, 19, 34, 38, 40, 45, 46, 47, 66, 67, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 179, 181, 193, 200, 201, 202, 205, 206], "convergence_control": [41, 45, 199, 207], "convergence_controller_class": [64, 199, 207], "convergencecontrol": [41, 45, 46, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84], "convergenceerror": [41, 47, 74], "convert": [27, 35, 106, 180, 182, 186, 198, 202, 203, 207], "cool": 128, "coordin": [94, 100, 101, 115, 124, 143], "copi": [0, 5, 7, 49, 53, 150, 156, 192, 198, 205, 207], "core": [0, 2, 3, 4, 5, 7, 10, 13, 156, 175, 176, 177, 178, 179, 181, 184, 200, 201, 202, 203, 206], "correct": [0, 3, 4, 9, 10, 11, 12, 34, 35, 36, 38, 42, 49, 50, 66, 67, 92, 105, 108, 109, 126, 131, 152, 157, 163, 164, 184, 193, 203, 206], "correctli": 12, "correctt": 36, "correl": 70, "corresp": 52, "correspond": [3, 5, 10, 12, 13, 19, 26, 27, 28, 34, 35, 46, 52, 60, 69, 153, 154, 155, 156, 157, 158, 159, 160, 161, 203, 205, 207], "corrupt": 79, "cosimo": [112, 114, 118], "cost": 38, "costli": [19, 203], "could": [0, 1, 4, 12, 19, 69, 70, 78, 151, 152], "coulomb": 126, "count": [17, 18, 20, 27, 28, 30, 31, 35, 37, 45, 46, 50, 95, 103, 107, 123, 126, 130, 145, 180, 189, 190, 192, 198, 201, 202, 203, 204, 205, 207], "count_switch": [103, 107], "counter": [95, 178, 201], "counterpart": [36, 205], "coupl": [36, 52, 86, 88, 200, 206], "cours": [6, 46, 203], "cover": [5, 9, 31, 37], "coverag": 10, "coverage_process_start": [192, 205], "cp": 99, "cpp": 36, "cpu": 206, "cr": 103, "cranknicolson": [149, 151], "crash": [40, 68], "crash_at_maxit": 130, "crashbas": [68, 74], "creat": [3, 5, 9, 10, 11, 13, 16, 17, 20, 22, 26, 29, 30, 31, 33, 37, 40, 48, 61, 103, 113, 117, 131, 144, 173, 174, 175, 176, 177, 184, 195, 197, 200, 201, 202, 203, 204, 206, 207], "createbinari": [17, 20], "creation": [0, 13, 22, 26, 29, 33, 37, 178, 179, 180, 181, 182, 186, 193, 201, 202, 203, 206], "criteria": [46, 72], "criterion": [45, 70, 196, 206], "crop": 38, "crucial": [79, 203], "csc": [94, 111, 119, 120, 128, 144], "csc_matrix": [62, 92, 104, 129], "csv": 40, "ct": 7, "cube": 39, "cube_1d": 36, "cubic": 109, "cupi": [0, 5, 57, 78, 86, 97, 99, 120], "cupy_mesh": [85, 97, 99, 120], "cupymulticomponentmesh": [85, 86], "current": [3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 17, 19, 20, 22, 26, 29, 33, 37, 43, 46, 48, 49, 50, 52, 53, 62, 66, 67, 69, 70, 71, 72, 73, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 92, 93, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 126, 127, 128, 129, 130, 131, 143, 144, 145, 146, 147, 148, 150, 151, 156, 159, 192, 196, 201, 205, 206, 207], "curv": 10, "custom": [2, 6, 8, 15, 42, 48, 49, 50, 51, 52, 53, 153, 154, 157, 159, 160, 161, 163, 164, 166, 167, 169, 202], "custon": [168, 170, 172], "cute": 9, "cv": 128, "cwd": [13, 17, 20, 22, 26, 29, 33, 37, 192, 205], "cycl": [14, 106], "d": [12, 15, 29, 30, 36, 37, 86, 88, 92, 101, 102, 103, 104, 106, 107, 108, 109, 110, 112, 114, 115, 116, 117, 118, 121, 123, 125, 127, 129, 130, 143, 144, 147, 192, 205], "d_collocation_accuracy_check": [176, 200], "d_mlsdc_with_particl": [186, 203], "d_pysdc_with_pytorch": [196, 206], "d_u": [113, 114, 115], "d_upwind": 104, "d_v": [113, 114, 115], "d_w": [95, 100, 101], "da": [112, 114], "dae": 0, "dahlquist": 38, "dalcin": [112, 114, 115, 118, 124, 143], "damp": 116, "daniel": [38, 40, 126], "darl": 151, "dash": 19, "dat": [13, 17, 20, 22, 26, 33], "data": [0, 3, 10, 11, 12, 13, 17, 19, 20, 22, 26, 27, 28, 29, 30, 31, 33, 34, 35, 37, 40, 43, 46, 47, 48, 49, 50, 51, 55, 60, 61, 66, 67, 79, 86, 87, 88, 89, 90, 92, 126, 165, 166, 167, 168, 169, 170, 171, 172, 173, 174, 175, 176, 177, 178, 179, 180, 181, 182, 183, 184, 185, 186, 187, 188, 189, 190, 192, 193, 194, 195, 196, 197, 198, 199, 200, 202, 203, 204, 205, 206, 207], "datacheck": [54, 61], "dataerror": [41, 47], "datatyp": [7, 11, 12, 46, 86, 88, 169, 206], "datatype_class": [7, 11, 12, 64, 174, 179, 181, 200, 201, 202, 206], "date": 5, "david": [115, 124, 143], "dc": 14, "ddx": 93, "de": [116, 152], "dead": 207, "deal": [14, 175, 176, 192, 200, 201, 202, 203, 205], "dearli": [3, 10], "debug": [45, 46, 205], "decai": [113, 114, 115], "decemb": 0, "decent": [3, 10], "decid": [30, 31, 37, 77, 81, 103, 206], "decis": [0, 1], "declar": 0, "decomposit": [19, 37], "decompress": 19, "decoupl": [37, 200], "decrement": 50, "dedic": [4, 9, 106], "deem": 1, "deeper": 201, "deepli": 201, "def": [5, 6, 7, 8, 12, 13, 17, 20, 22, 26, 27, 28, 29, 30, 31, 33, 35, 37, 146, 147, 173, 174, 175, 176, 177, 178, 179, 180, 181, 182, 183, 184, 185, 186, 187, 188, 189, 190, 191, 192, 193, 194, 195, 196, 197, 198, 199, 200, 201, 202, 203, 204, 205, 206, 207], "default": [4, 9, 12, 17, 20, 31, 37, 46, 48, 50, 53, 58, 59, 66, 67, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 84, 113, 117, 123, 128, 131, 144, 146, 147, 150, 156, 201, 203], "default_hook": [179, 181, 201, 202], "defaultdict": [13, 22, 26, 33], "defaulthook": [179, 201], "defer": [0, 3, 10, 12, 34, 36, 38, 92, 108, 157], "defin": [0, 2, 3, 6, 7, 10, 12, 35, 39, 42, 43, 50, 52, 70, 71, 72, 74, 80, 82, 83, 84, 106, 107, 108, 111, 112, 113, 116, 117, 118, 131, 144, 146, 147, 153, 157, 179, 181, 189, 201, 202, 203, 204, 206], "definit": [0, 7, 156, 202], "degre": [27, 28, 30, 31, 35, 37, 57, 92, 111, 113, 114, 115, 117, 131, 144, 148, 173, 175, 177, 178, 179, 180, 183, 184, 185, 187, 188, 189, 190, 195, 200, 201, 202, 203, 204, 205, 206], "del": [31, 37, 192, 205], "delet": [5, 77], "delta": [2, 12, 30, 31, 37, 95, 96, 97, 98, 99, 100, 101, 105, 108, 111, 112, 113, 114, 115, 124, 131], "delta_m": 43, "demonstr": [1, 10, 11, 19, 195, 196, 197, 201, 202, 203, 204, 205, 206, 207], "denot": [12, 36, 76, 93, 95, 98, 99, 100, 101, 103, 108, 115, 116, 117, 119, 120, 124, 131, 143], "dens": 77, "densiti": 128, "depend": [0, 3, 9, 10, 11, 14, 36, 46, 58, 70, 71, 72, 73, 75, 76, 79, 80, 83, 92, 93, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 112, 114, 115, 117, 118, 120, 122, 123, 124, 127, 128, 129, 130, 143, 144, 145, 146, 147, 148, 174, 176, 178, 179, 181, 195, 200, 201, 202, 203, 204, 206], "deprec": [0, 10], "der": [15, 16, 130, 146, 147], "deriv": [12, 15, 35, 45, 48, 57, 77, 86, 88, 92, 95, 104, 107, 116, 128, 131, 144, 145, 146, 147, 148, 157], "derogatori": 1, "describ": [8, 9, 10, 12, 17, 18, 20, 39, 103, 110, 113, 114, 115, 125, 180, 202], "descript": [0, 5, 10, 12, 13, 16, 17, 20, 22, 26, 27, 28, 29, 30, 31, 33, 35, 37, 41, 45, 46, 52, 59, 66, 67, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 177, 178, 179, 180, 181, 182, 184, 185, 186, 187, 188, 189, 190, 192, 193, 194, 195, 197, 198, 199, 201, 202, 203, 204, 205, 206, 207], "description_mlsdc": [185, 203], "description_mssdc": [198, 207], "description_pfasst": [198, 207], "description_sdc": [185, 203], "design": 16, "desir": [19, 70, 71, 76, 82], "dest": [46, 86, 88, 89, 90], "destroi": 128, "detail": [0, 2, 3, 5, 8, 9, 10, 11, 14, 70, 124, 126, 202, 205, 206], "detect": [5, 19, 79], "detector": [16, 79], "deterior": [6, 14], "determin": [0, 1, 11, 40, 46, 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"implementations.problem_classes.GeneralizedFisher_1D_PETSc.petsc_fisher_multiimplicit.get_sys_mat", false]], "get_transfer_matrix_q() (basetransfer static method)": [[42, "core.base_transfer.BaseTransfer.get_transfer_matrix_Q", false]], "get_update_order() (ark54 class method)": [[156, "implementations.sweeper_classes.Runge_Kutta.ARK54.get_update_order", false]], "get_update_order() (ark548l2sa class method)": [[156, "implementations.sweeper_classes.Runge_Kutta.ARK548L2SA.get_update_order", false]], "get_update_order() (ark548l2saerk class method)": [[156, "implementations.sweeper_classes.Runge_Kutta.ARK548L2SAERK.get_update_order", false]], "get_update_order() (ark548l2saesdirk2 class method)": [[156, "implementations.sweeper_classes.Runge_Kutta.ARK548L2SAESDIRK2.get_update_order", false]], "get_update_order() (cash_karp class method)": [[156, "implementations.sweeper_classes.Runge_Kutta.Cash_Karp.get_update_order", false]], "get_update_order() (dirk43 class method)": [[156, "implementations.sweeper_classes.Runge_Kutta.DIRK43.get_update_order", false]], "get_update_order() (esdirk43 class method)": [[156, "implementations.sweeper_classes.Runge_Kutta.ESDIRK43.get_update_order", false]], "get_update_order() (esdirk53 class method)": [[156, "implementations.sweeper_classes.Runge_Kutta.ESDIRK53.get_update_order", false]], "get_update_order() (heun_euler class method)": [[156, "implementations.sweeper_classes.Runge_Kutta.Heun_Euler.get_update_order", false]], "get_update_order() (rungekutta class method)": [[156, "implementations.sweeper_classes.Runge_Kutta.RungeKutta.get_update_order", false]], "get_wavenumbers() (chebychevhelper method)": [[60, "helpers.spectral_helper.ChebychevHelper.get_wavenumbers", false]], "get_wavenumbers() (ffthelper method)": [[60, "helpers.spectral_helper.FFTHelper.get_wavenumbers", false]], "get_wavenumbers() (spectralhelper method)": [[60, "helpers.spectral_helper.SpectralHelper.get_wavenumbers", false]], "get_wavenumbers() (spectralhelper1d method)": [[60, "helpers.spectral_helper.SpectralHelper1D.get_wavenumbers", false]], "get_zero() (spectralhelper1d method)": [[60, "helpers.spectral_helper.SpectralHelper1D.get_zero", false]], "getbchorizontal() (in module implementations.problem_classes.boussinesq_helpers.build2dfdmatrix)": [[141, "implementations.problem_classes.boussinesq_helpers.build2DFDMatrix.getBCHorizontal", false]], "getbcleft() (in module implementations.problem_classes.acoustic_helpers.buildfdmatrix)": [[137, "implementations.problem_classes.acoustic_helpers.buildFDMatrix.getBCLeft", false]], "getbcleft() (in module implementations.problem_classes.boussinesq_helpers.buildfdmatrix)": [[143, "implementations.problem_classes.boussinesq_helpers.buildFDMatrix.getBCLeft", false]], "getbcright() (in module implementations.problem_classes.acoustic_helpers.buildfdmatrix)": [[137, "implementations.problem_classes.acoustic_helpers.buildFDMatrix.getBCRight", false]], "getbcright() (in module implementations.problem_classes.boussinesq_helpers.buildfdmatrix)": [[143, "implementations.problem_classes.boussinesq_helpers.buildFDMatrix.getBCRight", false]], "getbcvertical() (in module implementations.problem_classes.boussinesq_helpers.build2dfdmatrix)": [[141, "implementations.problem_classes.boussinesq_helpers.build2DFDMatrix.getBCVertical", false]], "getboussinesq2dmatrix() (in module implementations.problem_classes.boussinesq_helpers.buildboussinesq2dmatrix)": [[142, "implementations.problem_classes.boussinesq_helpers.buildBoussinesq2DMatrix.getBoussinesq2DMatrix", false]], "getboussinesq2dupwindmatrix() (in module implementations.problem_classes.boussinesq_helpers.buildboussinesq2dmatrix)": [[142, "implementations.problem_classes.boussinesq_helpers.buildBoussinesq2DMatrix.getBoussinesq2DUpwindMatrix", false]], "getboussinesqbchorizontal() (in module implementations.problem_classes.boussinesq_helpers.buildboussinesq2dmatrix)": [[142, "implementations.problem_classes.boussinesq_helpers.buildBoussinesq2DMatrix.getBoussinesqBCHorizontal", false]], "getboussinesqbcvertical() (in module implementations.problem_classes.boussinesq_helpers.buildboussinesq2dmatrix)": [[142, "implementations.problem_classes.boussinesq_helpers.buildBoussinesq2DMatrix.getBoussinesqBCVertical", false]], "getcounter() (callback method)": [[144, "implementations.problem_classes.boussinesq_helpers.helper_classes.Callback.getcounter", false]], "gethorizontaldx() (in module implementations.problem_classes.acoustic_helpers.buildfdmatrix)": [[137, "implementations.problem_classes.acoustic_helpers.buildFDMatrix.getHorizontalDx", false]], "getmatrix() (in module implementations.problem_classes.acoustic_helpers.buildfdmatrix)": [[137, "implementations.problem_classes.acoustic_helpers.buildFDMatrix.getMatrix", false]], "getmatrix() (in module implementations.problem_classes.boussinesq_helpers.buildfdmatrix)": [[143, "implementations.problem_classes.boussinesq_helpers.buildFDMatrix.getMatrix", false]], "getresidual() (callback method)": [[144, "implementations.problem_classes.boussinesq_helpers.helper_classes.Callback.getresidual", false]], "getupwindmatrix() (in module implementations.problem_classes.boussinesq_helpers.buildfdmatrix)": [[143, "implementations.problem_classes.boussinesq_helpers.buildFDMatrix.getUpwindMatrix", false]], "getwave1dadvectionmatrix() (in module implementations.problem_classes.acoustic_helpers.buildwave1dmatrix)": [[138, "implementations.problem_classes.acoustic_helpers.buildWave1DMatrix.getWave1DAdvectionMatrix", false]], "getwave1dmatrix() (in module implementations.problem_classes.acoustic_helpers.buildwave1dmatrix)": [[138, "implementations.problem_classes.acoustic_helpers.buildWave1DMatrix.getWave1DMatrix", false]], "getwavebcleft() (in module implementations.problem_classes.acoustic_helpers.buildwave1dmatrix)": [[138, "implementations.problem_classes.acoustic_helpers.buildWave1DMatrix.getWaveBCLeft", false]], "getwavebcright() (in module implementations.problem_classes.acoustic_helpers.buildwave1dmatrix)": [[138, "implementations.problem_classes.acoustic_helpers.buildWave1DMatrix.getWaveBCRight", false]], "global_shape (spectralhelper attribute)": [[60, "helpers.spectral_helper.SpectralHelper.global_shape", false]], "gmres_logger (boussinesq_2d_imex attribute)": [[105, "implementations.problem_classes.Boussinesq_2D_FD_imex.boussinesq_2d_imex.gmres_logger", false]], "grayscott_imex_diffusion (class in implementations.problem_classes.grayscott_mpifft)": [[117, "implementations.problem_classes.GrayScott_MPIFFT.grayscott_imex_diffusion", false]], "grayscott_imex_linear (class in implementations.problem_classes.grayscott_mpifft)": [[117, "implementations.problem_classes.GrayScott_MPIFFT.grayscott_imex_linear", false]], "grayscott_mi_diffusion (class in implementations.problem_classes.grayscott_mpifft)": [[117, "implementations.problem_classes.GrayScott_MPIFFT.grayscott_mi_diffusion", false]], "grayscott_mi_linear (class in implementations.problem_classes.grayscott_mpifft)": [[117, "implementations.problem_classes.GrayScott_MPIFFT.grayscott_mi_linear", false]], "grids (genericndimfindiff property)": [[148, "implementations.problem_classes.generic_ND_FD.GenericNDimFinDiff.grids", false]], "gs_full (class in implementations.problem_classes.grayscott_2d_petsc_periodic)": [[116, "implementations.problem_classes.GrayScott_2D_PETSc_periodic.GS_full", false]], "gs_reaction (class in implementations.problem_classes.grayscott_2d_petsc_periodic)": [[116, "implementations.problem_classes.GrayScott_2D_PETSc_periodic.GS_reaction", false]], "h (boussinesq_2d_imex attribute)": [[105, "implementations.problem_classes.Boussinesq_2D_FD_imex.boussinesq_2d_imex.h", false]], "harmonic_oscillator (class in implementations.problem_classes.harmonicoscillator)": [[118, "implementations.problem_classes.HarmonicOscillator.harmonic_oscillator", false]], "heat1dchebychev (class in implementations.problem_classes.heatequation_chebychev)": [[121, "implementations.problem_classes.HeatEquation_Chebychev.Heat1DChebychev", false]], "heat1dultraspherical (class in implementations.problem_classes.heatequation_chebychev)": [[121, "implementations.problem_classes.HeatEquation_Chebychev.Heat1DUltraspherical", false]], "heat2d_petsc_forced (class in implementations.problem_classes.heatequation_2d_petsc_forced)": [[120, "implementations.problem_classes.HeatEquation_2D_PETSc_forced.heat2d_petsc_forced", false]], "heat2dchebychev (class in implementations.problem_classes.heatequation_chebychev)": [[121, "implementations.problem_classes.HeatEquation_Chebychev.Heat2DChebychev", false]], "heat2dultraspherical (class in implementations.problem_classes.heatequation_chebychev)": [[121, "implementations.problem_classes.HeatEquation_Chebychev.Heat2DUltraspherical", false]], "heatnd_forced (class in implementations.problem_classes.heatequation_nd_fd)": [[122, "implementations.problem_classes.HeatEquation_ND_FD.heatNd_forced", false]], "heatnd_forced (class in implementations.problem_classes.heatequation_nd_fd_cupy)": [[123, "implementations.problem_classes.HeatEquation_ND_FD_CuPy.heatNd_forced", false]], "heatnd_unforced (class in implementations.problem_classes.heatequation_nd_fd)": [[122, "implementations.problem_classes.HeatEquation_ND_FD.heatNd_unforced", false]], "heatnd_unforced (class in implementations.problem_classes.heatequation_nd_fd_cupy)": [[123, "implementations.problem_classes.HeatEquation_ND_FD_CuPy.heatNd_unforced", false]], "helpers": [[54, "module-helpers", false]], "helpers.nccl_communicator": [[55, "module-helpers.NCCL_communicator", false]], "helpers.plot_helper": [[56, "module-helpers.plot_helper", false]], "helpers.problem_helper": [[57, "module-helpers.problem_helper", false]], "helpers.pysdc_helper": [[58, "module-helpers.pysdc_helper", false]], "helpers.setup_helper": [[59, "module-helpers.setup_helper", false]], "helpers.spectral_helper": [[60, "module-helpers.spectral_helper", false]], "helpers.stats_helper": [[61, "module-helpers.stats_helper", false]], "helpers.testing": [[62, "module-helpers.testing", false]], "helpers.transfer_helper": [[63, "module-helpers.transfer_helper", false]], "helpers.visualization_tools": [[64, "module-helpers.visualization_tools", false]], "henon_heiles (class in implementations.problem_classes.henonheiles)": [[124, "implementations.problem_classes.HenonHeiles.henon_heiles", false]], "heun_euler (class in implementations.sweeper_classes.runge_kutta)": [[156, "implementations.sweeper_classes.Runge_Kutta.Heun_Euler", false]], "hooks (class in core.hooks)": [[48, "core.hooks.Hooks", false]], "hooks (controller property)": [[45, "core.controller.Controller.hooks", false]], "hotrod (class in implementations.convergence_controller_classes.hotrod)": [[80, "implementations.convergence_controller_classes.hotrod.HotRod", false]], "id (acoustic_1d_imex attribute)": [[93, "implementations.problem_classes.AcousticAdvection_1D_FD_imex.acoustic_1d_imex.Id", false]], "id (advectionnd attribute)": [[95, "implementations.problem_classes.AdvectionEquation_ND_FD.advectionNd.Id", false]], "id (boussinesq_2d_imex attribute)": [[105, "implementations.problem_classes.Boussinesq_2D_FD_imex.boussinesq_2d_imex.Id", false]], "id (genericndimfindiff attribute)": [[148, "implementations.problem_classes.generic_ND_FD.GenericNDimFinDiff.Id", false]], "id (heat2d_petsc_forced attribute)": [[120, "implementations.problem_classes.HeatEquation_2D_PETSc_forced.heat2d_petsc_forced.Id", false]], "id (heatnd_forced attribute)": [[123, "implementations.problem_classes.HeatEquation_ND_FD_CuPy.heatNd_forced.Id", false]], "id (heatnd_unforced attribute)": [[122, "implementations.problem_classes.HeatEquation_ND_FD.heatNd_unforced.Id", false]], "id (petsc_grayscott_multiimplicit attribute)": [[116, "implementations.problem_classes.GrayScott_2D_PETSc_periodic.petsc_grayscott_multiimplicit.Id", false]], "id (quench attribute)": [[131, "implementations.problem_classes.Quench.Quench.Id", false]], "imex_1st_order (class in implementations.sweeper_classes.imex_1st_order)": [[162, "implementations.sweeper_classes.imex_1st_order.imex_1st_order", false]], "imex_1st_order_mass (class in implementations.sweeper_classes.imex_1st_order_mass)": [[164, "implementations.sweeper_classes.imex_1st_order_mass.imex_1st_order_mass", false]], "imex_1st_order_mpi (class in implementations.sweeper_classes.imex_1st_order_mpi)": [[163, "implementations.sweeper_classes.imex_1st_order_MPI.imex_1st_order_MPI", false]], "imex_cupy_mesh (class in implementations.datatype_classes.cupy_mesh)": [[87, "implementations.datatype_classes.cupy_mesh.imex_cupy_mesh", false]], "imex_laplacian_mpifft (class in implementations.problem_classes.generic_mpifft_laplacian)": [[147, "implementations.problem_classes.generic_MPIFFT_Laplacian.IMEX_Laplacian_MPIFFT", false]], "imex_mesh (class in implementations.datatype_classes.mesh)": [[89, "implementations.datatype_classes.mesh.imex_mesh", false]], "impl (petsc_vec_comp2 attribute)": [[91, "implementations.datatype_classes.petsc_vec.petsc_vec_comp2.impl", false]], "impl (petsc_vec_imex attribute)": [[91, "implementations.datatype_classes.petsc_vec.petsc_vec_imex.impl", false]], "impl (rhs_fenics_mesh attribute)": [[88, "implementations.datatype_classes.fenics_mesh.rhs_fenics_mesh.impl", false]], "implementations": [[65, "module-implementations", false]], "implementations.controller_classes": [[66, "module-implementations.controller_classes", false]], "implementations.controller_classes.controller_mpi": [[67, "module-implementations.controller_classes.controller_MPI", false]], "implementations.controller_classes.controller_nonmpi": [[68, "module-implementations.controller_classes.controller_nonMPI", false]], "implementations.convergence_controller_classes": [[69, "module-implementations.convergence_controller_classes", false]], "implementations.convergence_controller_classes.adaptive_collocation": [[70, "module-implementations.convergence_controller_classes.adaptive_collocation", false]], "implementations.convergence_controller_classes.adaptivity": [[71, "module-implementations.convergence_controller_classes.adaptivity", false]], "implementations.convergence_controller_classes.basic_restarting": [[72, "module-implementations.convergence_controller_classes.basic_restarting", false]], "implementations.convergence_controller_classes.check_convergence": [[73, "module-implementations.convergence_controller_classes.check_convergence", false]], "implementations.convergence_controller_classes.check_iteration_estimator": [[74, "module-implementations.convergence_controller_classes.check_iteration_estimator", false]], "implementations.convergence_controller_classes.crash": [[75, "module-implementations.convergence_controller_classes.crash", false]], "implementations.convergence_controller_classes.estimate_contraction_factor": [[76, "module-implementations.convergence_controller_classes.estimate_contraction_factor", false]], "implementations.convergence_controller_classes.estimate_embedded_error": [[77, "module-implementations.convergence_controller_classes.estimate_embedded_error", false]], "implementations.convergence_controller_classes.estimate_extrapolation_error": [[78, "module-implementations.convergence_controller_classes.estimate_extrapolation_error", false]], "implementations.convergence_controller_classes.estimate_polynomial_error": [[79, "module-implementations.convergence_controller_classes.estimate_polynomial_error", false]], "implementations.convergence_controller_classes.hotrod": [[80, "module-implementations.convergence_controller_classes.hotrod", false]], "implementations.convergence_controller_classes.inexactness": [[81, "module-implementations.convergence_controller_classes.inexactness", false]], "implementations.convergence_controller_classes.interpolate_between_restarts": [[82, "module-implementations.convergence_controller_classes.interpolate_between_restarts", false]], "implementations.convergence_controller_classes.spread_step_sizes": [[83, "module-implementations.convergence_controller_classes.spread_step_sizes", false]], "implementations.convergence_controller_classes.step_size_limiter": [[84, "module-implementations.convergence_controller_classes.step_size_limiter", false]], "implementations.convergence_controller_classes.store_uold": [[85, "module-implementations.convergence_controller_classes.store_uold", false]], "implementations.datatype_classes": [[86, "module-implementations.datatype_classes", false]], "implementations.datatype_classes.cupy_mesh": [[87, "module-implementations.datatype_classes.cupy_mesh", false]], "implementations.datatype_classes.fenics_mesh": [[88, "module-implementations.datatype_classes.fenics_mesh", false]], "implementations.datatype_classes.mesh": [[89, "module-implementations.datatype_classes.mesh", false]], "implementations.datatype_classes.particles": [[90, "module-implementations.datatype_classes.particles", false]], "implementations.datatype_classes.petsc_vec": [[91, "module-implementations.datatype_classes.petsc_vec", false]], "implementations.problem_classes": [[92, "module-implementations.problem_classes", false]], "implementations.problem_classes.acoustic_helpers": [[136, "module-implementations.problem_classes.acoustic_helpers", false]], "implementations.problem_classes.acoustic_helpers.buildfdmatrix": [[137, "module-implementations.problem_classes.acoustic_helpers.buildFDMatrix", false]], "implementations.problem_classes.acoustic_helpers.buildwave1dmatrix": [[138, "module-implementations.problem_classes.acoustic_helpers.buildWave1DMatrix", false]], "implementations.problem_classes.acoustic_helpers.standard_integrators": [[139, "module-implementations.problem_classes.acoustic_helpers.standard_integrators", false]], "implementations.problem_classes.acousticadvection_1d_fd_imex": [[93, "module-implementations.problem_classes.AcousticAdvection_1D_FD_imex", false]], "implementations.problem_classes.advectiondiffusionequation_1d_fft": [[94, "module-implementations.problem_classes.AdvectionDiffusionEquation_1D_FFT", false]], "implementations.problem_classes.advectionequation_nd_fd": [[95, "module-implementations.problem_classes.AdvectionEquation_ND_FD", false]], "implementations.problem_classes.allencahn_1d_fd": [[96, "module-implementations.problem_classes.AllenCahn_1D_FD", false]], "implementations.problem_classes.allencahn_2d_fd": [[97, "module-implementations.problem_classes.AllenCahn_2D_FD", false]], "implementations.problem_classes.allencahn_2d_fd_gpu": [[98, "module-implementations.problem_classes.AllenCahn_2D_FD_gpu", false]], "implementations.problem_classes.allencahn_2d_fft": [[99, "module-implementations.problem_classes.AllenCahn_2D_FFT", false]], "implementations.problem_classes.allencahn_2d_fft_gpu": [[100, "module-implementations.problem_classes.AllenCahn_2D_FFT_gpu", false]], "implementations.problem_classes.allencahn_mpifft": [[101, "module-implementations.problem_classes.AllenCahn_MPIFFT", false]], "implementations.problem_classes.allencahn_temp_mpifft": [[102, "module-implementations.problem_classes.AllenCahn_Temp_MPIFFT", false]], "implementations.problem_classes.auzinger_implicit": [[103, "module-implementations.problem_classes.Auzinger_implicit", false]], "implementations.problem_classes.battery": [[104, "module-implementations.problem_classes.Battery", false]], "implementations.problem_classes.boussinesq_2d_fd_imex": [[105, "module-implementations.problem_classes.Boussinesq_2D_FD_imex", false]], "implementations.problem_classes.boussinesq_helpers": [[140, "module-implementations.problem_classes.boussinesq_helpers", false]], "implementations.problem_classes.boussinesq_helpers.build2dfdmatrix": [[141, "module-implementations.problem_classes.boussinesq_helpers.build2DFDMatrix", false]], "implementations.problem_classes.boussinesq_helpers.buildboussinesq2dmatrix": [[142, "module-implementations.problem_classes.boussinesq_helpers.buildBoussinesq2DMatrix", false]], "implementations.problem_classes.boussinesq_helpers.buildfdmatrix": [[143, "module-implementations.problem_classes.boussinesq_helpers.buildFDMatrix", false]], "implementations.problem_classes.boussinesq_helpers.helper_classes": [[144, "module-implementations.problem_classes.boussinesq_helpers.helper_classes", false]], "implementations.problem_classes.boussinesq_helpers.standard_integrators": [[145, "module-implementations.problem_classes.boussinesq_helpers.standard_integrators", false]], "implementations.problem_classes.boussinesq_helpers.unflatten": [[146, "module-implementations.problem_classes.boussinesq_helpers.unflatten", false]], "implementations.problem_classes.brusselator": [[106, "module-implementations.problem_classes.Brusselator", false]], "implementations.problem_classes.buckconverter": [[107, "module-implementations.problem_classes.BuckConverter", false]], "implementations.problem_classes.burgers": [[108, "module-implementations.problem_classes.Burgers", false]], "implementations.problem_classes.discontinuoustestode": [[109, "module-implementations.problem_classes.DiscontinuousTestODE", false]], "implementations.problem_classes.fastwaveslowwave_0d": [[110, "module-implementations.problem_classes.FastWaveSlowWave_0D", false]], "implementations.problem_classes.fermipastaulamtsingou": [[111, "module-implementations.problem_classes.FermiPastaUlamTsingou", false]], "implementations.problem_classes.fullsolarsystem": [[112, "module-implementations.problem_classes.FullSolarSystem", false]], "implementations.problem_classes.generalizedfisher_1d_fd_implicit": [[113, "module-implementations.problem_classes.GeneralizedFisher_1D_FD_implicit", false]], "implementations.problem_classes.generalizedfisher_1d_petsc": [[114, "module-implementations.problem_classes.GeneralizedFisher_1D_PETSc", false]], "implementations.problem_classes.generic_mpifft_laplacian": [[147, "module-implementations.problem_classes.generic_MPIFFT_Laplacian", false]], "implementations.problem_classes.generic_nd_fd": [[148, "module-implementations.problem_classes.generic_ND_FD", false]], "implementations.problem_classes.generic_spectral": [[149, "module-implementations.problem_classes.generic_spectral", false]], "implementations.problem_classes.grayscott_1d_fenics_implicit": [[115, "module-implementations.problem_classes.GrayScott_1D_FEniCS_implicit", false]], "implementations.problem_classes.grayscott_2d_petsc_periodic": [[116, "module-implementations.problem_classes.GrayScott_2D_PETSc_periodic", false]], 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11, 13, 17, 19, 20, 22, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 37, 40, 60, 75, 87, 89, 132, 178, 179, 180, 181, 182, 183, 184, 185, 186, 187, 188, 189, 190, 191, 192, 193, 194, 195, 196, 197, 198, 199, 200, 201, 202, 203, 204, 205, 206, 207, 208, 209, 210, 211, 212], "code_of_conduct": 1, "codecov": [0, 5], "coef": 60, "coeff": [60, 78, 148], "coeffic": 104, "coeffici": [2, 27, 28, 35, 43, 53, 60, 78, 94, 104, 107, 119, 120, 122, 123, 125, 130, 135, 178, 179, 180, 181, 182, 183, 184, 185, 188, 189, 190, 192, 193, 194, 195, 200, 203, 204, 205, 206, 207, 208, 209, 210, 211, 212], "coil": 14, "col": 114, "coll": [53, 180, 181, 189, 205, 208], "coll_err": [204, 212], "coll_error": [204, 212], "coll_error_after_step": [204, 212], "collabor": [0, 19], "collbas": [41, 43, 53, 180, 181, 184, 186, 205, 206, 207], "collect": [0, 5, 13, 22, 26, 30, 31, 33, 37, 111, 179, 181, 188, 205, 208], "colloc": [0, 2, 3, 10, 15, 19, 34, 35, 36, 37, 41, 47, 53, 70, 71, 77, 78, 79, 82, 83, 104, 109, 149, 155, 156, 157, 158, 159, 160, 161, 162, 163, 164, 165, 166, 180, 181, 183, 184, 186, 187, 201, 204, 206, 208, 211, 212], "collocation_class": [156, 157, 205], "collocation_nod": [201, 211], "collocationerror": [41, 47], "collumn": 53, "color": [5, 19, 29, 30, 31, 37, 179, 181, 200, 205, 211], "color_list": [29, 30, 31, 37], "column": [8, 78, 114, 162, 180, 181, 205], "com": [2, 3, 10, 106, 157], "combin": [5, 12, 13, 36, 60, 78, 197, 200, 208, 210, 211], "come": [0, 4, 5, 11, 16, 19, 29, 37, 46, 205, 210, 212], "comm": [31, 37, 46, 55, 60, 61, 67, 72, 73, 75, 83, 87, 89, 90, 91, 101, 102, 114, 116, 117, 120, 127, 132, 147, 149, 161, 197, 199, 200, 210, 211], "comm_world": [31, 37, 114, 116, 117, 120, 127, 147, 197, 199, 200, 210, 211], "command": [5, 16, 36], "comment": [1, 4, 5, 38], "commerci": 37, "commit": [1, 2, 4, 5, 9, 16], "commit_msg_fil": 5, "common": [1, 12, 14, 41, 60], "commonli": 12, "commun": [0, 1, 3, 10, 11, 19, 46, 47, 48, 55, 60, 61, 67, 70, 72, 73, 74, 75, 77, 79, 83, 87, 89, 90, 91, 101, 102, 114, 116, 117, 120, 127, 132, 147, 161, 197, 200, 210, 211], "communicate_converg": 73, "communicate_crash": 75, "communicate_tau_correction_for_full_interv": 161, "communicationerror": [41, 47], "comp": 108, "comp1": [87, 89], "comp2": [87, 89], "comp2_cupy_mesh": [86, 87, 98], "comp2_mesh": [86, 89, 96, 97, 117], "compact": 36, "compactli": 35, "companion": 2, "compar": [5, 11, 14, 17, 20, 27, 28, 29, 34, 35, 36, 37, 62, 70, 71, 78, 79, 80, 99, 100, 178, 179, 183, 184, 186, 187, 188, 190, 191, 193, 194, 195, 197, 198, 200, 201, 202, 203, 205, 206, 207, 208, 209, 210, 211, 212], "compare_control": [27, 28, 35], "compare_to_matrixbas": [27, 35], "compare_to_propag": [28, 35], "comparison": 17, "comparison_matrix_vs_nomat_detail": [27, 35], "comparison_matrix_vs_propagator_detail": [28, 35], "compat": [0, 3, 4, 10, 46, 70, 71, 74, 77, 78, 79, 80], "compil": [3, 10, 36], "complaint": [0, 1], "complet": [0, 5, 67, 201, 209, 211], "complex": [7, 10, 13, 34, 36, 55, 60, 89, 207], "complianc": 5, "complic": [207, 209], "compon": [2, 5, 11, 12, 14, 36, 60, 87, 88, 89, 91, 103, 117, 125, 126, 134, 149, 150, 151, 152, 165, 212], "composit": [35, 46, 83], "comprehens": 70, "compress": [0, 10], "compressor": 19, "compromis": 6, "comput": [0, 3, 7, 8, 10, 11, 12, 13, 15, 17, 19, 20, 22, 26, 29, 33, 37, 38, 40, 42, 46, 50, 53, 60, 62, 67, 70, 71, 77, 78, 84, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 131, 132, 133, 134, 135, 147, 149, 150, 151, 152, 153, 155, 156, 157, 158, 159, 160, 161, 162, 163, 164, 165, 166, 168, 169, 178, 179, 180, 181, 183, 184, 186, 188, 190, 193, 194, 195, 197, 198, 200, 201, 202, 203, 205, 206, 207, 208, 209, 210, 211, 212], "compute_buoyancy_gener": 132, "compute_end_point": [53, 155, 156, 157, 158, 159, 160, 161, 162, 163, 164, 165, 166, 183, 206], "compute_max_step_s": 132, 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191, 192, 193, 194, 195, 198, 199, 202, 203, 204, 206, 207, 208, 209, 210, 211, 212], "controller_param": [13, 17, 20, 22, 26, 27, 28, 29, 30, 31, 33, 35, 37, 45, 67, 68, 184, 185, 186, 187, 190, 191, 192, 193, 194, 195, 197, 198, 199, 200, 202, 203, 204, 206, 207, 208, 209, 210, 211, 212], "controller_paramet": 206, "controller_params_g": [203, 212], "controller_params_jac": [203, 212], "controller_pfasst": [203, 212], "controller_sdc": [190, 208], "controllererror": [41, 47], "contruct": 63, "conv": [40, 60], "conv_test_to0": 18, "conv_test_toinf": 18, "convect": [126, 132], "conveni": [16, 49, 205], "convent": [2, 5, 7], "converg": [0, 10, 12, 13, 14, 19, 34, 38, 40, 45, 46, 47, 67, 68, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 132, 184, 186, 198, 205, 206, 207, 210, 211], "convergence_control": [41, 45, 204, 212], "convergence_controller_class": [65, 204, 212], "convergencecontrol": [41, 45, 46, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 132], "convergenceerror": [41, 47, 75], "convers": [60, 149], "convert": [27, 35, 60, 107, 185, 187, 191, 203, 207, 208, 212], "convert_operator_matrix_to_oper": 60, "cool": 131, "coordin": [60, 95, 101, 102, 117, 127, 147], "copi": [0, 5, 7, 49, 53, 155, 161, 197, 203, 210, 212], "core": [0, 2, 3, 4, 5, 7, 10, 13, 161, 180, 181, 182, 183, 184, 186, 189, 205, 206, 207, 208, 211], "correct": [0, 3, 4, 9, 10, 11, 12, 34, 35, 36, 38, 42, 49, 50, 60, 67, 68, 93, 106, 108, 110, 111, 129, 135, 157, 162, 168, 169, 189, 198, 208, 211], "correctli": 12, "correctt": 36, "correl": 71, "corresp": 52, "correspond": [3, 5, 10, 12, 13, 19, 26, 27, 28, 34, 35, 46, 52, 60, 61, 70, 158, 159, 160, 161, 162, 163, 164, 165, 166, 208, 210, 212], "corrupt": 80, "cosimo": [114, 116, 120], "cosin": 60, "cost": [38, 60], "costli": [19, 208], "could": [0, 1, 4, 12, 19, 70, 71, 79, 156, 157], "coulomb": 129, "count": [17, 18, 20, 27, 28, 30, 31, 35, 37, 38, 45, 46, 50, 96, 104, 109, 126, 129, 134, 149, 150, 185, 194, 195, 197, 203, 206, 207, 208, 209, 210, 212], "count_switch": [104, 109], "counter": [96, 183, 206], "counterpart": [36, 60, 210], "coupl": [36, 52, 87, 89, 205, 211], "cours": [6, 46, 208], "cover": [5, 9, 31, 37], "coverag": 10, "coverage_process_start": [197, 210], "cp": 100, "cpp": 36, "cpu": 211, "cr": 104, "cranknicolson": [154, 156], "crash": [40, 69], "crash_at_maxit": 134, "crashbas": [69, 75], "creat": [3, 5, 9, 10, 11, 13, 16, 17, 20, 22, 26, 29, 30, 31, 33, 37, 40, 48, 62, 104, 115, 119, 135, 148, 178, 179, 180, 181, 182, 189, 200, 202, 205, 206, 207, 208, 209, 211, 212], "createbinari": [17, 20], "creation": [0, 13, 22, 26, 29, 33, 37, 183, 184, 185, 186, 187, 191, 198, 206, 207, 208, 211], "criteria": [46, 73, 132], "criterion": [45, 71, 201, 211], "crop": 38, "crucial": [80, 208], "csc": [95, 113, 122, 123, 131, 148], "csc_matrix": [63, 93, 105, 133], "csv": 40, "ct": 7, "cube": 39, "cube_1d": 36, "cubic": 111, "cupi": [0, 5, 57, 79, 87, 98, 100, 123], "cupy_mesh": [86, 98, 100, 123], "cupymulticomponentmesh": [86, 87], "current": [3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 17, 19, 20, 22, 26, 29, 33, 37, 43, 46, 48, 49, 50, 52, 53, 60, 63, 67, 68, 70, 71, 72, 73, 74, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 93, 94, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 129, 130, 131, 132, 133, 134, 135, 147, 148, 149, 150, 151, 152, 153, 155, 156, 161, 164, 197, 201, 206, 210, 211, 212], "curv": 10, "custom": [2, 6, 8, 15, 42, 48, 49, 50, 51, 52, 53, 158, 159, 162, 164, 165, 166, 168, 169, 171, 172, 174, 207], "custon": [173, 175, 177], "cute": 9, "cv": 131, "cwd": [13, 17, 20, 22, 26, 29, 33, 37, 197, 210], "cycl": [14, 107], "d": [12, 15, 29, 30, 36, 37, 60, 87, 89, 93, 102, 103, 104, 105, 107, 109, 110, 111, 112, 114, 116, 117, 118, 119, 120, 124, 126, 128, 130, 133, 134, 147, 148, 152, 197, 210], "d_collocation_accuracy_check": [181, 205], "d_mlsdc_with_particl": [191, 208], "d_pysdc_with_pytorch": [201, 211], "d_u": [115, 116, 117], "d_upwind": 105, "d_v": [115, 116, 117], "d_w": [96, 101, 102], "da": [114, 116], "dae": [0, 149], "dahlquist": 38, "dalcin": [114, 116, 117, 120, 127, 147], "damp": 118, "daniel": [38, 40, 129], "darl": 156, "dash": 19, "dat": [13, 17, 20, 22, 26, 33], "data": [0, 3, 10, 11, 12, 13, 17, 19, 20, 22, 26, 27, 28, 29, 30, 31, 33, 34, 35, 37, 40, 43, 46, 47, 48, 49, 50, 51, 55, 60, 61, 62, 67, 68, 80, 87, 88, 89, 90, 91, 93, 129, 170, 171, 172, 173, 174, 175, 176, 177, 178, 179, 180, 181, 182, 183, 184, 185, 186, 187, 188, 189, 190, 191, 192, 193, 194, 195, 197, 198, 199, 200, 201, 202, 203, 204, 205, 207, 208, 209, 210, 211, 212], "datacheck": [54, 62], "dataerror": [41, 47], "datatyp": [7, 11, 12, 46, 87, 89, 149, 174, 211], "datatype_class": [7, 11, 12, 65, 179, 184, 186, 205, 206, 207, 211], "date": 5, "david": [117, 127, 147], "dc": 14, "dct": 60, "ddx": 94, "de": [118, 157], "dead": 212, 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"implementations.transfer_classes.BaseTransferMPI module", "implementations.transfer_classes.BaseTransfer_mass module", "implementations.transfer_classes.TransferFenicsMesh module", "implementations.transfer_classes.TransferMesh module", "implementations.transfer_classes.TransferMesh_FFT module", "implementations.transfer_classes.TransferMesh_FFT2D module", "implementations.transfer_classes.TransferMesh_MPIFFT module", "implementations.transfer_classes.TransferMesh_NoCoarse module", "implementations.transfer_classes.TransferPETScDMDA module", "implementations.transfer_classes.TransferParticles_NoCoarse module", "&lt;no title&gt;", "&lt;no title&gt;", "&lt;no title&gt;", "&lt;no title&gt;", "&lt;no title&gt;", "&lt;no title&gt;", "&lt;no title&gt;", "&lt;no title&gt;", "&lt;no title&gt;", "&lt;no title&gt;", "&lt;no title&gt;", "&lt;no title&gt;", "&lt;no title&gt;", "&lt;no title&gt;", "&lt;no title&gt;", "&lt;no title&gt;", "&lt;no title&gt;", "&lt;no title&gt;", "&lt;no title&gt;", "&lt;no title&gt;", "&lt;no title&gt;", "&lt;no title&gt;", "&lt;no title&gt;", "&lt;no title&gt;", "&lt;no title&gt;", "&lt;no title&gt;", "&lt;no title&gt;", "Step-1: A first spatial problem", "Step-2: Data structures and my first sweeper", "Step-3: Statistics and a new sweeper", "Step-4: Multilevel SDC", "Step-5: PFASST", "Step-6: Advanced PFASST controllers", "Step-7: pySDC with external libraries", "Step-8: Advanced topics"], "titleterms": {"": 206, "1": [1, 205], "2": [1, 206], "3": [1, 207], "4": [1, 208], "5": 209, "6": 210, "7": 211, "8": 212, "A": [205, 206, 207, 208, 209, 210, 211, 212], "The": [36, 210], "To": 212, "abstract": 78, "accuraci": 205, "acknowledg": [3, 10], "acoust": 34, "acoustic_help": [136, 137, 138, 139], "acousticadvection_1d_fd_imex": 93, "ad": [9, 207], "adapt": [16, 71], "adaptive_colloc": 70, "add": 9, "advanc": [210, 212], "advect": [34, 209], "advectiondiffusionequation_1d_fft": 94, "advectionequation_nd_fd": 95, "algebra": 12, "allen": 11, 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\ No newline at end of file
diff --git a/tutorial/doc_step_1_B.html b/tutorial/doc_step_1_B.html
index f7c5b210e5..70c7769761 100644
--- a/tutorial/doc_step_1_B.html
+++ b/tutorial/doc_step_1_B.html
@@ -241,7 +241,7 @@ <h3>Navigation</h3>
 <span class="n">Expected</span> <span class="n">order</span><span class="p">:</span>  <span class="mi">2</span> <span class="o">--</span> <span class="n">Computed</span> <span class="n">order</span> <span class="mf">1.982</span>
 </pre></div>
 </div>
-<a class="reference internal image-reference" href="../_images/step_1_accuracy_test_space.png"><img alt="../_images/step_1_accuracy_test_space.png" src="../_images/step_1_accuracy_test_space.png" style="width: 522.5px; height: 352.0px;" />
+<a class="reference internal image-reference" href="../_images/step_1_accuracy_test_space.png"><img alt="../_images/step_1_accuracy_test_space.png" src="../_images/step_1_accuracy_test_space.png" style="width: 605.5px; height: 405.5px;" />
 </a>
 
 
diff --git a/tutorial/doc_step_1_D.html b/tutorial/doc_step_1_D.html
index 3d2cf61c19..10bcc13741 100644
--- a/tutorial/doc_step_1_D.html
+++ b/tutorial/doc_step_1_D.html
@@ -245,7 +245,7 @@ <h3>Navigation</h3>
 <span class="n">Expected</span> <span class="n">order</span><span class="p">:</span>  <span class="mi">5</span> <span class="o">--</span> <span class="n">Computed</span> <span class="n">order</span> <span class="mf">5.671</span>
 </pre></div>
 </div>
-<a class="reference internal image-reference" href="../_images/step_1_accuracy_test_coll.png"><img alt="../_images/step_1_accuracy_test_coll.png" src="../_images/step_1_accuracy_test_coll.png" style="width: 522.5px; height: 352.0px;" />
+<a class="reference internal image-reference" href="../_images/step_1_accuracy_test_coll.png"><img alt="../_images/step_1_accuracy_test_coll.png" src="../_images/step_1_accuracy_test_coll.png" style="width: 605.5px; height: 405.5px;" />
 </a>
 
 
diff --git a/tutorial/doc_step_2_C.html b/tutorial/doc_step_2_C.html
index 23896d90d9..b1b8fc208a 100644
--- a/tutorial/doc_step_2_C.html
+++ b/tutorial/doc_step_2_C.html
@@ -118,7 +118,7 @@ <h3>Navigation</h3>
 </pre></div>
 </div>
 <p>Results:</p>
-<div class="highlight-default notranslate"><div class="highlight"><pre><span></span>2024-09-19 08:57:00,351 - controller - controller - welcome_message - 146 - INFO: Welcome to the one and only, really very astonishing and 87.3% bug free
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span>2024-09-20 16:22:50,373 - controller - controller - welcome_message - 146 - INFO: Welcome to the one and only, really very astonishing and 87.3% bug free
                                  _____ _____   _____ 
                                 / ____|  __ \ / ____|
                     _ __  _   _| (___ | |  | | |     
@@ -128,8 +128,8 @@ <h3>Navigation</h3>
                    | |     __/ |                     
                    |_|    |___/                      
                                                      
-2024-09-19 08:57:00,351 - controller - controller - dump_setup - 160 - INFO: Setup overview (--&gt; user-defined, -&gt; dependency) -- BEGIN
-2024-09-19 08:57:00,352 - controller - controller - dump_setup - 227 - INFO: ----------------------------------------------------------------------------------------------------
+2024-09-20 16:22:50,373 - controller - controller - dump_setup - 160 - INFO: Setup overview (--&gt; user-defined, -&gt; dependency) -- BEGIN
+2024-09-20 16:22:50,373 - controller - controller - dump_setup - 227 - INFO: ----------------------------------------------------------------------------------------------------
 
 Controller: &lt;class &#39;pySDC.implementations.controller_classes.controller_nonMPI.controller_nonMPI&#39;&gt;
     all_to_done = False
@@ -182,34 +182,34 @@ <h3>Navigation</h3>
  -&gt; |  1 |   100 | SpreadStepSizesBlockwiseNonMPI
     |  2 |   200 | CheckConvergence
 
-2024-09-19 08:57:00,352 - controller - controller - dump_setup - 230 - INFO: ----------------------------------------------------------------------------------------------------
-2024-09-19 08:57:00,352 - controller - controller - dump_setup - 232 - INFO: Setup overview (--&gt; user-defined, -&gt; dependency) -- END
+2024-09-20 16:22:50,373 - controller - controller - dump_setup - 230 - INFO: ----------------------------------------------------------------------------------------------------
+2024-09-20 16:22:50,373 - controller - controller - dump_setup - 232 - INFO: Setup overview (--&gt; user-defined, -&gt; dependency) -- END
 
-2024-09-19 08:57:00,356 - hooks - default_hook - post_sweep - 170 - INFO: Process  0 on time 0.100000 at stage         IT_FINE: Level: 0 -- Iteration:  1 -- Sweep:  1 -- residual: 4.11190756e-03
-2024-09-19 08:57:00,359 - hooks - default_hook - post_sweep - 170 - INFO: Process  0 on time 0.100000 at stage         IT_FINE: Level: 0 -- Iteration:  2 -- Sweep:  1 -- residual: 6.68442667e-04
-2024-09-19 08:57:00,362 - hooks - default_hook - post_sweep - 170 - INFO: Process  0 on time 0.100000 at stage         IT_FINE: Level: 0 -- Iteration:  3 -- Sweep:  1 -- residual: 8.80377591e-05
-2024-09-19 08:57:00,365 - hooks - default_hook - post_sweep - 170 - INFO: Process  0 on time 0.100000 at stage         IT_FINE: Level: 0 -- Iteration:  4 -- Sweep:  1 -- residual: 1.21707909e-05
-2024-09-19 08:57:00,368 - hooks - default_hook - post_sweep - 170 - INFO: Process  0 on time 0.100000 at stage         IT_FINE: Level: 0 -- Iteration:  5 -- Sweep:  1 -- residual: 1.38272147e-06
-2024-09-19 08:57:00,372 - hooks - default_hook - post_sweep - 170 - INFO: Process  0 on time 0.100000 at stage         IT_FINE: Level: 0 -- Iteration:  6 -- Sweep:  1 -- residual: 6.36445413e-07
-2024-09-19 08:57:00,375 - hooks - default_hook - post_sweep - 170 - INFO: Process  0 on time 0.100000 at stage         IT_FINE: Level: 0 -- Iteration:  7 -- Sweep:  1 -- residual: 1.68953216e-07
-2024-09-19 08:57:00,378 - hooks - default_hook - post_sweep - 170 - INFO: Process  0 on time 0.100000 at stage         IT_FINE: Level: 0 -- Iteration:  8 -- Sweep:  1 -- residual: 3.52601840e-08
-2024-09-19 08:57:00,381 - hooks - default_hook - post_sweep - 170 - INFO: Process  0 on time 0.100000 at stage         IT_FINE: Level: 0 -- Iteration:  9 -- Sweep:  1 -- residual: 6.07249025e-09
-2024-09-19 08:57:00,384 - hooks - default_hook - post_sweep - 170 - INFO: Process  0 on time 0.100000 at stage         IT_FINE: Level: 0 -- Iteration: 10 -- Sweep:  1 -- residual: 8.27343378e-10
-2024-09-19 08:57:00,388 - hooks - default_hook - post_sweep - 170 - INFO: Process  0 on time 0.100000 at stage         IT_FINE: Level: 0 -- Iteration: 11 -- Sweep:  1 -- residual: 1.18931339e-10
-2024-09-19 08:57:00,391 - hooks - default_hook - post_sweep - 170 - INFO: Process  0 on time 0.100000 at stage         IT_FINE: Level: 0 -- Iteration: 12 -- Sweep:  1 -- residual: 1.48499772e-11
-2024-09-19 08:57:00,395 - hooks - default_hook - post_sweep - 170 - INFO: Process  0 on time 0.200000 at stage         IT_FINE: Level: 0 -- Iteration:  1 -- Sweep:  1 -- residual: 6.69984764e-03
-2024-09-19 08:57:00,398 - hooks - default_hook - post_sweep - 170 - INFO: Process  0 on time 0.200000 at stage         IT_FINE: Level: 0 -- Iteration:  2 -- Sweep:  1 -- residual: 1.05518433e-03
-2024-09-19 08:57:00,401 - hooks - default_hook - post_sweep - 170 - INFO: Process  0 on time 0.200000 at stage         IT_FINE: Level: 0 -- Iteration:  3 -- Sweep:  1 -- residual: 1.40642621e-04
-2024-09-19 08:57:00,404 - hooks - default_hook - post_sweep - 170 - INFO: Process  0 on time 0.200000 at stage         IT_FINE: Level: 0 -- Iteration:  4 -- Sweep:  1 -- residual: 1.85982063e-05
-2024-09-19 08:57:00,408 - hooks - default_hook - post_sweep - 170 - INFO: Process  0 on time 0.200000 at stage         IT_FINE: Level: 0 -- Iteration:  5 -- Sweep:  1 -- residual: 2.79216702e-06
-2024-09-19 08:57:00,411 - hooks - default_hook - post_sweep - 170 - INFO: Process  0 on time 0.200000 at stage         IT_FINE: Level: 0 -- Iteration:  6 -- Sweep:  1 -- residual: 1.12278839e-06
-2024-09-19 08:57:00,414 - hooks - default_hook - post_sweep - 170 - INFO: Process  0 on time 0.200000 at stage         IT_FINE: Level: 0 -- Iteration:  7 -- Sweep:  1 -- residual: 2.85495353e-07
-2024-09-19 08:57:00,417 - hooks - default_hook - post_sweep - 170 - INFO: Process  0 on time 0.200000 at stage         IT_FINE: Level: 0 -- Iteration:  8 -- Sweep:  1 -- residual: 5.78947003e-08
-2024-09-19 08:57:00,421 - hooks - default_hook - post_sweep - 170 - INFO: Process  0 on time 0.200000 at stage         IT_FINE: Level: 0 -- Iteration:  9 -- Sweep:  1 -- residual: 9.68230621e-09
-2024-09-19 08:57:00,424 - hooks - default_hook - post_sweep - 170 - INFO: Process  0 on time 0.200000 at stage         IT_FINE: Level: 0 -- Iteration: 10 -- Sweep:  1 -- residual: 1.26313315e-09
-2024-09-19 08:57:00,427 - hooks - default_hook - post_sweep - 170 - INFO: Process  0 on time 0.200000 at stage         IT_FINE: Level: 0 -- Iteration: 11 -- Sweep:  1 -- residual: 1.82951499e-10
-2024-09-19 08:57:00,430 - hooks - default_hook - post_sweep - 170 - INFO: Process  0 on time 0.200000 at stage         IT_FINE: Level: 0 -- Iteration: 12 -- Sweep:  1 -- residual: 1.99691114e-11
-2024-09-19 08:57:00,431 - hooks - default_hook - post_run - 340 - INFO: Finished run after 0.08s
+2024-09-20 16:22:50,377 - hooks - default_hook - post_sweep - 170 - INFO: Process  0 on time 0.100000 at stage         IT_FINE: Level: 0 -- Iteration:  1 -- Sweep:  1 -- residual: 4.11190756e-03
+2024-09-20 16:22:50,380 - hooks - default_hook - post_sweep - 170 - INFO: Process  0 on time 0.100000 at stage         IT_FINE: Level: 0 -- Iteration:  2 -- Sweep:  1 -- residual: 6.68442667e-04
+2024-09-20 16:22:50,384 - hooks - default_hook - post_sweep - 170 - INFO: Process  0 on time 0.100000 at stage         IT_FINE: Level: 0 -- Iteration:  3 -- Sweep:  1 -- residual: 8.80377591e-05
+2024-09-20 16:22:50,387 - hooks - default_hook - post_sweep - 170 - INFO: Process  0 on time 0.100000 at stage         IT_FINE: Level: 0 -- Iteration:  4 -- Sweep:  1 -- residual: 1.21707909e-05
+2024-09-20 16:22:50,390 - hooks - default_hook - post_sweep - 170 - INFO: Process  0 on time 0.100000 at stage         IT_FINE: Level: 0 -- Iteration:  5 -- Sweep:  1 -- residual: 1.38272147e-06
+2024-09-20 16:22:50,393 - hooks - default_hook - post_sweep - 170 - INFO: Process  0 on time 0.100000 at stage         IT_FINE: Level: 0 -- Iteration:  6 -- Sweep:  1 -- residual: 6.36445413e-07
+2024-09-20 16:22:50,397 - hooks - default_hook - post_sweep - 170 - INFO: Process  0 on time 0.100000 at stage         IT_FINE: Level: 0 -- Iteration:  7 -- Sweep:  1 -- residual: 1.68953216e-07
+2024-09-20 16:22:50,400 - hooks - default_hook - post_sweep - 170 - INFO: Process  0 on time 0.100000 at stage         IT_FINE: Level: 0 -- Iteration:  8 -- Sweep:  1 -- residual: 3.52601840e-08
+2024-09-20 16:22:50,403 - hooks - default_hook - post_sweep - 170 - INFO: Process  0 on time 0.100000 at stage         IT_FINE: Level: 0 -- Iteration:  9 -- Sweep:  1 -- residual: 6.07249025e-09
+2024-09-20 16:22:50,407 - hooks - default_hook - post_sweep - 170 - INFO: Process  0 on time 0.100000 at stage         IT_FINE: Level: 0 -- Iteration: 10 -- Sweep:  1 -- residual: 8.27343378e-10
+2024-09-20 16:22:50,410 - hooks - default_hook - post_sweep - 170 - INFO: Process  0 on time 0.100000 at stage         IT_FINE: Level: 0 -- Iteration: 11 -- Sweep:  1 -- residual: 1.18931339e-10
+2024-09-20 16:22:50,413 - hooks - default_hook - post_sweep - 170 - INFO: Process  0 on time 0.100000 at stage         IT_FINE: Level: 0 -- Iteration: 12 -- Sweep:  1 -- residual: 1.48499772e-11
+2024-09-20 16:22:50,417 - hooks - default_hook - post_sweep - 170 - INFO: Process  0 on time 0.200000 at stage         IT_FINE: Level: 0 -- Iteration:  1 -- Sweep:  1 -- residual: 6.69984764e-03
+2024-09-20 16:22:50,420 - hooks - default_hook - post_sweep - 170 - INFO: Process  0 on time 0.200000 at stage         IT_FINE: Level: 0 -- Iteration:  2 -- Sweep:  1 -- residual: 1.05518433e-03
+2024-09-20 16:22:50,424 - hooks - default_hook - post_sweep - 170 - INFO: Process  0 on time 0.200000 at stage         IT_FINE: Level: 0 -- Iteration:  3 -- Sweep:  1 -- residual: 1.40642621e-04
+2024-09-20 16:22:50,427 - hooks - default_hook - post_sweep - 170 - INFO: Process  0 on time 0.200000 at stage         IT_FINE: Level: 0 -- Iteration:  4 -- Sweep:  1 -- residual: 1.85982063e-05
+2024-09-20 16:22:50,430 - hooks - default_hook - post_sweep - 170 - INFO: Process  0 on time 0.200000 at stage         IT_FINE: Level: 0 -- Iteration:  5 -- Sweep:  1 -- residual: 2.79216702e-06
+2024-09-20 16:22:50,433 - hooks - default_hook - post_sweep - 170 - INFO: Process  0 on time 0.200000 at stage         IT_FINE: Level: 0 -- Iteration:  6 -- Sweep:  1 -- residual: 1.12278839e-06
+2024-09-20 16:22:50,437 - hooks - default_hook - post_sweep - 170 - INFO: Process  0 on time 0.200000 at stage         IT_FINE: Level: 0 -- Iteration:  7 -- Sweep:  1 -- residual: 2.85495353e-07
+2024-09-20 16:22:50,440 - hooks - default_hook - post_sweep - 170 - INFO: Process  0 on time 0.200000 at stage         IT_FINE: Level: 0 -- Iteration:  8 -- Sweep:  1 -- residual: 5.78947003e-08
+2024-09-20 16:22:50,443 - hooks - default_hook - post_sweep - 170 - INFO: Process  0 on time 0.200000 at stage         IT_FINE: Level: 0 -- Iteration:  9 -- Sweep:  1 -- residual: 9.68230621e-09
+2024-09-20 16:22:50,446 - hooks - default_hook - post_sweep - 170 - INFO: Process  0 on time 0.200000 at stage         IT_FINE: Level: 0 -- Iteration: 10 -- Sweep:  1 -- residual: 1.26313315e-09
+2024-09-20 16:22:50,449 - hooks - default_hook - post_sweep - 170 - INFO: Process  0 on time 0.200000 at stage         IT_FINE: Level: 0 -- Iteration: 11 -- Sweep:  1 -- residual: 1.82951499e-10
+2024-09-20 16:22:50,453 - hooks - default_hook - post_sweep - 170 - INFO: Process  0 on time 0.200000 at stage         IT_FINE: Level: 0 -- Iteration: 12 -- Sweep:  1 -- residual: 1.99691114e-11
+2024-09-20 16:22:50,453 - hooks - default_hook - post_run - 340 - INFO: Finished run after 0.08s
 Error after SDC iterations: 1.166689e-05
 </pre></div>
 </div>
diff --git a/tutorial/doc_step_3_B.html b/tutorial/doc_step_3_B.html
index b7bac415df..c34a814e11 100644
--- a/tutorial/doc_step_3_B.html
+++ b/tutorial/doc_step_3_B.html
@@ -145,7 +145,7 @@ <h3>Navigation</h3>
 </pre></div>
 </div>
 <p>Results:</p>
-<div class="highlight-default notranslate"><div class="highlight"><pre><span></span>2024-09-19 08:57:00,996 - controller - controller - welcome_message - 146 - INFO: Welcome to the one and only, really very astonishing and 87.3% bug free
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span>2024-09-20 16:22:50,966 - controller - controller - welcome_message - 146 - INFO: Welcome to the one and only, really very astonishing and 87.3% bug free
                                  _____ _____   _____ 
                                 / ____|  __ \ / ____|
                     _ __  _   _| (___ | |  | | |     
@@ -155,8 +155,8 @@ <h3>Navigation</h3>
                    | |     __/ |                     
                    |_|    |___/                      
                                                      
-2024-09-19 08:57:00,996 - controller - controller - dump_setup - 160 - INFO: Setup overview (--&gt; user-defined, -&gt; dependency) -- BEGIN
-2024-09-19 08:57:00,996 - controller - controller - dump_setup - 227 - INFO: ----------------------------------------------------------------------------------------------------
+2024-09-20 16:22:50,966 - controller - controller - dump_setup - 160 - INFO: Setup overview (--&gt; user-defined, -&gt; dependency) -- BEGIN
+2024-09-20 16:22:50,966 - controller - controller - dump_setup - 227 - INFO: ----------------------------------------------------------------------------------------------------
 
 Controller: &lt;class &#39;pySDC.implementations.controller_classes.controller_nonMPI.controller_nonMPI&#39;&gt;
     all_to_done = False
@@ -204,21 +204,21 @@ <h3>Navigation</h3>
  -&gt; |  1 |   100 | SpreadStepSizesBlockwiseNonMPI
     |  2 |   200 | CheckConvergence
 
-2024-09-19 08:57:00,996 - controller - controller - dump_setup - 230 - INFO: ----------------------------------------------------------------------------------------------------
-2024-09-19 08:57:00,996 - controller - controller - dump_setup - 232 - INFO: Setup overview (--&gt; user-defined, -&gt; dependency) -- END
-
-2024-09-19 08:57:02,351 - hooks - default_hook - post_sweep - 170 - INFO: Process  0 on time 0.000000 at stage         IT_FINE: Level: 0 -- Iteration:  1 -- Sweep:  1 -- residual: 3.53203678e+00
-2024-09-19 08:57:02,358 - hooks - default_hook - post_sweep - 170 - INFO: Process  0 on time 0.000000 at stage         IT_FINE: Level: 0 -- Iteration:  2 -- Sweep:  1 -- residual: 2.09852117e-01
-2024-09-19 08:57:02,365 - hooks - default_hook - post_sweep - 170 - INFO: Process  0 on time 0.000000 at stage         IT_FINE: Level: 0 -- Iteration:  3 -- Sweep:  1 -- residual: 3.50301513e-02
-2024-09-19 08:57:02,372 - hooks - default_hook - post_sweep - 170 - INFO: Process  0 on time 0.000000 at stage         IT_FINE: Level: 0 -- Iteration:  4 -- Sweep:  1 -- residual: 4.67724741e-03
-2024-09-19 08:57:02,379 - hooks - default_hook - post_sweep - 170 - INFO: Process  0 on time 0.000000 at stage         IT_FINE: Level: 0 -- Iteration:  5 -- Sweep:  1 -- residual: 7.95583202e-04
-2024-09-19 08:57:02,386 - hooks - default_hook - post_sweep - 170 - INFO: Process  0 on time 0.000000 at stage         IT_FINE: Level: 0 -- Iteration:  6 -- Sweep:  1 -- residual: 1.11405073e-04
-2024-09-19 08:57:02,393 - hooks - default_hook - post_sweep - 170 - INFO: Process  0 on time 0.000000 at stage         IT_FINE: Level: 0 -- Iteration:  7 -- Sweep:  1 -- residual: 1.26902403e-05
-2024-09-19 08:57:02,400 - hooks - default_hook - post_sweep - 170 - INFO: Process  0 on time 0.000000 at stage         IT_FINE: Level: 0 -- Iteration:  8 -- Sweep:  1 -- residual: 1.16534547e-06
-2024-09-19 08:57:02,406 - hooks - default_hook - post_sweep - 170 - INFO: Process  0 on time 0.000000 at stage         IT_FINE: Level: 0 -- Iteration:  9 -- Sweep:  1 -- residual: 1.66968022e-07
-2024-09-19 08:57:02,413 - hooks - default_hook - post_sweep - 170 - INFO: Process  0 on time 0.000000 at stage         IT_FINE: Level: 0 -- Iteration: 10 -- Sweep:  1 -- residual: 2.09408171e-08
-2024-09-19 08:57:02,420 - hooks - default_hook - post_sweep - 170 - INFO: Process  0 on time 0.000000 at stage         IT_FINE: Level: 0 -- Iteration: 11 -- Sweep:  1 -- residual: 2.17123386e-09
-2024-09-19 08:57:02,423 - hooks - default_hook - post_run - 340 - INFO: Finished run after 1.43s
+2024-09-20 16:22:50,967 - controller - controller - dump_setup - 230 - INFO: ----------------------------------------------------------------------------------------------------
+2024-09-20 16:22:50,967 - controller - controller - dump_setup - 232 - INFO: Setup overview (--&gt; user-defined, -&gt; dependency) -- END
+
+2024-09-20 16:22:52,320 - hooks - default_hook - post_sweep - 170 - INFO: Process  0 on time 0.000000 at stage         IT_FINE: Level: 0 -- Iteration:  1 -- Sweep:  1 -- residual: 3.53203678e+00
+2024-09-20 16:22:52,327 - hooks - default_hook - post_sweep - 170 - INFO: Process  0 on time 0.000000 at stage         IT_FINE: Level: 0 -- Iteration:  2 -- Sweep:  1 -- residual: 2.09852117e-01
+2024-09-20 16:22:52,334 - hooks - default_hook - post_sweep - 170 - INFO: Process  0 on time 0.000000 at stage         IT_FINE: Level: 0 -- Iteration:  3 -- Sweep:  1 -- residual: 3.50301513e-02
+2024-09-20 16:22:52,341 - hooks - default_hook - post_sweep - 170 - INFO: Process  0 on time 0.000000 at stage         IT_FINE: Level: 0 -- Iteration:  4 -- Sweep:  1 -- residual: 4.67724741e-03
+2024-09-20 16:22:52,347 - hooks - default_hook - post_sweep - 170 - INFO: Process  0 on time 0.000000 at stage         IT_FINE: Level: 0 -- Iteration:  5 -- Sweep:  1 -- residual: 7.95583202e-04
+2024-09-20 16:22:52,354 - hooks - default_hook - post_sweep - 170 - INFO: Process  0 on time 0.000000 at stage         IT_FINE: Level: 0 -- Iteration:  6 -- Sweep:  1 -- residual: 1.11405073e-04
+2024-09-20 16:22:52,361 - hooks - default_hook - post_sweep - 170 - INFO: Process  0 on time 0.000000 at stage         IT_FINE: Level: 0 -- Iteration:  7 -- Sweep:  1 -- residual: 1.26902403e-05
+2024-09-20 16:22:52,368 - hooks - default_hook - post_sweep - 170 - INFO: Process  0 on time 0.000000 at stage         IT_FINE: Level: 0 -- Iteration:  8 -- Sweep:  1 -- residual: 1.16534547e-06
+2024-09-20 16:22:52,375 - hooks - default_hook - post_sweep - 170 - INFO: Process  0 on time 0.000000 at stage         IT_FINE: Level: 0 -- Iteration:  9 -- Sweep:  1 -- residual: 1.66968022e-07
+2024-09-20 16:22:52,382 - hooks - default_hook - post_sweep - 170 - INFO: Process  0 on time 0.000000 at stage         IT_FINE: Level: 0 -- Iteration: 10 -- Sweep:  1 -- residual: 2.09408171e-08
+2024-09-20 16:22:52,389 - hooks - default_hook - post_sweep - 170 - INFO: Process  0 on time 0.000000 at stage         IT_FINE: Level: 0 -- Iteration: 11 -- Sweep:  1 -- residual: 2.17123386e-09
+2024-09-20 16:22:52,392 - hooks - default_hook - post_run - 340 - INFO: Finished run after 1.42s
 Total energy and deviation in iteration  0: 8799.5000000000 -- 0.00000000e+00
 Total energy and deviation in iteration 11: 8785.0038936088 -- 1.44961064e+01
 </pre></div>
diff --git a/tutorial/doc_step_4_D.html b/tutorial/doc_step_4_D.html
index 707669b32f..1dcbdda104 100644
--- a/tutorial/doc_step_4_D.html
+++ b/tutorial/doc_step_4_D.html
@@ -191,7 +191,7 @@ <h3>Navigation</h3>
 </pre></div>
 </div>
 <p>Results:</p>
-<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">Timings</span> <span class="k">for</span> <span class="n">SDC</span><span class="p">,</span> <span class="n">MLSDC</span> <span class="ow">and</span> <span class="n">MLSDC</span><span class="o">+</span><span class="n">finter</span><span class="p">:</span>   <span class="mf">5.93434594</span> <span class="o">--</span>   <span class="mf">6.60096954</span> <span class="o">--</span>   <span class="mf">7.66078035</span>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">Timings</span> <span class="k">for</span> <span class="n">SDC</span><span class="p">,</span> <span class="n">MLSDC</span> <span class="ow">and</span> <span class="n">MLSDC</span><span class="o">+</span><span class="n">finter</span><span class="p">:</span>   <span class="mf">6.00803794</span> <span class="o">--</span>   <span class="mf">6.64845455</span> <span class="o">--</span>   <span class="mf">7.71512253</span>
 <span class="n">Total</span> <span class="n">energy</span> <span class="ow">and</span> <span class="n">relative</span> <span class="n">deviation</span> <span class="ow">in</span> <span class="n">iteration</span>  <span class="mi">0</span><span class="p">:</span> <span class="mf">407936.7556966486</span> <span class="o">--</span> <span class="mf">0.00000000e+00</span>
 <span class="n">Total</span> <span class="n">energy</span> <span class="ow">and</span> <span class="n">relative</span> <span class="n">deviation</span> <span class="ow">in</span> <span class="n">iteration</span> <span class="mi">12</span><span class="p">:</span> <span class="mf">406977.9425667246</span> <span class="o">--</span> <span class="mf">2.35039652e-03</span>
 <span class="n">Total</span> <span class="n">energy</span> <span class="ow">and</span> <span class="n">relative</span> <span class="n">deviation</span> <span class="ow">in</span> <span class="n">iteration</span>  <span class="mi">0</span><span class="p">:</span> <span class="mf">407936.7556966486</span> <span class="o">--</span> <span class="mf">0.00000000e+00</span>
diff --git a/tutorial/doc_step_7_A.html b/tutorial/doc_step_7_A.html
index 44a7ac22ec..c1837603fc 100644
--- a/tutorial/doc_step_7_A.html
+++ b/tutorial/doc_step_7_A.html
@@ -232,49 +232,49 @@ <h3>Navigation</h3>
    <span class="n">Range</span> <span class="n">of</span> <span class="n">values</span> <span class="k">for</span> <span class="n">number</span> <span class="n">of</span> <span class="n">iterations</span><span class="p">:</span>  <span class="mi">0</span> 
    <span class="n">Position</span> <span class="n">of</span> <span class="nb">max</span><span class="o">/</span><span class="nb">min</span> <span class="n">number</span> <span class="n">of</span> <span class="n">iterations</span><span class="p">:</span>  <span class="mi">0</span> <span class="o">--</span>  <span class="mi">0</span>
    <span class="n">Std</span> <span class="ow">and</span> <span class="n">var</span> <span class="k">for</span> <span class="n">number</span> <span class="n">of</span> <span class="n">iterations</span><span class="p">:</span> <span class="mf">0.00</span> <span class="o">--</span> <span class="mf">0.00</span>
-<span class="n">Time</span> <span class="n">to</span> <span class="n">solution</span><span class="p">:</span> <span class="mf">1.7961</span> <span class="n">sec</span><span class="o">.</span>
+<span class="n">Time</span> <span class="n">to</span> <span class="n">solution</span><span class="p">:</span> <span class="mf">1.7330</span> <span class="n">sec</span><span class="o">.</span>
 
 <span class="n">Variant</span> <span class="n">mass</span> <span class="k">with</span> <span class="n">ml</span><span class="o">=</span><span class="kc">False</span> <span class="ow">and</span> <span class="n">num_procs</span><span class="o">=</span><span class="mi">1</span> <span class="o">--</span> <span class="n">error</span> <span class="n">at</span> <span class="n">time</span> <span class="mf">1.0</span><span class="p">:</span> <span class="mf">1.1387594756569534e-08</span>
    <span class="n">Mean</span> <span class="n">number</span> <span class="n">of</span> <span class="n">iterations</span><span class="p">:</span> <span class="mf">6.00</span>
    <span class="n">Range</span> <span class="n">of</span> <span class="n">values</span> <span class="k">for</span> <span class="n">number</span> <span class="n">of</span> <span class="n">iterations</span><span class="p">:</span>  <span class="mi">0</span> 
    <span class="n">Position</span> <span class="n">of</span> <span class="nb">max</span><span class="o">/</span><span class="nb">min</span> <span class="n">number</span> <span class="n">of</span> <span class="n">iterations</span><span class="p">:</span>  <span class="mi">0</span> <span class="o">--</span>  <span class="mi">0</span>
    <span class="n">Std</span> <span class="ow">and</span> <span class="n">var</span> <span class="k">for</span> <span class="n">number</span> <span class="n">of</span> <span class="n">iterations</span><span class="p">:</span> <span class="mf">0.00</span> <span class="o">--</span> <span class="mf">0.00</span>
-<span class="n">Time</span> <span class="n">to</span> <span class="n">solution</span><span class="p">:</span> <span class="mf">1.4084</span> <span class="n">sec</span><span class="o">.</span>
+<span class="n">Time</span> <span class="n">to</span> <span class="n">solution</span><span class="p">:</span> <span class="mf">1.3967</span> <span class="n">sec</span><span class="o">.</span>
 
 <span class="n">Variant</span> <span class="n">mass_timebc</span> <span class="k">with</span> <span class="n">ml</span><span class="o">=</span><span class="kc">False</span> <span class="ow">and</span> <span class="n">num_procs</span><span class="o">=</span><span class="mi">1</span> <span class="o">--</span> <span class="n">error</span> <span class="n">at</span> <span class="n">time</span> <span class="mf">1.0</span><span class="p">:</span> <span class="mf">3.2473562155116167e-07</span>
    <span class="n">Mean</span> <span class="n">number</span> <span class="n">of</span> <span class="n">iterations</span><span class="p">:</span> <span class="mf">6.00</span>
    <span class="n">Range</span> <span class="n">of</span> <span class="n">values</span> <span class="k">for</span> <span class="n">number</span> <span class="n">of</span> <span class="n">iterations</span><span class="p">:</span>  <span class="mi">3</span> 
    <span class="n">Position</span> <span class="n">of</span> <span class="nb">max</span><span class="o">/</span><span class="nb">min</span> <span class="n">number</span> <span class="n">of</span> <span class="n">iterations</span><span class="p">:</span>  <span class="mi">3</span> <span class="o">--</span>  <span class="mi">0</span>
    <span class="n">Std</span> <span class="ow">and</span> <span class="n">var</span> <span class="k">for</span> <span class="n">number</span> <span class="n">of</span> <span class="n">iterations</span><span class="p">:</span> <span class="mf">1.10</span> <span class="o">--</span> <span class="mf">1.20</span>
-<span class="n">Time</span> <span class="n">to</span> <span class="n">solution</span><span class="p">:</span> <span class="mf">1.4094</span> <span class="n">sec</span><span class="o">.</span>
+<span class="n">Time</span> <span class="n">to</span> <span class="n">solution</span><span class="p">:</span> <span class="mf">1.3952</span> <span class="n">sec</span><span class="o">.</span>
 
 <span class="n">Variant</span> <span class="n">mass_inv</span> <span class="k">with</span> <span class="n">ml</span><span class="o">=</span><span class="kc">True</span> <span class="ow">and</span> <span class="n">num_procs</span><span class="o">=</span><span class="mi">1</span> <span class="o">--</span> <span class="n">error</span> <span class="n">at</span> <span class="n">time</span> <span class="mf">1.0</span><span class="p">:</span> <span class="mf">1.138768636885694e-08</span>
    <span class="n">Mean</span> <span class="n">number</span> <span class="n">of</span> <span class="n">iterations</span><span class="p">:</span> <span class="mf">6.00</span>
    <span class="n">Range</span> <span class="n">of</span> <span class="n">values</span> <span class="k">for</span> <span class="n">number</span> <span class="n">of</span> <span class="n">iterations</span><span class="p">:</span>  <span class="mi">0</span> 
    <span class="n">Position</span> <span class="n">of</span> <span class="nb">max</span><span class="o">/</span><span class="nb">min</span> <span class="n">number</span> <span class="n">of</span> <span class="n">iterations</span><span class="p">:</span>  <span class="mi">0</span> <span class="o">--</span>  <span class="mi">0</span>
    <span class="n">Std</span> <span class="ow">and</span> <span class="n">var</span> <span class="k">for</span> <span class="n">number</span> <span class="n">of</span> <span class="n">iterations</span><span class="p">:</span> <span class="mf">0.00</span> <span class="o">--</span> <span class="mf">0.00</span>
-<span class="n">Time</span> <span class="n">to</span> <span class="n">solution</span><span class="p">:</span> <span class="mf">2.5440</span> <span class="n">sec</span><span class="o">.</span>
+<span class="n">Time</span> <span class="n">to</span> <span class="n">solution</span><span class="p">:</span> <span class="mf">2.5014</span> <span class="n">sec</span><span class="o">.</span>
 
 <span class="n">Variant</span> <span class="n">mass</span> <span class="k">with</span> <span class="n">ml</span><span class="o">=</span><span class="kc">True</span> <span class="ow">and</span> <span class="n">num_procs</span><span class="o">=</span><span class="mi">1</span> <span class="o">--</span> <span class="n">error</span> <span class="n">at</span> <span class="n">time</span> <span class="mf">1.0</span><span class="p">:</span> <span class="mf">1.1387216566052821e-08</span>
    <span class="n">Mean</span> <span class="n">number</span> <span class="n">of</span> <span class="n">iterations</span><span class="p">:</span> <span class="mf">6.00</span>
    <span class="n">Range</span> <span class="n">of</span> <span class="n">values</span> <span class="k">for</span> <span class="n">number</span> <span class="n">of</span> <span class="n">iterations</span><span class="p">:</span>  <span class="mi">0</span> 
    <span class="n">Position</span> <span class="n">of</span> <span class="nb">max</span><span class="o">/</span><span class="nb">min</span> <span class="n">number</span> <span class="n">of</span> <span class="n">iterations</span><span class="p">:</span>  <span class="mi">0</span> <span class="o">--</span>  <span class="mi">0</span>
    <span class="n">Std</span> <span class="ow">and</span> <span class="n">var</span> <span class="k">for</span> <span class="n">number</span> <span class="n">of</span> <span class="n">iterations</span><span class="p">:</span> <span class="mf">0.00</span> <span class="o">--</span> <span class="mf">0.00</span>
-<span class="n">Time</span> <span class="n">to</span> <span class="n">solution</span><span class="p">:</span> <span class="mf">2.1717</span> <span class="n">sec</span><span class="o">.</span>
+<span class="n">Time</span> <span class="n">to</span> <span class="n">solution</span><span class="p">:</span> <span class="mf">2.1059</span> <span class="n">sec</span><span class="o">.</span>
 
 <span class="n">Variant</span> <span class="n">mass_timebc</span> <span class="k">with</span> <span class="n">ml</span><span class="o">=</span><span class="kc">True</span> <span class="ow">and</span> <span class="n">num_procs</span><span class="o">=</span><span class="mi">1</span> <span class="o">--</span> <span class="n">error</span> <span class="n">at</span> <span class="n">time</span> <span class="mf">1.0</span><span class="p">:</span> <span class="mf">3.2473561574763597e-07</span>
    <span class="n">Mean</span> <span class="n">number</span> <span class="n">of</span> <span class="n">iterations</span><span class="p">:</span> <span class="mf">6.00</span>
    <span class="n">Range</span> <span class="n">of</span> <span class="n">values</span> <span class="k">for</span> <span class="n">number</span> <span class="n">of</span> <span class="n">iterations</span><span class="p">:</span>  <span class="mi">3</span> 
    <span class="n">Position</span> <span class="n">of</span> <span class="nb">max</span><span class="o">/</span><span class="nb">min</span> <span class="n">number</span> <span class="n">of</span> <span class="n">iterations</span><span class="p">:</span>  <span class="mi">3</span> <span class="o">--</span>  <span class="mi">0</span>
    <span class="n">Std</span> <span class="ow">and</span> <span class="n">var</span> <span class="k">for</span> <span class="n">number</span> <span class="n">of</span> <span class="n">iterations</span><span class="p">:</span> <span class="mf">1.10</span> <span class="o">--</span> <span class="mf">1.20</span>
-<span class="n">Time</span> <span class="n">to</span> <span class="n">solution</span><span class="p">:</span> <span class="mf">2.1695</span> <span class="n">sec</span><span class="o">.</span>
+<span class="n">Time</span> <span class="n">to</span> <span class="n">solution</span><span class="p">:</span> <span class="mf">2.1560</span> <span class="n">sec</span><span class="o">.</span>
 
 <span class="n">Variant</span> <span class="n">mass_inv</span> <span class="k">with</span> <span class="n">ml</span><span class="o">=</span><span class="kc">True</span> <span class="ow">and</span> <span class="n">num_procs</span><span class="o">=</span><span class="mi">5</span> <span class="o">--</span> <span class="n">error</span> <span class="n">at</span> <span class="n">time</span> <span class="mf">1.0</span><span class="p">:</span> <span class="mf">1.1150087179536389e-08</span>
    <span class="n">Mean</span> <span class="n">number</span> <span class="n">of</span> <span class="n">iterations</span><span class="p">:</span> <span class="mf">11.60</span>
    <span class="n">Range</span> <span class="n">of</span> <span class="n">values</span> <span class="k">for</span> <span class="n">number</span> <span class="n">of</span> <span class="n">iterations</span><span class="p">:</span>  <span class="mi">9</span> 
    <span class="n">Position</span> <span class="n">of</span> <span class="nb">max</span><span class="o">/</span><span class="nb">min</span> <span class="n">number</span> <span class="n">of</span> <span class="n">iterations</span><span class="p">:</span>  <span class="mi">4</span> <span class="o">--</span>  <span class="mi">0</span>
    <span class="n">Std</span> <span class="ow">and</span> <span class="n">var</span> <span class="k">for</span> <span class="n">number</span> <span class="n">of</span> <span class="n">iterations</span><span class="p">:</span> <span class="mf">3.26</span> <span class="o">--</span> <span class="mf">10.64</span>
-<span class="n">Time</span> <span class="n">to</span> <span class="n">solution</span><span class="p">:</span> <span class="mf">4.1983</span> <span class="n">sec</span><span class="o">.</span>
+<span class="n">Time</span> <span class="n">to</span> <span class="n">solution</span><span class="p">:</span> <span class="mf">4.1015</span> <span class="n">sec</span><span class="o">.</span>
 
 </pre></div>
 </div>
diff --git a/tutorial/doc_step_7_B.html b/tutorial/doc_step_7_B.html
index b40cad185d..d60e775f34 100644
--- a/tutorial/doc_step_7_B.html
+++ b/tutorial/doc_step_7_B.html
@@ -200,7 +200,7 @@ <h3>Navigation</h3>
    <span class="n">Position</span> <span class="n">of</span> <span class="nb">max</span><span class="o">/</span><span class="nb">min</span> <span class="n">number</span> <span class="n">of</span> <span class="n">iterations</span><span class="p">:</span>  <span class="mi">0</span> <span class="o">--</span> <span class="mi">19</span>
    <span class="n">Std</span> <span class="ow">and</span> <span class="n">var</span> <span class="k">for</span> <span class="n">number</span> <span class="n">of</span> <span class="n">iterations</span><span class="p">:</span> <span class="mf">2.10</span> <span class="o">--</span> <span class="mf">4.41</span>
 <span class="n">Error</span><span class="p">:</span> <span class="mf">1.1321e-05</span>
-<span class="n">Time</span> <span class="n">to</span> <span class="n">solution</span><span class="p">:</span> <span class="mf">1.4530</span> <span class="n">sec</span><span class="o">.</span>
+<span class="n">Time</span> <span class="n">to</span> <span class="n">solution</span><span class="p">:</span> <span class="mf">1.3354</span> <span class="n">sec</span><span class="o">.</span>
 
 <span class="n">Running</span> <span class="k">with</span> <span class="n">ml</span><span class="o">=</span><span class="kc">False</span> <span class="ow">and</span> <span class="n">num_procs</span><span class="o">=</span><span class="mf">1.</span><span class="o">..</span>
    <span class="n">Min</span><span class="o">/</span><span class="n">Mean</span><span class="o">/</span><span class="n">Max</span> <span class="n">number</span> <span class="n">of</span> <span class="n">iterations</span><span class="p">:</span> <span class="mf">8.00</span> <span class="o">/</span> <span class="mf">11.40</span> <span class="o">/</span> <span class="mf">15.00</span>
@@ -208,7 +208,7 @@ <h3>Navigation</h3>
    <span class="n">Position</span> <span class="n">of</span> <span class="nb">max</span><span class="o">/</span><span class="nb">min</span> <span class="n">number</span> <span class="n">of</span> <span class="n">iterations</span><span class="p">:</span>  <span class="mi">0</span> <span class="o">--</span> <span class="mi">19</span>
    <span class="n">Std</span> <span class="ow">and</span> <span class="n">var</span> <span class="k">for</span> <span class="n">number</span> <span class="n">of</span> <span class="n">iterations</span><span class="p">:</span> <span class="mf">2.03</span> <span class="o">--</span> <span class="mf">4.14</span>
 <span class="n">Error</span><span class="p">:</span> <span class="mf">4.1749e-06</span>
-<span class="n">Time</span> <span class="n">to</span> <span class="n">solution</span><span class="p">:</span> <span class="mf">1.1877</span> <span class="n">sec</span><span class="o">.</span>
+<span class="n">Time</span> <span class="n">to</span> <span class="n">solution</span><span class="p">:</span> <span class="mf">1.0773</span> <span class="n">sec</span><span class="o">.</span>
 
 <span class="n">Running</span> <span class="k">with</span> <span class="n">ml</span><span class="o">=</span><span class="kc">True</span> <span class="ow">and</span> <span class="n">num_procs</span><span class="o">=</span><span class="mf">1.</span><span class="o">..</span>
    <span class="n">Min</span><span class="o">/</span><span class="n">Mean</span><span class="o">/</span><span class="n">Max</span> <span class="n">number</span> <span class="n">of</span> <span class="n">iterations</span><span class="p">:</span> <span class="mf">5.00</span> <span class="o">/</span> <span class="mf">6.60</span> <span class="o">/</span> <span class="mf">8.00</span>
@@ -216,7 +216,7 @@ <h3>Navigation</h3>
    <span class="n">Position</span> <span class="n">of</span> <span class="nb">max</span><span class="o">/</span><span class="nb">min</span> <span class="n">number</span> <span class="n">of</span> <span class="n">iterations</span><span class="p">:</span>  <span class="mi">0</span> <span class="o">--</span> <span class="mi">16</span>
    <span class="n">Std</span> <span class="ow">and</span> <span class="n">var</span> <span class="k">for</span> <span class="n">number</span> <span class="n">of</span> <span class="n">iterations</span><span class="p">:</span> <span class="mf">1.07</span> <span class="o">--</span> <span class="mf">1.14</span>
 <span class="n">Error</span><span class="p">:</span> <span class="mf">1.1316e-05</span>
-<span class="n">Time</span> <span class="n">to</span> <span class="n">solution</span><span class="p">:</span> <span class="mf">1.4027</span> <span class="n">sec</span><span class="o">.</span>
+<span class="n">Time</span> <span class="n">to</span> <span class="n">solution</span><span class="p">:</span> <span class="mf">1.3054</span> <span class="n">sec</span><span class="o">.</span>
 
 <span class="n">Running</span> <span class="k">with</span> <span class="n">ml</span><span class="o">=</span><span class="kc">True</span> <span class="ow">and</span> <span class="n">num_procs</span><span class="o">=</span><span class="mf">1.</span><span class="o">..</span>
    <span class="n">Min</span><span class="o">/</span><span class="n">Mean</span><span class="o">/</span><span class="n">Max</span> <span class="n">number</span> <span class="n">of</span> <span class="n">iterations</span><span class="p">:</span> <span class="mf">4.00</span> <span class="o">/</span> <span class="mf">5.95</span> <span class="o">/</span> <span class="mf">8.00</span>
@@ -224,7 +224,7 @@ <h3>Navigation</h3>
    <span class="n">Position</span> <span class="n">of</span> <span class="nb">max</span><span class="o">/</span><span class="nb">min</span> <span class="n">number</span> <span class="n">of</span> <span class="n">iterations</span><span class="p">:</span>  <span class="mi">0</span> <span class="o">--</span> <span class="mi">19</span>
    <span class="n">Std</span> <span class="ow">and</span> <span class="n">var</span> <span class="k">for</span> <span class="n">number</span> <span class="n">of</span> <span class="n">iterations</span><span class="p">:</span> <span class="mf">1.02</span> <span class="o">--</span> <span class="mf">1.05</span>
 <span class="n">Error</span><span class="p">:</span> <span class="mf">4.1744e-06</span>
-<span class="n">Time</span> <span class="n">to</span> <span class="n">solution</span><span class="p">:</span> <span class="mf">1.2926</span> <span class="n">sec</span><span class="o">.</span>
+<span class="n">Time</span> <span class="n">to</span> <span class="n">solution</span><span class="p">:</span> <span class="mf">1.2137</span> <span class="n">sec</span><span class="o">.</span>
 
 <span class="n">Running</span> <span class="k">with</span> <span class="n">ml</span><span class="o">=</span><span class="kc">True</span> <span class="ow">and</span> <span class="n">num_procs</span><span class="o">=</span><span class="mf">10.</span><span class="o">..</span>
    <span class="n">Min</span><span class="o">/</span><span class="n">Mean</span><span class="o">/</span><span class="n">Max</span> <span class="n">number</span> <span class="n">of</span> <span class="n">iterations</span><span class="p">:</span> <span class="mf">7.00</span> <span class="o">/</span> <span class="mf">12.45</span> <span class="o">/</span> <span class="mf">18.00</span>
@@ -232,7 +232,7 @@ <h3>Navigation</h3>
    <span class="n">Position</span> <span class="n">of</span> <span class="nb">max</span><span class="o">/</span><span class="nb">min</span> <span class="n">number</span> <span class="n">of</span> <span class="n">iterations</span><span class="p">:</span>  <span class="mi">9</span> <span class="o">--</span> <span class="mi">10</span>
    <span class="n">Std</span> <span class="ow">and</span> <span class="n">var</span> <span class="k">for</span> <span class="n">number</span> <span class="n">of</span> <span class="n">iterations</span><span class="p">:</span> <span class="mf">3.11</span> <span class="o">--</span> <span class="mf">9.65</span>
 <span class="n">Error</span><span class="p">:</span> <span class="mf">1.1306e-05</span>
-<span class="n">Time</span> <span class="n">to</span> <span class="n">solution</span><span class="p">:</span> <span class="mf">2.9081</span> <span class="n">sec</span><span class="o">.</span>
+<span class="n">Time</span> <span class="n">to</span> <span class="n">solution</span><span class="p">:</span> <span class="mf">2.7650</span> <span class="n">sec</span><span class="o">.</span>
 
 <span class="n">Running</span> <span class="k">with</span> <span class="n">ml</span><span class="o">=</span><span class="kc">True</span> <span class="ow">and</span> <span class="n">num_procs</span><span class="o">=</span><span class="mf">10.</span><span class="o">..</span>
    <span class="n">Min</span><span class="o">/</span><span class="n">Mean</span><span class="o">/</span><span class="n">Max</span> <span class="n">number</span> <span class="n">of</span> <span class="n">iterations</span><span class="p">:</span> <span class="mf">6.00</span> <span class="o">/</span> <span class="mf">11.50</span> <span class="o">/</span> <span class="mf">17.00</span>
@@ -240,7 +240,7 @@ <h3>Navigation</h3>
    <span class="n">Position</span> <span class="n">of</span> <span class="nb">max</span><span class="o">/</span><span class="nb">min</span> <span class="n">number</span> <span class="n">of</span> <span class="n">iterations</span><span class="p">:</span>  <span class="mi">9</span> <span class="o">--</span> <span class="mi">10</span>
    <span class="n">Std</span> <span class="ow">and</span> <span class="n">var</span> <span class="k">for</span> <span class="n">number</span> <span class="n">of</span> <span class="n">iterations</span><span class="p">:</span> <span class="mf">3.04</span> <span class="o">--</span> <span class="mf">9.25</span>
 <span class="n">Error</span><span class="p">:</span> <span class="mf">4.1688e-06</span>
-<span class="n">Time</span> <span class="n">to</span> <span class="n">solution</span><span class="p">:</span> <span class="mf">2.7314</span> <span class="n">sec</span><span class="o">.</span>
+<span class="n">Time</span> <span class="n">to</span> <span class="n">solution</span><span class="p">:</span> <span class="mf">2.6021</span> <span class="n">sec</span><span class="o">.</span>
 
 </pre></div>
 </div>
diff --git a/tutorial/doc_step_7_C.html b/tutorial/doc_step_7_C.html
index b244667cc2..86823b99f9 100644
--- a/tutorial/doc_step_7_C.html
+++ b/tutorial/doc_step_7_C.html
@@ -215,7 +215,7 @@ <h3>Navigation</h3>
    <span class="n">Range</span> <span class="n">of</span> <span class="n">values</span> <span class="k">for</span> <span class="n">number</span> <span class="n">of</span> <span class="n">iterations</span><span class="p">:</span>  <span class="mi">0</span> 
    <span class="n">Position</span> <span class="n">of</span> <span class="nb">max</span><span class="o">/</span><span class="nb">min</span> <span class="n">number</span> <span class="n">of</span> <span class="n">iterations</span><span class="p">:</span>  <span class="mi">0</span> <span class="o">--</span>  <span class="mi">0</span>
    <span class="n">Std</span> <span class="ow">and</span> <span class="n">var</span> <span class="k">for</span> <span class="n">number</span> <span class="n">of</span> <span class="n">iterations</span><span class="p">:</span> <span class="mf">0.00</span> <span class="o">--</span> <span class="mf">0.00</span>
-<span class="n">Time</span> <span class="n">to</span> <span class="n">solution</span><span class="p">:</span> <span class="mf">2.8030</span> <span class="n">sec</span><span class="o">.</span>
+<span class="n">Time</span> <span class="n">to</span> <span class="n">solution</span><span class="p">:</span> <span class="mf">2.0782</span> <span class="n">sec</span><span class="o">.</span>
 <span class="n">Error</span> <span class="n">vs</span><span class="o">.</span> <span class="n">PDE</span> <span class="n">solution</span><span class="p">:</span> <span class="mf">1.9479e-04</span>
 </pre></div>
 </div>
@@ -227,26 +227,26 @@ <h3>Navigation</h3>
    <span class="n">Range</span> <span class="n">of</span> <span class="n">values</span> <span class="k">for</span> <span class="n">number</span> <span class="n">of</span> <span class="n">iterations</span><span class="p">:</span>  <span class="mi">0</span> 
    <span class="n">Position</span> <span class="n">of</span> <span class="nb">max</span><span class="o">/</span><span class="nb">min</span> <span class="n">number</span> <span class="n">of</span> <span class="n">iterations</span><span class="p">:</span>  <span class="mi">0</span> <span class="o">--</span>  <span class="mi">0</span>
    <span class="n">Std</span> <span class="ow">and</span> <span class="n">var</span> <span class="k">for</span> <span class="n">number</span> <span class="n">of</span> <span class="n">iterations</span><span class="p">:</span> <span class="mf">0.00</span> <span class="o">--</span> <span class="mf">0.00</span>
-<span class="n">Time</span> <span class="n">to</span> <span class="n">solution</span><span class="p">:</span> <span class="mf">0.9242</span> <span class="n">sec</span><span class="o">.</span>
+<span class="n">Time</span> <span class="n">to</span> <span class="n">solution</span><span class="p">:</span> <span class="mf">0.9128</span> <span class="n">sec</span><span class="o">.</span>
 <span class="n">Error</span> <span class="n">vs</span><span class="o">.</span> <span class="n">PDE</span> <span class="n">solution</span><span class="p">:</span> <span class="mf">1.9479e-04</span>
 </pre></div>
 </div>
 <p>2 processor in time, 2 processors in space</p>
-<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">This</span> <span class="ow">is</span> <span class="n">time</span><span class="o">-</span><span class="n">rank</span> <span class="mf">0.</span><span class="o">..</span>
-<span class="n">Number</span> <span class="n">of</span> <span class="n">iterations</span> <span class="k">for</span> <span class="n">time</span> <span class="mf">0.00</span><span class="p">:</span> <span class="mi">12</span>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">This</span> <span class="ow">is</span> <span class="n">time</span><span class="o">-</span><span class="n">rank</span> <span class="mf">1.</span><span class="o">..</span>
+<span class="n">Number</span> <span class="n">of</span> <span class="n">iterations</span> <span class="k">for</span> <span class="n">time</span> <span class="mf">0.12</span><span class="p">:</span> <span class="mi">12</span>
    <span class="n">Mean</span> <span class="n">number</span> <span class="n">of</span> <span class="n">iterations</span><span class="p">:</span> <span class="mf">12.00</span>
    <span class="n">Range</span> <span class="n">of</span> <span class="n">values</span> <span class="k">for</span> <span class="n">number</span> <span class="n">of</span> <span class="n">iterations</span><span class="p">:</span>  <span class="mi">0</span> 
    <span class="n">Position</span> <span class="n">of</span> <span class="nb">max</span><span class="o">/</span><span class="nb">min</span> <span class="n">number</span> <span class="n">of</span> <span class="n">iterations</span><span class="p">:</span>  <span class="mi">0</span> <span class="o">--</span>  <span class="mi">0</span>
    <span class="n">Std</span> <span class="ow">and</span> <span class="n">var</span> <span class="k">for</span> <span class="n">number</span> <span class="n">of</span> <span class="n">iterations</span><span class="p">:</span> <span class="mf">0.00</span> <span class="o">--</span> <span class="mf">0.00</span>
-<span class="n">Time</span> <span class="n">to</span> <span class="n">solution</span><span class="p">:</span> <span class="mf">0.8208</span> <span class="n">sec</span><span class="o">.</span>
+<span class="n">Time</span> <span class="n">to</span> <span class="n">solution</span><span class="p">:</span> <span class="mf">0.7904</span> <span class="n">sec</span><span class="o">.</span>
 <span class="n">Error</span> <span class="n">vs</span><span class="o">.</span> <span class="n">PDE</span> <span class="n">solution</span><span class="p">:</span> <span class="mf">1.9479e-04</span>
-<span class="n">This</span> <span class="ow">is</span> <span class="n">time</span><span class="o">-</span><span class="n">rank</span> <span class="mf">1.</span><span class="o">..</span>
-<span class="n">Number</span> <span class="n">of</span> <span class="n">iterations</span> <span class="k">for</span> <span class="n">time</span> <span class="mf">0.12</span><span class="p">:</span> <span class="mi">12</span>
+<span class="n">This</span> <span class="ow">is</span> <span class="n">time</span><span class="o">-</span><span class="n">rank</span> <span class="mf">0.</span><span class="o">..</span>
+<span class="n">Number</span> <span class="n">of</span> <span class="n">iterations</span> <span class="k">for</span> <span class="n">time</span> <span class="mf">0.00</span><span class="p">:</span> <span class="mi">12</span>
    <span class="n">Mean</span> <span class="n">number</span> <span class="n">of</span> <span class="n">iterations</span><span class="p">:</span> <span class="mf">12.00</span>
    <span class="n">Range</span> <span class="n">of</span> <span class="n">values</span> <span class="k">for</span> <span class="n">number</span> <span class="n">of</span> <span class="n">iterations</span><span class="p">:</span>  <span class="mi">0</span> 
    <span class="n">Position</span> <span class="n">of</span> <span class="nb">max</span><span class="o">/</span><span class="nb">min</span> <span class="n">number</span> <span class="n">of</span> <span class="n">iterations</span><span class="p">:</span>  <span class="mi">0</span> <span class="o">--</span>  <span class="mi">0</span>
    <span class="n">Std</span> <span class="ow">and</span> <span class="n">var</span> <span class="k">for</span> <span class="n">number</span> <span class="n">of</span> <span class="n">iterations</span><span class="p">:</span> <span class="mf">0.00</span> <span class="o">--</span> <span class="mf">0.00</span>
-<span class="n">Time</span> <span class="n">to</span> <span class="n">solution</span><span class="p">:</span> <span class="mf">0.8205</span> <span class="n">sec</span><span class="o">.</span>
+<span class="n">Time</span> <span class="n">to</span> <span class="n">solution</span><span class="p">:</span> <span class="mf">0.7905</span> <span class="n">sec</span><span class="o">.</span>
 <span class="n">Error</span> <span class="n">vs</span><span class="o">.</span> <span class="n">PDE</span> <span class="n">solution</span><span class="p">:</span> <span class="mf">1.9479e-04</span>
 </pre></div>
 </div>
diff --git a/tutorial/doc_step_7_D.html b/tutorial/doc_step_7_D.html
index 6cf812b835..15a09f9487 100644
--- a/tutorial/doc_step_7_D.html
+++ b/tutorial/doc_step_7_D.html
@@ -129,27 +129,27 @@ <h3>Navigation</h3>
 </pre></div>
 </div>
 <p>Results:</p>
-<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">Training</span> <span class="k">for</span> <span class="n">dt</span><span class="o">=</span><span class="mf">1.55e-03</span><span class="p">:</span> <span class="n">Epoch</span> <span class="p">[</span>  <span class="mi">50</span><span class="o">/</span> <span class="mi">250</span><span class="p">],</span> <span class="n">Loss</span><span class="p">:</span> <span class="mf">1.7391e-02</span>
-<span class="n">Training</span> <span class="k">for</span> <span class="n">dt</span><span class="o">=</span><span class="mf">1.55e-03</span><span class="p">:</span> <span class="n">Epoch</span> <span class="p">[</span> <span class="mi">100</span><span class="o">/</span> <span class="mi">250</span><span class="p">],</span> <span class="n">Loss</span><span class="p">:</span> <span class="mf">8.3421e-05</span>
-<span class="n">Training</span> <span class="k">for</span> <span class="n">dt</span><span class="o">=</span><span class="mf">1.55e-03</span><span class="p">:</span> <span class="n">Epoch</span> <span class="p">[</span> <span class="mi">150</span><span class="o">/</span> <span class="mi">250</span><span class="p">],</span> <span class="n">Loss</span><span class="p">:</span> <span class="mf">4.8431e-07</span>
-<span class="n">Training</span> <span class="k">for</span> <span class="n">dt</span><span class="o">=</span><span class="mf">1.55e-03</span><span class="p">:</span> <span class="n">Epoch</span> <span class="p">[</span> <span class="mi">200</span><span class="o">/</span> <span class="mi">250</span><span class="p">],</span> <span class="n">Loss</span><span class="p">:</span> <span class="mf">2.0377e-09</span>
-<span class="n">Training</span> <span class="k">for</span> <span class="n">dt</span><span class="o">=</span><span class="mf">1.55e-03</span><span class="p">:</span> <span class="n">Epoch</span> <span class="p">[</span> <span class="mi">250</span><span class="o">/</span> <span class="mi">250</span><span class="p">],</span> <span class="n">Loss</span><span class="p">:</span> <span class="mf">1.1031e-11</span>
-<span class="n">Error</span> <span class="n">of</span> <span class="n">prediction</span> <span class="n">at</span> <span class="mf">1.55e-03</span> <span class="n">during</span> <span class="n">training</span><span class="p">:</span> <span class="mf">1.82e-05</span>
-<span class="n">Training</span> <span class="k">for</span> <span class="n">dt</span><span class="o">=</span><span class="mf">6.45e-03</span><span class="p">:</span> <span class="n">Epoch</span> <span class="p">[</span>  <span class="mi">50</span><span class="o">/</span> <span class="mi">250</span><span class="p">],</span> <span class="n">Loss</span><span class="p">:</span> <span class="mf">6.7525e-05</span>
-<span class="n">Training</span> <span class="k">for</span> <span class="n">dt</span><span class="o">=</span><span class="mf">6.45e-03</span><span class="p">:</span> <span class="n">Epoch</span> <span class="p">[</span> <span class="mi">100</span><span class="o">/</span> <span class="mi">250</span><span class="p">],</span> <span class="n">Loss</span><span class="p">:</span> <span class="mf">1.0774e-06</span>
-<span class="n">Training</span> <span class="k">for</span> <span class="n">dt</span><span class="o">=</span><span class="mf">6.45e-03</span><span class="p">:</span> <span class="n">Epoch</span> <span class="p">[</span> <span class="mi">150</span><span class="o">/</span> <span class="mi">250</span><span class="p">],</span> <span class="n">Loss</span><span class="p">:</span> <span class="mf">4.2838e-09</span>
-<span class="n">Training</span> <span class="k">for</span> <span class="n">dt</span><span class="o">=</span><span class="mf">6.45e-03</span><span class="p">:</span> <span class="n">Epoch</span> <span class="p">[</span> <span class="mi">200</span><span class="o">/</span> <span class="mi">250</span><span class="p">],</span> <span class="n">Loss</span><span class="p">:</span> <span class="mf">2.6739e-11</span>
-<span class="n">Training</span> <span class="k">for</span> <span class="n">dt</span><span class="o">=</span><span class="mf">6.45e-03</span><span class="p">:</span> <span class="n">Epoch</span> <span class="p">[</span> <span class="mi">250</span><span class="o">/</span> <span class="mi">250</span><span class="p">],</span> <span class="n">Loss</span><span class="p">:</span> <span class="mf">1.1213e-13</span>
-<span class="n">Error</span> <span class="n">of</span> <span class="n">prediction</span> <span class="n">at</span> <span class="mf">6.45e-03</span> <span class="n">during</span> <span class="n">training</span><span class="p">:</span> <span class="mf">5.23e-07</span>
-<span class="n">Training</span> <span class="k">for</span> <span class="n">dt</span><span class="o">=</span><span class="mf">1.00e-02</span><span class="p">:</span> <span class="n">Epoch</span> <span class="p">[</span>  <span class="mi">50</span><span class="o">/</span> <span class="mi">250</span><span class="p">],</span> <span class="n">Loss</span><span class="p">:</span> <span class="mf">3.1718e-05</span>
-<span class="n">Training</span> <span class="k">for</span> <span class="n">dt</span><span class="o">=</span><span class="mf">1.00e-02</span><span class="p">:</span> <span class="n">Epoch</span> <span class="p">[</span> <span class="mi">100</span><span class="o">/</span> <span class="mi">250</span><span class="p">],</span> <span class="n">Loss</span><span class="p">:</span> <span class="mf">8.5910e-08</span>
-<span class="n">Training</span> <span class="k">for</span> <span class="n">dt</span><span class="o">=</span><span class="mf">1.00e-02</span><span class="p">:</span> <span class="n">Epoch</span> <span class="p">[</span> <span class="mi">150</span><span class="o">/</span> <span class="mi">250</span><span class="p">],</span> <span class="n">Loss</span><span class="p">:</span> <span class="mf">4.2362e-10</span>
-<span class="n">Training</span> <span class="k">for</span> <span class="n">dt</span><span class="o">=</span><span class="mf">1.00e-02</span><span class="p">:</span> <span class="n">Epoch</span> <span class="p">[</span> <span class="mi">200</span><span class="o">/</span> <span class="mi">250</span><span class="p">],</span> <span class="n">Loss</span><span class="p">:</span> <span class="mf">1.6118e-12</span>
-<span class="n">Training</span> <span class="k">for</span> <span class="n">dt</span><span class="o">=</span><span class="mf">1.00e-02</span><span class="p">:</span> <span class="n">Epoch</span> <span class="p">[</span> <span class="mi">250</span><span class="o">/</span> <span class="mi">250</span><span class="p">],</span> <span class="n">Loss</span><span class="p">:</span> <span class="mf">2.6642e-14</span>
-<span class="n">Error</span> <span class="n">of</span> <span class="n">prediction</span> <span class="n">at</span> <span class="mf">1.00e-02</span> <span class="n">during</span> <span class="n">training</span><span class="p">:</span> <span class="mf">4.06e-07</span>
-<span class="n">Error</span> <span class="n">of</span> <span class="n">prediction</span> <span class="n">at</span> <span class="mf">1.55e-03</span> <span class="n">after</span> <span class="n">training</span><span class="p">:</span> <span class="mf">4.25e-01</span>
-<span class="n">Error</span> <span class="n">of</span> <span class="n">prediction</span> <span class="n">at</span> <span class="mf">6.45e-03</span> <span class="n">after</span> <span class="n">training</span><span class="p">:</span> <span class="mf">1.12e-01</span>
-<span class="n">Error</span> <span class="n">of</span> <span class="n">prediction</span> <span class="n">at</span> <span class="mf">1.00e-02</span> <span class="n">after</span> <span class="n">training</span><span class="p">:</span> <span class="mf">4.06e-07</span>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">Training</span> <span class="k">for</span> <span class="n">dt</span><span class="o">=</span><span class="mf">1.55e-03</span><span class="p">:</span> <span class="n">Epoch</span> <span class="p">[</span>  <span class="mi">50</span><span class="o">/</span> <span class="mi">250</span><span class="p">],</span> <span class="n">Loss</span><span class="p">:</span> <span class="mf">8.1769e-03</span>
+<span class="n">Training</span> <span class="k">for</span> <span class="n">dt</span><span class="o">=</span><span class="mf">1.55e-03</span><span class="p">:</span> <span class="n">Epoch</span> <span class="p">[</span> <span class="mi">100</span><span class="o">/</span> <span class="mi">250</span><span class="p">],</span> <span class="n">Loss</span><span class="p">:</span> <span class="mf">3.0408e-05</span>
+<span class="n">Training</span> <span class="k">for</span> <span class="n">dt</span><span class="o">=</span><span class="mf">1.55e-03</span><span class="p">:</span> <span class="n">Epoch</span> <span class="p">[</span> <span class="mi">150</span><span class="o">/</span> <span class="mi">250</span><span class="p">],</span> <span class="n">Loss</span><span class="p">:</span> <span class="mf">1.7127e-07</span>
+<span class="n">Training</span> <span class="k">for</span> <span class="n">dt</span><span class="o">=</span><span class="mf">1.55e-03</span><span class="p">:</span> <span class="n">Epoch</span> <span class="p">[</span> <span class="mi">200</span><span class="o">/</span> <span class="mi">250</span><span class="p">],</span> <span class="n">Loss</span><span class="p">:</span> <span class="mf">9.1137e-10</span>
+<span class="n">Training</span> <span class="k">for</span> <span class="n">dt</span><span class="o">=</span><span class="mf">1.55e-03</span><span class="p">:</span> <span class="n">Epoch</span> <span class="p">[</span> <span class="mi">250</span><span class="o">/</span> <span class="mi">250</span><span class="p">],</span> <span class="n">Loss</span><span class="p">:</span> <span class="mf">4.6576e-12</span>
+<span class="n">Error</span> <span class="n">of</span> <span class="n">prediction</span> <span class="n">at</span> <span class="mf">1.55e-03</span> <span class="n">during</span> <span class="n">training</span><span class="p">:</span> <span class="mf">1.30e-05</span>
+<span class="n">Training</span> <span class="k">for</span> <span class="n">dt</span><span class="o">=</span><span class="mf">6.45e-03</span><span class="p">:</span> <span class="n">Epoch</span> <span class="p">[</span>  <span class="mi">50</span><span class="o">/</span> <span class="mi">250</span><span class="p">],</span> <span class="n">Loss</span><span class="p">:</span> <span class="mf">6.7436e-05</span>
+<span class="n">Training</span> <span class="k">for</span> <span class="n">dt</span><span class="o">=</span><span class="mf">6.45e-03</span><span class="p">:</span> <span class="n">Epoch</span> <span class="p">[</span> <span class="mi">100</span><span class="o">/</span> <span class="mi">250</span><span class="p">],</span> <span class="n">Loss</span><span class="p">:</span> <span class="mf">6.8079e-07</span>
+<span class="n">Training</span> <span class="k">for</span> <span class="n">dt</span><span class="o">=</span><span class="mf">6.45e-03</span><span class="p">:</span> <span class="n">Epoch</span> <span class="p">[</span> <span class="mi">150</span><span class="o">/</span> <span class="mi">250</span><span class="p">],</span> <span class="n">Loss</span><span class="p">:</span> <span class="mf">1.7740e-09</span>
+<span class="n">Training</span> <span class="k">for</span> <span class="n">dt</span><span class="o">=</span><span class="mf">6.45e-03</span><span class="p">:</span> <span class="n">Epoch</span> <span class="p">[</span> <span class="mi">200</span><span class="o">/</span> <span class="mi">250</span><span class="p">],</span> <span class="n">Loss</span><span class="p">:</span> <span class="mf">1.4685e-11</span>
+<span class="n">Training</span> <span class="k">for</span> <span class="n">dt</span><span class="o">=</span><span class="mf">6.45e-03</span><span class="p">:</span> <span class="n">Epoch</span> <span class="p">[</span> <span class="mi">250</span><span class="o">/</span> <span class="mi">250</span><span class="p">],</span> <span class="n">Loss</span><span class="p">:</span> <span class="mf">9.2464e-14</span>
+<span class="n">Error</span> <span class="n">of</span> <span class="n">prediction</span> <span class="n">at</span> <span class="mf">6.45e-03</span> <span class="n">during</span> <span class="n">training</span><span class="p">:</span> <span class="mf">8.52e-07</span>
+<span class="n">Training</span> <span class="k">for</span> <span class="n">dt</span><span class="o">=</span><span class="mf">1.00e-02</span><span class="p">:</span> <span class="n">Epoch</span> <span class="p">[</span>  <span class="mi">50</span><span class="o">/</span> <span class="mi">250</span><span class="p">],</span> <span class="n">Loss</span><span class="p">:</span> <span class="mf">2.0351e-05</span>
+<span class="n">Training</span> <span class="k">for</span> <span class="n">dt</span><span class="o">=</span><span class="mf">1.00e-02</span><span class="p">:</span> <span class="n">Epoch</span> <span class="p">[</span> <span class="mi">100</span><span class="o">/</span> <span class="mi">250</span><span class="p">],</span> <span class="n">Loss</span><span class="p">:</span> <span class="mf">3.8560e-08</span>
+<span class="n">Training</span> <span class="k">for</span> <span class="n">dt</span><span class="o">=</span><span class="mf">1.00e-02</span><span class="p">:</span> <span class="n">Epoch</span> <span class="p">[</span> <span class="mi">150</span><span class="o">/</span> <span class="mi">250</span><span class="p">],</span> <span class="n">Loss</span><span class="p">:</span> <span class="mf">5.8901e-10</span>
+<span class="n">Training</span> <span class="k">for</span> <span class="n">dt</span><span class="o">=</span><span class="mf">1.00e-02</span><span class="p">:</span> <span class="n">Epoch</span> <span class="p">[</span> <span class="mi">200</span><span class="o">/</span> <span class="mi">250</span><span class="p">],</span> <span class="n">Loss</span><span class="p">:</span> <span class="mf">2.8962e-12</span>
+<span class="n">Training</span> <span class="k">for</span> <span class="n">dt</span><span class="o">=</span><span class="mf">1.00e-02</span><span class="p">:</span> <span class="n">Epoch</span> <span class="p">[</span> <span class="mi">250</span><span class="o">/</span> <span class="mi">250</span><span class="p">],</span> <span class="n">Loss</span><span class="p">:</span> <span class="mf">1.5985e-14</span>
+<span class="n">Error</span> <span class="n">of</span> <span class="n">prediction</span> <span class="n">at</span> <span class="mf">1.00e-02</span> <span class="n">during</span> <span class="n">training</span><span class="p">:</span> <span class="mf">3.44e-07</span>
+<span class="n">Error</span> <span class="n">of</span> <span class="n">prediction</span> <span class="n">at</span> <span class="mf">1.55e-03</span> <span class="n">after</span> <span class="n">training</span><span class="p">:</span> <span class="mf">4.28e-01</span>
+<span class="n">Error</span> <span class="n">of</span> <span class="n">prediction</span> <span class="n">at</span> <span class="mf">6.45e-03</span> <span class="n">after</span> <span class="n">training</span><span class="p">:</span> <span class="mf">1.13e-01</span>
+<span class="n">Error</span> <span class="n">of</span> <span class="n">prediction</span> <span class="n">at</span> <span class="mf">1.00e-02</span> <span class="n">after</span> <span class="n">training</span><span class="p">:</span> <span class="mf">3.44e-07</span>
 </pre></div>
 </div>
 
diff --git a/tutorial/doc_step_8_A.html b/tutorial/doc_step_8_A.html
index 172d61e8db..9c995be557 100644
--- a/tutorial/doc_step_8_A.html
+++ b/tutorial/doc_step_8_A.html
@@ -115,7 +115,7 @@ <h3>Navigation</h3>
 <span class="n">Number</span> <span class="n">of</span> <span class="n">iterations</span> <span class="k">for</span> <span class="n">time</span> <span class="mf">0.88</span><span class="p">:</span> <span class="mi">7</span>
 </pre></div>
 </div>
-<a class="reference internal image-reference" href="../_images/step_8_residuals.png"><img alt="../_images/step_8_residuals.png" src="../_images/step_8_residuals.png" style="width: 614.0px; height: 433.5px;" />
+<a class="reference internal image-reference" href="../_images/step_8_residuals.png"><img alt="../_images/step_8_residuals.png" src="../_images/step_8_residuals.png" style="width: 754.0px; height: 505.5px;" />
 </a>
 
 
diff --git a/tutorial/doc_step_8_B.html b/tutorial/doc_step_8_B.html
index 2675715880..95b341b954 100644
--- a/tutorial/doc_step_8_B.html
+++ b/tutorial/doc_step_8_B.html
@@ -226,9 +226,9 @@ <h3>Navigation</h3>
 <span class="n">Number</span> <span class="n">of</span> <span class="n">iterations</span> <span class="k">for</span> <span class="n">time</span> <span class="mf">0.88</span> <span class="p">(</span><span class="n">PFASST</span><span class="o">/</span><span class="n">parMSSDC</span><span class="o">/</span><span class="n">serMSSDC</span><span class="p">):</span>  <span class="mi">8</span> <span class="o">/</span> <span class="mi">18</span> <span class="o">/</span> <span class="mi">11</span>
 </pre></div>
 </div>
-<a class="reference internal image-reference" href="../_images/step_8_residuals_mssdc_jac.png"><img alt="../_images/step_8_residuals_mssdc_jac.png" src="../_images/step_8_residuals_mssdc_jac.png" style="width: 614.0px; height: 433.5px;" />
+<a class="reference internal image-reference" href="../_images/step_8_residuals_mssdc_jac.png"><img alt="../_images/step_8_residuals_mssdc_jac.png" src="../_images/step_8_residuals_mssdc_jac.png" style="width: 754.0px; height: 505.5px;" />
 </a>
-<a class="reference internal image-reference" href="../_images/step_8_residuals_mssdc_gs.png"><img alt="../_images/step_8_residuals_mssdc_gs.png" src="../_images/step_8_residuals_mssdc_gs.png" style="width: 614.0px; height: 433.5px;" />
+<a class="reference internal image-reference" href="../_images/step_8_residuals_mssdc_gs.png"><img alt="../_images/step_8_residuals_mssdc_gs.png" src="../_images/step_8_residuals_mssdc_gs.png" style="width: 754.0px; height: 505.5px;" />
 </a>
 
 
diff --git a/tutorial/doc_step_8_C.html b/tutorial/doc_step_8_C.html
index b812ce20d5..8d4164a57d 100644
--- a/tutorial/doc_step_8_C.html
+++ b/tutorial/doc_step_8_C.html
@@ -355,7 +355,7 @@ <h3>Navigation</h3>
    Errors after step   0.8750 with 6 iterations: 3.6530e-08 / 2.0645e-13
    Errors after step   1.0000 with 6 iterations: 3.6478e-08 / 1.2759e-13
 
-...done, took 0.3308976680000342 seconds!
+...done, took 0.33569771999998466 seconds!
 -----------------------------------------------------------------------------
 Running diffusion in 1 dimensions with time-step size 0.125...
 
@@ -369,7 +369,7 @@ <h3>Navigation</h3>
    Errors after step   0.8750 with 3 iterations: 3.5501e-08 / 3.9957e-13
    Errors after step   1.0000 with 3 iterations: 3.6813e-08 / 1.6975e-13
 
-...done, took 0.7408457070000054 seconds!
+...done, took 0.7528066140000078 seconds!
 -----------------------------------------------------------------------------
 Running advection in 1 dimensions with time-step size 0.125...
 
@@ -383,7 +383,7 @@ <h3>Navigation</h3>
    Errors after step   0.8750 with 4 iterations: 2.5517e-08 / 3.3307e-16
    Errors after step   1.0000 with 4 iterations: 2.9147e-08 / 4.4409e-16
 
-...done, took 0.21992689800003973 seconds!
+...done, took 0.22154832900000088 seconds!
 -----------------------------------------------------------------------------
 Running advection in 1 dimensions with time-step size 0.125...
 
@@ -397,7 +397,7 @@ <h3>Navigation</h3>
    Errors after step   0.8750 with 2 iterations: 2.5987e-08 / 3.1841e-13
    Errors after step   1.0000 with 2 iterations: 2.9666e-08 / 3.1819e-13
 
-...done, took 0.6080810290000045 seconds!
+...done, took 0.6121239379999963 seconds!
 -----------------------------------------------------------------------------
 Running auzinger in 1 dimensions with time-step size 0.125...
 
@@ -411,7 +411,7 @@ <h3>Navigation</h3>
    Errors after step   0.8750 with 6 iterations: 2.8266e-08 / 0.0000e+00
    Errors after step   1.0000 with 6 iterations: 2.8338e-08 / 0.0000e+00
 
-...done, took 0.13996988500002772 seconds!
+...done, took 0.14012889900001824 seconds!
 -----------------------------------------------------------------------------
 Running auzinger in 1 dimensions with time-step size 0.125...
 
@@ -425,7 +425,7 @@ <h3>Navigation</h3>
    Errors after step   0.8750 with 4 iterations: 3.1486e-08 / 0.0000e+00
    Errors after step   1.0000 with 4 iterations: 3.3802e-08 / 0.0000e+00
 
-...done, took 0.1660917489999747 seconds!
+...done, took 0.16614925400000402 seconds!
 -----------------------------------------------------------------------------
 </pre></div>
 </div>
diff --git a/tutorial/step_1.html b/tutorial/step_1.html
index cdf51c5664..9ecfa8413b 100644
--- a/tutorial/step_1.html
+++ b/tutorial/step_1.html
@@ -361,7 +361,7 @@ <h2>Part B: Spatial accuracy check<a class="headerlink" href="#part-b-spatial-ac
 <span class="n">Expected</span> <span class="n">order</span><span class="p">:</span>  <span class="mi">2</span> <span class="o">--</span> <span class="n">Computed</span> <span class="n">order</span> <span class="mf">1.982</span>
 </pre></div>
 </div>
-<a class="reference internal image-reference" href="../_images/step_1_accuracy_test_space.png"><img alt="../_images/step_1_accuracy_test_space.png" src="../_images/step_1_accuracy_test_space.png" style="width: 522.5px; height: 352.0px;" />
+<a class="reference internal image-reference" href="../_images/step_1_accuracy_test_space.png"><img alt="../_images/step_1_accuracy_test_space.png" src="../_images/step_1_accuracy_test_space.png" style="width: 605.5px; height: 405.5px;" />
 </a>
 </section>
 <section id="part-c-collocation-problem-setup">
@@ -694,7 +694,7 @@ <h2>Part D: Collocation accuracy test<a class="headerlink" href="#part-d-colloca
 <span class="n">Expected</span> <span class="n">order</span><span class="p">:</span>  <span class="mi">5</span> <span class="o">--</span> <span class="n">Computed</span> <span class="n">order</span> <span class="mf">5.671</span>
 </pre></div>
 </div>
-<a class="reference internal image-reference" href="../_images/step_1_accuracy_test_coll.png"><img alt="../_images/step_1_accuracy_test_coll.png" src="../_images/step_1_accuracy_test_coll.png" style="width: 522.5px; height: 352.0px;" />
+<a class="reference internal image-reference" href="../_images/step_1_accuracy_test_coll.png"><img alt="../_images/step_1_accuracy_test_coll.png" src="../_images/step_1_accuracy_test_coll.png" style="width: 605.5px; height: 405.5px;" />
 </a>
 </section>
 </section>
diff --git a/tutorial/step_2.html b/tutorial/step_2.html
index 8f4568b448..5f27e3d249 100644
--- a/tutorial/step_2.html
+++ b/tutorial/step_2.html
@@ -417,7 +417,7 @@ <h2>Part C: Using pySDC’s frontend<a class="headerlink" href="#part-c-using-py
 </pre></div>
 </div>
 <p>Results:</p>
-<div class="highlight-default notranslate"><div class="highlight"><pre><span></span>2024-09-19 08:57:00,351 - controller - controller - welcome_message - 146 - INFO: Welcome to the one and only, really very astonishing and 87.3% bug free
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span>2024-09-20 16:22:50,373 - controller - controller - welcome_message - 146 - INFO: Welcome to the one and only, really very astonishing and 87.3% bug free
                                  _____ _____   _____ 
                                 / ____|  __ \ / ____|
                     _ __  _   _| (___ | |  | | |     
@@ -427,8 +427,8 @@ <h2>Part C: Using pySDC’s frontend<a class="headerlink" href="#part-c-using-py
                    | |     __/ |                     
                    |_|    |___/                      
                                                      
-2024-09-19 08:57:00,351 - controller - controller - dump_setup - 160 - INFO: Setup overview (--&gt; user-defined, -&gt; dependency) -- BEGIN
-2024-09-19 08:57:00,352 - controller - controller - dump_setup - 227 - INFO: ----------------------------------------------------------------------------------------------------
+2024-09-20 16:22:50,373 - controller - controller - dump_setup - 160 - INFO: Setup overview (--&gt; user-defined, -&gt; dependency) -- BEGIN
+2024-09-20 16:22:50,373 - controller - controller - dump_setup - 227 - INFO: ----------------------------------------------------------------------------------------------------
 
 Controller: &lt;class &#39;pySDC.implementations.controller_classes.controller_nonMPI.controller_nonMPI&#39;&gt;
     all_to_done = False
@@ -481,34 +481,34 @@ <h2>Part C: Using pySDC’s frontend<a class="headerlink" href="#part-c-using-py
  -&gt; |  1 |   100 | SpreadStepSizesBlockwiseNonMPI
     |  2 |   200 | CheckConvergence
 
-2024-09-19 08:57:00,352 - controller - controller - dump_setup - 230 - INFO: ----------------------------------------------------------------------------------------------------
-2024-09-19 08:57:00,352 - controller - controller - dump_setup - 232 - INFO: Setup overview (--&gt; user-defined, -&gt; dependency) -- END
-
-2024-09-19 08:57:00,356 - hooks - default_hook - post_sweep - 170 - INFO: Process  0 on time 0.100000 at stage         IT_FINE: Level: 0 -- Iteration:  1 -- Sweep:  1 -- residual: 4.11190756e-03
-2024-09-19 08:57:00,359 - hooks - default_hook - post_sweep - 170 - INFO: Process  0 on time 0.100000 at stage         IT_FINE: Level: 0 -- Iteration:  2 -- Sweep:  1 -- residual: 6.68442667e-04
-2024-09-19 08:57:00,362 - hooks - default_hook - post_sweep - 170 - INFO: Process  0 on time 0.100000 at stage         IT_FINE: Level: 0 -- Iteration:  3 -- Sweep:  1 -- residual: 8.80377591e-05
-2024-09-19 08:57:00,365 - hooks - default_hook - post_sweep - 170 - INFO: Process  0 on time 0.100000 at stage         IT_FINE: Level: 0 -- Iteration:  4 -- Sweep:  1 -- residual: 1.21707909e-05
-2024-09-19 08:57:00,368 - hooks - default_hook - post_sweep - 170 - INFO: Process  0 on time 0.100000 at stage         IT_FINE: Level: 0 -- Iteration:  5 -- Sweep:  1 -- residual: 1.38272147e-06
-2024-09-19 08:57:00,372 - hooks - default_hook - post_sweep - 170 - INFO: Process  0 on time 0.100000 at stage         IT_FINE: Level: 0 -- Iteration:  6 -- Sweep:  1 -- residual: 6.36445413e-07
-2024-09-19 08:57:00,375 - hooks - default_hook - post_sweep - 170 - INFO: Process  0 on time 0.100000 at stage         IT_FINE: Level: 0 -- Iteration:  7 -- Sweep:  1 -- residual: 1.68953216e-07
-2024-09-19 08:57:00,378 - hooks - default_hook - post_sweep - 170 - INFO: Process  0 on time 0.100000 at stage         IT_FINE: Level: 0 -- Iteration:  8 -- Sweep:  1 -- residual: 3.52601840e-08
-2024-09-19 08:57:00,381 - hooks - default_hook - post_sweep - 170 - INFO: Process  0 on time 0.100000 at stage         IT_FINE: Level: 0 -- Iteration:  9 -- Sweep:  1 -- residual: 6.07249025e-09
-2024-09-19 08:57:00,384 - hooks - default_hook - post_sweep - 170 - INFO: Process  0 on time 0.100000 at stage         IT_FINE: Level: 0 -- Iteration: 10 -- Sweep:  1 -- residual: 8.27343378e-10
-2024-09-19 08:57:00,388 - hooks - default_hook - post_sweep - 170 - INFO: Process  0 on time 0.100000 at stage         IT_FINE: Level: 0 -- Iteration: 11 -- Sweep:  1 -- residual: 1.18931339e-10
-2024-09-19 08:57:00,391 - hooks - default_hook - post_sweep - 170 - INFO: Process  0 on time 0.100000 at stage         IT_FINE: Level: 0 -- Iteration: 12 -- Sweep:  1 -- residual: 1.48499772e-11
-2024-09-19 08:57:00,395 - hooks - default_hook - post_sweep - 170 - INFO: Process  0 on time 0.200000 at stage         IT_FINE: Level: 0 -- Iteration:  1 -- Sweep:  1 -- residual: 6.69984764e-03
-2024-09-19 08:57:00,398 - hooks - default_hook - post_sweep - 170 - INFO: Process  0 on time 0.200000 at stage         IT_FINE: Level: 0 -- Iteration:  2 -- Sweep:  1 -- residual: 1.05518433e-03
-2024-09-19 08:57:00,401 - hooks - default_hook - post_sweep - 170 - INFO: Process  0 on time 0.200000 at stage         IT_FINE: Level: 0 -- Iteration:  3 -- Sweep:  1 -- residual: 1.40642621e-04
-2024-09-19 08:57:00,404 - hooks - default_hook - post_sweep - 170 - INFO: Process  0 on time 0.200000 at stage         IT_FINE: Level: 0 -- Iteration:  4 -- Sweep:  1 -- residual: 1.85982063e-05
-2024-09-19 08:57:00,408 - hooks - default_hook - post_sweep - 170 - INFO: Process  0 on time 0.200000 at stage         IT_FINE: Level: 0 -- Iteration:  5 -- Sweep:  1 -- residual: 2.79216702e-06
-2024-09-19 08:57:00,411 - hooks - default_hook - post_sweep - 170 - INFO: Process  0 on time 0.200000 at stage         IT_FINE: Level: 0 -- Iteration:  6 -- Sweep:  1 -- residual: 1.12278839e-06
-2024-09-19 08:57:00,414 - hooks - default_hook - post_sweep - 170 - INFO: Process  0 on time 0.200000 at stage         IT_FINE: Level: 0 -- Iteration:  7 -- Sweep:  1 -- residual: 2.85495353e-07
-2024-09-19 08:57:00,417 - hooks - default_hook - post_sweep - 170 - INFO: Process  0 on time 0.200000 at stage         IT_FINE: Level: 0 -- Iteration:  8 -- Sweep:  1 -- residual: 5.78947003e-08
-2024-09-19 08:57:00,421 - hooks - default_hook - post_sweep - 170 - INFO: Process  0 on time 0.200000 at stage         IT_FINE: Level: 0 -- Iteration:  9 -- Sweep:  1 -- residual: 9.68230621e-09
-2024-09-19 08:57:00,424 - hooks - default_hook - post_sweep - 170 - INFO: Process  0 on time 0.200000 at stage         IT_FINE: Level: 0 -- Iteration: 10 -- Sweep:  1 -- residual: 1.26313315e-09
-2024-09-19 08:57:00,427 - hooks - default_hook - post_sweep - 170 - INFO: Process  0 on time 0.200000 at stage         IT_FINE: Level: 0 -- Iteration: 11 -- Sweep:  1 -- residual: 1.82951499e-10
-2024-09-19 08:57:00,430 - hooks - default_hook - post_sweep - 170 - INFO: Process  0 on time 0.200000 at stage         IT_FINE: Level: 0 -- Iteration: 12 -- Sweep:  1 -- residual: 1.99691114e-11
-2024-09-19 08:57:00,431 - hooks - default_hook - post_run - 340 - INFO: Finished run after 0.08s
+2024-09-20 16:22:50,373 - controller - controller - dump_setup - 230 - INFO: ----------------------------------------------------------------------------------------------------
+2024-09-20 16:22:50,373 - controller - controller - dump_setup - 232 - INFO: Setup overview (--&gt; user-defined, -&gt; dependency) -- END
+
+2024-09-20 16:22:50,377 - hooks - default_hook - post_sweep - 170 - INFO: Process  0 on time 0.100000 at stage         IT_FINE: Level: 0 -- Iteration:  1 -- Sweep:  1 -- residual: 4.11190756e-03
+2024-09-20 16:22:50,380 - hooks - default_hook - post_sweep - 170 - INFO: Process  0 on time 0.100000 at stage         IT_FINE: Level: 0 -- Iteration:  2 -- Sweep:  1 -- residual: 6.68442667e-04
+2024-09-20 16:22:50,384 - hooks - default_hook - post_sweep - 170 - INFO: Process  0 on time 0.100000 at stage         IT_FINE: Level: 0 -- Iteration:  3 -- Sweep:  1 -- residual: 8.80377591e-05
+2024-09-20 16:22:50,387 - hooks - default_hook - post_sweep - 170 - INFO: Process  0 on time 0.100000 at stage         IT_FINE: Level: 0 -- Iteration:  4 -- Sweep:  1 -- residual: 1.21707909e-05
+2024-09-20 16:22:50,390 - hooks - default_hook - post_sweep - 170 - INFO: Process  0 on time 0.100000 at stage         IT_FINE: Level: 0 -- Iteration:  5 -- Sweep:  1 -- residual: 1.38272147e-06
+2024-09-20 16:22:50,393 - hooks - default_hook - post_sweep - 170 - INFO: Process  0 on time 0.100000 at stage         IT_FINE: Level: 0 -- Iteration:  6 -- Sweep:  1 -- residual: 6.36445413e-07
+2024-09-20 16:22:50,397 - hooks - default_hook - post_sweep - 170 - INFO: Process  0 on time 0.100000 at stage         IT_FINE: Level: 0 -- Iteration:  7 -- Sweep:  1 -- residual: 1.68953216e-07
+2024-09-20 16:22:50,400 - hooks - default_hook - post_sweep - 170 - INFO: Process  0 on time 0.100000 at stage         IT_FINE: Level: 0 -- Iteration:  8 -- Sweep:  1 -- residual: 3.52601840e-08
+2024-09-20 16:22:50,403 - hooks - default_hook - post_sweep - 170 - INFO: Process  0 on time 0.100000 at stage         IT_FINE: Level: 0 -- Iteration:  9 -- Sweep:  1 -- residual: 6.07249025e-09
+2024-09-20 16:22:50,407 - hooks - default_hook - post_sweep - 170 - INFO: Process  0 on time 0.100000 at stage         IT_FINE: Level: 0 -- Iteration: 10 -- Sweep:  1 -- residual: 8.27343378e-10
+2024-09-20 16:22:50,410 - hooks - default_hook - post_sweep - 170 - INFO: Process  0 on time 0.100000 at stage         IT_FINE: Level: 0 -- Iteration: 11 -- Sweep:  1 -- residual: 1.18931339e-10
+2024-09-20 16:22:50,413 - hooks - default_hook - post_sweep - 170 - INFO: Process  0 on time 0.100000 at stage         IT_FINE: Level: 0 -- Iteration: 12 -- Sweep:  1 -- residual: 1.48499772e-11
+2024-09-20 16:22:50,417 - hooks - default_hook - post_sweep - 170 - INFO: Process  0 on time 0.200000 at stage         IT_FINE: Level: 0 -- Iteration:  1 -- Sweep:  1 -- residual: 6.69984764e-03
+2024-09-20 16:22:50,420 - hooks - default_hook - post_sweep - 170 - INFO: Process  0 on time 0.200000 at stage         IT_FINE: Level: 0 -- Iteration:  2 -- Sweep:  1 -- residual: 1.05518433e-03
+2024-09-20 16:22:50,424 - hooks - default_hook - post_sweep - 170 - INFO: Process  0 on time 0.200000 at stage         IT_FINE: Level: 0 -- Iteration:  3 -- Sweep:  1 -- residual: 1.40642621e-04
+2024-09-20 16:22:50,427 - hooks - default_hook - post_sweep - 170 - INFO: Process  0 on time 0.200000 at stage         IT_FINE: Level: 0 -- Iteration:  4 -- Sweep:  1 -- residual: 1.85982063e-05
+2024-09-20 16:22:50,430 - hooks - default_hook - post_sweep - 170 - INFO: Process  0 on time 0.200000 at stage         IT_FINE: Level: 0 -- Iteration:  5 -- Sweep:  1 -- residual: 2.79216702e-06
+2024-09-20 16:22:50,433 - hooks - default_hook - post_sweep - 170 - INFO: Process  0 on time 0.200000 at stage         IT_FINE: Level: 0 -- Iteration:  6 -- Sweep:  1 -- residual: 1.12278839e-06
+2024-09-20 16:22:50,437 - hooks - default_hook - post_sweep - 170 - INFO: Process  0 on time 0.200000 at stage         IT_FINE: Level: 0 -- Iteration:  7 -- Sweep:  1 -- residual: 2.85495353e-07
+2024-09-20 16:22:50,440 - hooks - default_hook - post_sweep - 170 - INFO: Process  0 on time 0.200000 at stage         IT_FINE: Level: 0 -- Iteration:  8 -- Sweep:  1 -- residual: 5.78947003e-08
+2024-09-20 16:22:50,443 - hooks - default_hook - post_sweep - 170 - INFO: Process  0 on time 0.200000 at stage         IT_FINE: Level: 0 -- Iteration:  9 -- Sweep:  1 -- residual: 9.68230621e-09
+2024-09-20 16:22:50,446 - hooks - default_hook - post_sweep - 170 - INFO: Process  0 on time 0.200000 at stage         IT_FINE: Level: 0 -- Iteration: 10 -- Sweep:  1 -- residual: 1.26313315e-09
+2024-09-20 16:22:50,449 - hooks - default_hook - post_sweep - 170 - INFO: Process  0 on time 0.200000 at stage         IT_FINE: Level: 0 -- Iteration: 11 -- Sweep:  1 -- residual: 1.82951499e-10
+2024-09-20 16:22:50,453 - hooks - default_hook - post_sweep - 170 - INFO: Process  0 on time 0.200000 at stage         IT_FINE: Level: 0 -- Iteration: 12 -- Sweep:  1 -- residual: 1.99691114e-11
+2024-09-20 16:22:50,453 - hooks - default_hook - post_run - 340 - INFO: Finished run after 0.08s
 Error after SDC iterations: 1.166689e-05
 </pre></div>
 </div>
diff --git a/tutorial/step_3.html b/tutorial/step_3.html
index af37013d9e..c8dfba478d 100644
--- a/tutorial/step_3.html
+++ b/tutorial/step_3.html
@@ -344,7 +344,7 @@ <h2>Part B: Adding statistics<a class="headerlink" href="#part-b-adding-statisti
 </pre></div>
 </div>
 <p>Results:</p>
-<div class="highlight-default notranslate"><div class="highlight"><pre><span></span>2024-09-19 08:57:00,996 - controller - controller - welcome_message - 146 - INFO: Welcome to the one and only, really very astonishing and 87.3% bug free
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span>2024-09-20 16:22:50,966 - controller - controller - welcome_message - 146 - INFO: Welcome to the one and only, really very astonishing and 87.3% bug free
                                  _____ _____   _____ 
                                 / ____|  __ \ / ____|
                     _ __  _   _| (___ | |  | | |     
@@ -354,8 +354,8 @@ <h2>Part B: Adding statistics<a class="headerlink" href="#part-b-adding-statisti
                    | |     __/ |                     
                    |_|    |___/                      
                                                      
-2024-09-19 08:57:00,996 - controller - controller - dump_setup - 160 - INFO: Setup overview (--&gt; user-defined, -&gt; dependency) -- BEGIN
-2024-09-19 08:57:00,996 - controller - controller - dump_setup - 227 - INFO: ----------------------------------------------------------------------------------------------------
+2024-09-20 16:22:50,966 - controller - controller - dump_setup - 160 - INFO: Setup overview (--&gt; user-defined, -&gt; dependency) -- BEGIN
+2024-09-20 16:22:50,966 - controller - controller - dump_setup - 227 - INFO: ----------------------------------------------------------------------------------------------------
 
 Controller: &lt;class &#39;pySDC.implementations.controller_classes.controller_nonMPI.controller_nonMPI&#39;&gt;
     all_to_done = False
@@ -403,21 +403,21 @@ <h2>Part B: Adding statistics<a class="headerlink" href="#part-b-adding-statisti
  -&gt; |  1 |   100 | SpreadStepSizesBlockwiseNonMPI
     |  2 |   200 | CheckConvergence
 
-2024-09-19 08:57:00,996 - controller - controller - dump_setup - 230 - INFO: ----------------------------------------------------------------------------------------------------
-2024-09-19 08:57:00,996 - controller - controller - dump_setup - 232 - INFO: Setup overview (--&gt; user-defined, -&gt; dependency) -- END
-
-2024-09-19 08:57:02,351 - hooks - default_hook - post_sweep - 170 - INFO: Process  0 on time 0.000000 at stage         IT_FINE: Level: 0 -- Iteration:  1 -- Sweep:  1 -- residual: 3.53203678e+00
-2024-09-19 08:57:02,358 - hooks - default_hook - post_sweep - 170 - INFO: Process  0 on time 0.000000 at stage         IT_FINE: Level: 0 -- Iteration:  2 -- Sweep:  1 -- residual: 2.09852117e-01
-2024-09-19 08:57:02,365 - hooks - default_hook - post_sweep - 170 - INFO: Process  0 on time 0.000000 at stage         IT_FINE: Level: 0 -- Iteration:  3 -- Sweep:  1 -- residual: 3.50301513e-02
-2024-09-19 08:57:02,372 - hooks - default_hook - post_sweep - 170 - INFO: Process  0 on time 0.000000 at stage         IT_FINE: Level: 0 -- Iteration:  4 -- Sweep:  1 -- residual: 4.67724741e-03
-2024-09-19 08:57:02,379 - hooks - default_hook - post_sweep - 170 - INFO: Process  0 on time 0.000000 at stage         IT_FINE: Level: 0 -- Iteration:  5 -- Sweep:  1 -- residual: 7.95583202e-04
-2024-09-19 08:57:02,386 - hooks - default_hook - post_sweep - 170 - INFO: Process  0 on time 0.000000 at stage         IT_FINE: Level: 0 -- Iteration:  6 -- Sweep:  1 -- residual: 1.11405073e-04
-2024-09-19 08:57:02,393 - hooks - default_hook - post_sweep - 170 - INFO: Process  0 on time 0.000000 at stage         IT_FINE: Level: 0 -- Iteration:  7 -- Sweep:  1 -- residual: 1.26902403e-05
-2024-09-19 08:57:02,400 - hooks - default_hook - post_sweep - 170 - INFO: Process  0 on time 0.000000 at stage         IT_FINE: Level: 0 -- Iteration:  8 -- Sweep:  1 -- residual: 1.16534547e-06
-2024-09-19 08:57:02,406 - hooks - default_hook - post_sweep - 170 - INFO: Process  0 on time 0.000000 at stage         IT_FINE: Level: 0 -- Iteration:  9 -- Sweep:  1 -- residual: 1.66968022e-07
-2024-09-19 08:57:02,413 - hooks - default_hook - post_sweep - 170 - INFO: Process  0 on time 0.000000 at stage         IT_FINE: Level: 0 -- Iteration: 10 -- Sweep:  1 -- residual: 2.09408171e-08
-2024-09-19 08:57:02,420 - hooks - default_hook - post_sweep - 170 - INFO: Process  0 on time 0.000000 at stage         IT_FINE: Level: 0 -- Iteration: 11 -- Sweep:  1 -- residual: 2.17123386e-09
-2024-09-19 08:57:02,423 - hooks - default_hook - post_run - 340 - INFO: Finished run after 1.43s
+2024-09-20 16:22:50,967 - controller - controller - dump_setup - 230 - INFO: ----------------------------------------------------------------------------------------------------
+2024-09-20 16:22:50,967 - controller - controller - dump_setup - 232 - INFO: Setup overview (--&gt; user-defined, -&gt; dependency) -- END
+
+2024-09-20 16:22:52,320 - hooks - default_hook - post_sweep - 170 - INFO: Process  0 on time 0.000000 at stage         IT_FINE: Level: 0 -- Iteration:  1 -- Sweep:  1 -- residual: 3.53203678e+00
+2024-09-20 16:22:52,327 - hooks - default_hook - post_sweep - 170 - INFO: Process  0 on time 0.000000 at stage         IT_FINE: Level: 0 -- Iteration:  2 -- Sweep:  1 -- residual: 2.09852117e-01
+2024-09-20 16:22:52,334 - hooks - default_hook - post_sweep - 170 - INFO: Process  0 on time 0.000000 at stage         IT_FINE: Level: 0 -- Iteration:  3 -- Sweep:  1 -- residual: 3.50301513e-02
+2024-09-20 16:22:52,341 - hooks - default_hook - post_sweep - 170 - INFO: Process  0 on time 0.000000 at stage         IT_FINE: Level: 0 -- Iteration:  4 -- Sweep:  1 -- residual: 4.67724741e-03
+2024-09-20 16:22:52,347 - hooks - default_hook - post_sweep - 170 - INFO: Process  0 on time 0.000000 at stage         IT_FINE: Level: 0 -- Iteration:  5 -- Sweep:  1 -- residual: 7.95583202e-04
+2024-09-20 16:22:52,354 - hooks - default_hook - post_sweep - 170 - INFO: Process  0 on time 0.000000 at stage         IT_FINE: Level: 0 -- Iteration:  6 -- Sweep:  1 -- residual: 1.11405073e-04
+2024-09-20 16:22:52,361 - hooks - default_hook - post_sweep - 170 - INFO: Process  0 on time 0.000000 at stage         IT_FINE: Level: 0 -- Iteration:  7 -- Sweep:  1 -- residual: 1.26902403e-05
+2024-09-20 16:22:52,368 - hooks - default_hook - post_sweep - 170 - INFO: Process  0 on time 0.000000 at stage         IT_FINE: Level: 0 -- Iteration:  8 -- Sweep:  1 -- residual: 1.16534547e-06
+2024-09-20 16:22:52,375 - hooks - default_hook - post_sweep - 170 - INFO: Process  0 on time 0.000000 at stage         IT_FINE: Level: 0 -- Iteration:  9 -- Sweep:  1 -- residual: 1.66968022e-07
+2024-09-20 16:22:52,382 - hooks - default_hook - post_sweep - 170 - INFO: Process  0 on time 0.000000 at stage         IT_FINE: Level: 0 -- Iteration: 10 -- Sweep:  1 -- residual: 2.09408171e-08
+2024-09-20 16:22:52,389 - hooks - default_hook - post_sweep - 170 - INFO: Process  0 on time 0.000000 at stage         IT_FINE: Level: 0 -- Iteration: 11 -- Sweep:  1 -- residual: 2.17123386e-09
+2024-09-20 16:22:52,392 - hooks - default_hook - post_run - 340 - INFO: Finished run after 1.42s
 Total energy and deviation in iteration  0: 8799.5000000000 -- 0.00000000e+00
 Total energy and deviation in iteration 11: 8785.0038936088 -- 1.44961064e+01
 </pre></div>
diff --git a/tutorial/step_4.html b/tutorial/step_4.html
index 8ff6da526b..1ddab5c7c7 100644
--- a/tutorial/step_4.html
+++ b/tutorial/step_4.html
@@ -619,7 +619,7 @@ <h2>Part D: MLSDC with particles<a class="headerlink" href="#part-d-mlsdc-with-p
 </pre></div>
 </div>
 <p>Results:</p>
-<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">Timings</span> <span class="k">for</span> <span class="n">SDC</span><span class="p">,</span> <span class="n">MLSDC</span> <span class="ow">and</span> <span class="n">MLSDC</span><span class="o">+</span><span class="n">finter</span><span class="p">:</span>   <span class="mf">5.93434594</span> <span class="o">--</span>   <span class="mf">6.60096954</span> <span class="o">--</span>   <span class="mf">7.66078035</span>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">Timings</span> <span class="k">for</span> <span class="n">SDC</span><span class="p">,</span> <span class="n">MLSDC</span> <span class="ow">and</span> <span class="n">MLSDC</span><span class="o">+</span><span class="n">finter</span><span class="p">:</span>   <span class="mf">6.00803794</span> <span class="o">--</span>   <span class="mf">6.64845455</span> <span class="o">--</span>   <span class="mf">7.71512253</span>
 <span class="n">Total</span> <span class="n">energy</span> <span class="ow">and</span> <span class="n">relative</span> <span class="n">deviation</span> <span class="ow">in</span> <span class="n">iteration</span>  <span class="mi">0</span><span class="p">:</span> <span class="mf">407936.7556966486</span> <span class="o">--</span> <span class="mf">0.00000000e+00</span>
 <span class="n">Total</span> <span class="n">energy</span> <span class="ow">and</span> <span class="n">relative</span> <span class="n">deviation</span> <span class="ow">in</span> <span class="n">iteration</span> <span class="mi">12</span><span class="p">:</span> <span class="mf">406977.9425667246</span> <span class="o">--</span> <span class="mf">2.35039652e-03</span>
 <span class="n">Total</span> <span class="n">energy</span> <span class="ow">and</span> <span class="n">relative</span> <span class="n">deviation</span> <span class="ow">in</span> <span class="n">iteration</span>  <span class="mi">0</span><span class="p">:</span> <span class="mf">407936.7556966486</span> <span class="o">--</span> <span class="mf">0.00000000e+00</span>
diff --git a/tutorial/step_7.html b/tutorial/step_7.html
index f7005d744a..aa29f9249c 100644
--- a/tutorial/step_7.html
+++ b/tutorial/step_7.html
@@ -256,49 +256,49 @@ <h2>Part A: pySDC and FEniCS<a class="headerlink" href="#part-a-pysdc-and-fenics
    <span class="n">Range</span> <span class="n">of</span> <span class="n">values</span> <span class="k">for</span> <span class="n">number</span> <span class="n">of</span> <span class="n">iterations</span><span class="p">:</span>  <span class="mi">0</span> 
    <span class="n">Position</span> <span class="n">of</span> <span class="nb">max</span><span class="o">/</span><span class="nb">min</span> <span class="n">number</span> <span class="n">of</span> <span class="n">iterations</span><span class="p">:</span>  <span class="mi">0</span> <span class="o">--</span>  <span class="mi">0</span>
    <span class="n">Std</span> <span class="ow">and</span> <span class="n">var</span> <span class="k">for</span> <span class="n">number</span> <span class="n">of</span> <span class="n">iterations</span><span class="p">:</span> <span class="mf">0.00</span> <span class="o">--</span> <span class="mf">0.00</span>
-<span class="n">Time</span> <span class="n">to</span> <span class="n">solution</span><span class="p">:</span> <span class="mf">1.7961</span> <span class="n">sec</span><span class="o">.</span>
+<span class="n">Time</span> <span class="n">to</span> <span class="n">solution</span><span class="p">:</span> <span class="mf">1.7330</span> <span class="n">sec</span><span class="o">.</span>
 
 <span class="n">Variant</span> <span class="n">mass</span> <span class="k">with</span> <span class="n">ml</span><span class="o">=</span><span class="kc">False</span> <span class="ow">and</span> <span class="n">num_procs</span><span class="o">=</span><span class="mi">1</span> <span class="o">--</span> <span class="n">error</span> <span class="n">at</span> <span class="n">time</span> <span class="mf">1.0</span><span class="p">:</span> <span class="mf">1.1387594756569534e-08</span>
    <span class="n">Mean</span> <span class="n">number</span> <span class="n">of</span> <span class="n">iterations</span><span class="p">:</span> <span class="mf">6.00</span>
    <span class="n">Range</span> <span class="n">of</span> <span class="n">values</span> <span class="k">for</span> <span class="n">number</span> <span class="n">of</span> <span class="n">iterations</span><span class="p">:</span>  <span class="mi">0</span> 
    <span class="n">Position</span> <span class="n">of</span> <span class="nb">max</span><span class="o">/</span><span class="nb">min</span> <span class="n">number</span> <span class="n">of</span> <span class="n">iterations</span><span class="p">:</span>  <span class="mi">0</span> <span class="o">--</span>  <span class="mi">0</span>
    <span class="n">Std</span> <span class="ow">and</span> <span class="n">var</span> <span class="k">for</span> <span class="n">number</span> <span class="n">of</span> <span class="n">iterations</span><span class="p">:</span> <span class="mf">0.00</span> <span class="o">--</span> <span class="mf">0.00</span>
-<span class="n">Time</span> <span class="n">to</span> <span class="n">solution</span><span class="p">:</span> <span class="mf">1.4084</span> <span class="n">sec</span><span class="o">.</span>
+<span class="n">Time</span> <span class="n">to</span> <span class="n">solution</span><span class="p">:</span> <span class="mf">1.3967</span> <span class="n">sec</span><span class="o">.</span>
 
 <span class="n">Variant</span> <span class="n">mass_timebc</span> <span class="k">with</span> <span class="n">ml</span><span class="o">=</span><span class="kc">False</span> <span class="ow">and</span> <span class="n">num_procs</span><span class="o">=</span><span class="mi">1</span> <span class="o">--</span> <span class="n">error</span> <span class="n">at</span> <span class="n">time</span> <span class="mf">1.0</span><span class="p">:</span> <span class="mf">3.2473562155116167e-07</span>
    <span class="n">Mean</span> <span class="n">number</span> <span class="n">of</span> <span class="n">iterations</span><span class="p">:</span> <span class="mf">6.00</span>
    <span class="n">Range</span> <span class="n">of</span> <span class="n">values</span> <span class="k">for</span> <span class="n">number</span> <span class="n">of</span> <span class="n">iterations</span><span class="p">:</span>  <span class="mi">3</span> 
    <span class="n">Position</span> <span class="n">of</span> <span class="nb">max</span><span class="o">/</span><span class="nb">min</span> <span class="n">number</span> <span class="n">of</span> <span class="n">iterations</span><span class="p">:</span>  <span class="mi">3</span> <span class="o">--</span>  <span class="mi">0</span>
    <span class="n">Std</span> <span class="ow">and</span> <span class="n">var</span> <span class="k">for</span> <span class="n">number</span> <span class="n">of</span> <span class="n">iterations</span><span class="p">:</span> <span class="mf">1.10</span> <span class="o">--</span> <span class="mf">1.20</span>
-<span class="n">Time</span> <span class="n">to</span> <span class="n">solution</span><span class="p">:</span> <span class="mf">1.4094</span> <span class="n">sec</span><span class="o">.</span>
+<span class="n">Time</span> <span class="n">to</span> <span class="n">solution</span><span class="p">:</span> <span class="mf">1.3952</span> <span class="n">sec</span><span class="o">.</span>
 
 <span class="n">Variant</span> <span class="n">mass_inv</span> <span class="k">with</span> <span class="n">ml</span><span class="o">=</span><span class="kc">True</span> <span class="ow">and</span> <span class="n">num_procs</span><span class="o">=</span><span class="mi">1</span> <span class="o">--</span> <span class="n">error</span> <span class="n">at</span> <span class="n">time</span> <span class="mf">1.0</span><span class="p">:</span> <span class="mf">1.138768636885694e-08</span>
    <span class="n">Mean</span> <span class="n">number</span> <span class="n">of</span> <span class="n">iterations</span><span class="p">:</span> <span class="mf">6.00</span>
    <span class="n">Range</span> <span class="n">of</span> <span class="n">values</span> <span class="k">for</span> <span class="n">number</span> <span class="n">of</span> <span class="n">iterations</span><span class="p">:</span>  <span class="mi">0</span> 
    <span class="n">Position</span> <span class="n">of</span> <span class="nb">max</span><span class="o">/</span><span class="nb">min</span> <span class="n">number</span> <span class="n">of</span> <span class="n">iterations</span><span class="p">:</span>  <span class="mi">0</span> <span class="o">--</span>  <span class="mi">0</span>
    <span class="n">Std</span> <span class="ow">and</span> <span class="n">var</span> <span class="k">for</span> <span class="n">number</span> <span class="n">of</span> <span class="n">iterations</span><span class="p">:</span> <span class="mf">0.00</span> <span class="o">--</span> <span class="mf">0.00</span>
-<span class="n">Time</span> <span class="n">to</span> <span class="n">solution</span><span class="p">:</span> <span class="mf">2.5440</span> <span class="n">sec</span><span class="o">.</span>
+<span class="n">Time</span> <span class="n">to</span> <span class="n">solution</span><span class="p">:</span> <span class="mf">2.5014</span> <span class="n">sec</span><span class="o">.</span>
 
 <span class="n">Variant</span> <span class="n">mass</span> <span class="k">with</span> <span class="n">ml</span><span class="o">=</span><span class="kc">True</span> <span class="ow">and</span> <span class="n">num_procs</span><span class="o">=</span><span class="mi">1</span> <span class="o">--</span> <span class="n">error</span> <span class="n">at</span> <span class="n">time</span> <span class="mf">1.0</span><span class="p">:</span> <span class="mf">1.1387216566052821e-08</span>
    <span class="n">Mean</span> <span class="n">number</span> <span class="n">of</span> <span class="n">iterations</span><span class="p">:</span> <span class="mf">6.00</span>
    <span class="n">Range</span> <span class="n">of</span> <span class="n">values</span> <span class="k">for</span> <span class="n">number</span> <span class="n">of</span> <span class="n">iterations</span><span class="p">:</span>  <span class="mi">0</span> 
    <span class="n">Position</span> <span class="n">of</span> <span class="nb">max</span><span class="o">/</span><span class="nb">min</span> <span class="n">number</span> <span class="n">of</span> <span class="n">iterations</span><span class="p">:</span>  <span class="mi">0</span> <span class="o">--</span>  <span class="mi">0</span>
    <span class="n">Std</span> <span class="ow">and</span> <span class="n">var</span> <span class="k">for</span> <span class="n">number</span> <span class="n">of</span> <span class="n">iterations</span><span class="p">:</span> <span class="mf">0.00</span> <span class="o">--</span> <span class="mf">0.00</span>
-<span class="n">Time</span> <span class="n">to</span> <span class="n">solution</span><span class="p">:</span> <span class="mf">2.1717</span> <span class="n">sec</span><span class="o">.</span>
+<span class="n">Time</span> <span class="n">to</span> <span class="n">solution</span><span class="p">:</span> <span class="mf">2.1059</span> <span class="n">sec</span><span class="o">.</span>
 
 <span class="n">Variant</span> <span class="n">mass_timebc</span> <span class="k">with</span> <span class="n">ml</span><span class="o">=</span><span class="kc">True</span> <span class="ow">and</span> <span class="n">num_procs</span><span class="o">=</span><span class="mi">1</span> <span class="o">--</span> <span class="n">error</span> <span class="n">at</span> <span class="n">time</span> <span class="mf">1.0</span><span class="p">:</span> <span class="mf">3.2473561574763597e-07</span>
    <span class="n">Mean</span> <span class="n">number</span> <span class="n">of</span> <span class="n">iterations</span><span class="p">:</span> <span class="mf">6.00</span>
    <span class="n">Range</span> <span class="n">of</span> <span class="n">values</span> <span class="k">for</span> <span class="n">number</span> <span class="n">of</span> <span class="n">iterations</span><span class="p">:</span>  <span class="mi">3</span> 
    <span class="n">Position</span> <span class="n">of</span> <span class="nb">max</span><span class="o">/</span><span class="nb">min</span> <span class="n">number</span> <span class="n">of</span> <span class="n">iterations</span><span class="p">:</span>  <span class="mi">3</span> <span class="o">--</span>  <span class="mi">0</span>
    <span class="n">Std</span> <span class="ow">and</span> <span class="n">var</span> <span class="k">for</span> <span class="n">number</span> <span class="n">of</span> <span class="n">iterations</span><span class="p">:</span> <span class="mf">1.10</span> <span class="o">--</span> <span class="mf">1.20</span>
-<span class="n">Time</span> <span class="n">to</span> <span class="n">solution</span><span class="p">:</span> <span class="mf">2.1695</span> <span class="n">sec</span><span class="o">.</span>
+<span class="n">Time</span> <span class="n">to</span> <span class="n">solution</span><span class="p">:</span> <span class="mf">2.1560</span> <span class="n">sec</span><span class="o">.</span>
 
 <span class="n">Variant</span> <span class="n">mass_inv</span> <span class="k">with</span> <span class="n">ml</span><span class="o">=</span><span class="kc">True</span> <span class="ow">and</span> <span class="n">num_procs</span><span class="o">=</span><span class="mi">5</span> <span class="o">--</span> <span class="n">error</span> <span class="n">at</span> <span class="n">time</span> <span class="mf">1.0</span><span class="p">:</span> <span class="mf">1.1150087179536389e-08</span>
    <span class="n">Mean</span> <span class="n">number</span> <span class="n">of</span> <span class="n">iterations</span><span class="p">:</span> <span class="mf">11.60</span>
    <span class="n">Range</span> <span class="n">of</span> <span class="n">values</span> <span class="k">for</span> <span class="n">number</span> <span class="n">of</span> <span class="n">iterations</span><span class="p">:</span>  <span class="mi">9</span> 
    <span class="n">Position</span> <span class="n">of</span> <span class="nb">max</span><span class="o">/</span><span class="nb">min</span> <span class="n">number</span> <span class="n">of</span> <span class="n">iterations</span><span class="p">:</span>  <span class="mi">4</span> <span class="o">--</span>  <span class="mi">0</span>
    <span class="n">Std</span> <span class="ow">and</span> <span class="n">var</span> <span class="k">for</span> <span class="n">number</span> <span class="n">of</span> <span class="n">iterations</span><span class="p">:</span> <span class="mf">3.26</span> <span class="o">--</span> <span class="mf">10.64</span>
-<span class="n">Time</span> <span class="n">to</span> <span class="n">solution</span><span class="p">:</span> <span class="mf">4.1983</span> <span class="n">sec</span><span class="o">.</span>
+<span class="n">Time</span> <span class="n">to</span> <span class="n">solution</span><span class="p">:</span> <span class="mf">4.1015</span> <span class="n">sec</span><span class="o">.</span>
 
 </pre></div>
 </div>
@@ -479,7 +479,7 @@ <h2>Part B: mpi4py-fft for parallel Fourier transforms<a class="headerlink" href
    <span class="n">Position</span> <span class="n">of</span> <span class="nb">max</span><span class="o">/</span><span class="nb">min</span> <span class="n">number</span> <span class="n">of</span> <span class="n">iterations</span><span class="p">:</span>  <span class="mi">0</span> <span class="o">--</span> <span class="mi">19</span>
    <span class="n">Std</span> <span class="ow">and</span> <span class="n">var</span> <span class="k">for</span> <span class="n">number</span> <span class="n">of</span> <span class="n">iterations</span><span class="p">:</span> <span class="mf">2.10</span> <span class="o">--</span> <span class="mf">4.41</span>
 <span class="n">Error</span><span class="p">:</span> <span class="mf">1.1321e-05</span>
-<span class="n">Time</span> <span class="n">to</span> <span class="n">solution</span><span class="p">:</span> <span class="mf">1.4530</span> <span class="n">sec</span><span class="o">.</span>
+<span class="n">Time</span> <span class="n">to</span> <span class="n">solution</span><span class="p">:</span> <span class="mf">1.3354</span> <span class="n">sec</span><span class="o">.</span>
 
 <span class="n">Running</span> <span class="k">with</span> <span class="n">ml</span><span class="o">=</span><span class="kc">False</span> <span class="ow">and</span> <span class="n">num_procs</span><span class="o">=</span><span class="mf">1.</span><span class="o">..</span>
    <span class="n">Min</span><span class="o">/</span><span class="n">Mean</span><span class="o">/</span><span class="n">Max</span> <span class="n">number</span> <span class="n">of</span> <span class="n">iterations</span><span class="p">:</span> <span class="mf">8.00</span> <span class="o">/</span> <span class="mf">11.40</span> <span class="o">/</span> <span class="mf">15.00</span>
@@ -487,7 +487,7 @@ <h2>Part B: mpi4py-fft for parallel Fourier transforms<a class="headerlink" href
    <span class="n">Position</span> <span class="n">of</span> <span class="nb">max</span><span class="o">/</span><span class="nb">min</span> <span class="n">number</span> <span class="n">of</span> <span class="n">iterations</span><span class="p">:</span>  <span class="mi">0</span> <span class="o">--</span> <span class="mi">19</span>
    <span class="n">Std</span> <span class="ow">and</span> <span class="n">var</span> <span class="k">for</span> <span class="n">number</span> <span class="n">of</span> <span class="n">iterations</span><span class="p">:</span> <span class="mf">2.03</span> <span class="o">--</span> <span class="mf">4.14</span>
 <span class="n">Error</span><span class="p">:</span> <span class="mf">4.1749e-06</span>
-<span class="n">Time</span> <span class="n">to</span> <span class="n">solution</span><span class="p">:</span> <span class="mf">1.1877</span> <span class="n">sec</span><span class="o">.</span>
+<span class="n">Time</span> <span class="n">to</span> <span class="n">solution</span><span class="p">:</span> <span class="mf">1.0773</span> <span class="n">sec</span><span class="o">.</span>
 
 <span class="n">Running</span> <span class="k">with</span> <span class="n">ml</span><span class="o">=</span><span class="kc">True</span> <span class="ow">and</span> <span class="n">num_procs</span><span class="o">=</span><span class="mf">1.</span><span class="o">..</span>
    <span class="n">Min</span><span class="o">/</span><span class="n">Mean</span><span class="o">/</span><span class="n">Max</span> <span class="n">number</span> <span class="n">of</span> <span class="n">iterations</span><span class="p">:</span> <span class="mf">5.00</span> <span class="o">/</span> <span class="mf">6.60</span> <span class="o">/</span> <span class="mf">8.00</span>
@@ -495,7 +495,7 @@ <h2>Part B: mpi4py-fft for parallel Fourier transforms<a class="headerlink" href
    <span class="n">Position</span> <span class="n">of</span> <span class="nb">max</span><span class="o">/</span><span class="nb">min</span> <span class="n">number</span> <span class="n">of</span> <span class="n">iterations</span><span class="p">:</span>  <span class="mi">0</span> <span class="o">--</span> <span class="mi">16</span>
    <span class="n">Std</span> <span class="ow">and</span> <span class="n">var</span> <span class="k">for</span> <span class="n">number</span> <span class="n">of</span> <span class="n">iterations</span><span class="p">:</span> <span class="mf">1.07</span> <span class="o">--</span> <span class="mf">1.14</span>
 <span class="n">Error</span><span class="p">:</span> <span class="mf">1.1316e-05</span>
-<span class="n">Time</span> <span class="n">to</span> <span class="n">solution</span><span class="p">:</span> <span class="mf">1.4027</span> <span class="n">sec</span><span class="o">.</span>
+<span class="n">Time</span> <span class="n">to</span> <span class="n">solution</span><span class="p">:</span> <span class="mf">1.3054</span> <span class="n">sec</span><span class="o">.</span>
 
 <span class="n">Running</span> <span class="k">with</span> <span class="n">ml</span><span class="o">=</span><span class="kc">True</span> <span class="ow">and</span> <span class="n">num_procs</span><span class="o">=</span><span class="mf">1.</span><span class="o">..</span>
    <span class="n">Min</span><span class="o">/</span><span class="n">Mean</span><span class="o">/</span><span class="n">Max</span> <span class="n">number</span> <span class="n">of</span> <span class="n">iterations</span><span class="p">:</span> <span class="mf">4.00</span> <span class="o">/</span> <span class="mf">5.95</span> <span class="o">/</span> <span class="mf">8.00</span>
@@ -503,7 +503,7 @@ <h2>Part B: mpi4py-fft for parallel Fourier transforms<a class="headerlink" href
    <span class="n">Position</span> <span class="n">of</span> <span class="nb">max</span><span class="o">/</span><span class="nb">min</span> <span class="n">number</span> <span class="n">of</span> <span class="n">iterations</span><span class="p">:</span>  <span class="mi">0</span> <span class="o">--</span> <span class="mi">19</span>
    <span class="n">Std</span> <span class="ow">and</span> <span class="n">var</span> <span class="k">for</span> <span class="n">number</span> <span class="n">of</span> <span class="n">iterations</span><span class="p">:</span> <span class="mf">1.02</span> <span class="o">--</span> <span class="mf">1.05</span>
 <span class="n">Error</span><span class="p">:</span> <span class="mf">4.1744e-06</span>
-<span class="n">Time</span> <span class="n">to</span> <span class="n">solution</span><span class="p">:</span> <span class="mf">1.2926</span> <span class="n">sec</span><span class="o">.</span>
+<span class="n">Time</span> <span class="n">to</span> <span class="n">solution</span><span class="p">:</span> <span class="mf">1.2137</span> <span class="n">sec</span><span class="o">.</span>
 
 <span class="n">Running</span> <span class="k">with</span> <span class="n">ml</span><span class="o">=</span><span class="kc">True</span> <span class="ow">and</span> <span class="n">num_procs</span><span class="o">=</span><span class="mf">10.</span><span class="o">..</span>
    <span class="n">Min</span><span class="o">/</span><span class="n">Mean</span><span class="o">/</span><span class="n">Max</span> <span class="n">number</span> <span class="n">of</span> <span class="n">iterations</span><span class="p">:</span> <span class="mf">7.00</span> <span class="o">/</span> <span class="mf">12.45</span> <span class="o">/</span> <span class="mf">18.00</span>
@@ -511,7 +511,7 @@ <h2>Part B: mpi4py-fft for parallel Fourier transforms<a class="headerlink" href
    <span class="n">Position</span> <span class="n">of</span> <span class="nb">max</span><span class="o">/</span><span class="nb">min</span> <span class="n">number</span> <span class="n">of</span> <span class="n">iterations</span><span class="p">:</span>  <span class="mi">9</span> <span class="o">--</span> <span class="mi">10</span>
    <span class="n">Std</span> <span class="ow">and</span> <span class="n">var</span> <span class="k">for</span> <span class="n">number</span> <span class="n">of</span> <span class="n">iterations</span><span class="p">:</span> <span class="mf">3.11</span> <span class="o">--</span> <span class="mf">9.65</span>
 <span class="n">Error</span><span class="p">:</span> <span class="mf">1.1306e-05</span>
-<span class="n">Time</span> <span class="n">to</span> <span class="n">solution</span><span class="p">:</span> <span class="mf">2.9081</span> <span class="n">sec</span><span class="o">.</span>
+<span class="n">Time</span> <span class="n">to</span> <span class="n">solution</span><span class="p">:</span> <span class="mf">2.7650</span> <span class="n">sec</span><span class="o">.</span>
 
 <span class="n">Running</span> <span class="k">with</span> <span class="n">ml</span><span class="o">=</span><span class="kc">True</span> <span class="ow">and</span> <span class="n">num_procs</span><span class="o">=</span><span class="mf">10.</span><span class="o">..</span>
    <span class="n">Min</span><span class="o">/</span><span class="n">Mean</span><span class="o">/</span><span class="n">Max</span> <span class="n">number</span> <span class="n">of</span> <span class="n">iterations</span><span class="p">:</span> <span class="mf">6.00</span> <span class="o">/</span> <span class="mf">11.50</span> <span class="o">/</span> <span class="mf">17.00</span>
@@ -519,7 +519,7 @@ <h2>Part B: mpi4py-fft for parallel Fourier transforms<a class="headerlink" href
    <span class="n">Position</span> <span class="n">of</span> <span class="nb">max</span><span class="o">/</span><span class="nb">min</span> <span class="n">number</span> <span class="n">of</span> <span class="n">iterations</span><span class="p">:</span>  <span class="mi">9</span> <span class="o">--</span> <span class="mi">10</span>
    <span class="n">Std</span> <span class="ow">and</span> <span class="n">var</span> <span class="k">for</span> <span class="n">number</span> <span class="n">of</span> <span class="n">iterations</span><span class="p">:</span> <span class="mf">3.04</span> <span class="o">--</span> <span class="mf">9.25</span>
 <span class="n">Error</span><span class="p">:</span> <span class="mf">4.1688e-06</span>
-<span class="n">Time</span> <span class="n">to</span> <span class="n">solution</span><span class="p">:</span> <span class="mf">2.7314</span> <span class="n">sec</span><span class="o">.</span>
+<span class="n">Time</span> <span class="n">to</span> <span class="n">solution</span><span class="p">:</span> <span class="mf">2.6021</span> <span class="n">sec</span><span class="o">.</span>
 
 </pre></div>
 </div>
@@ -714,7 +714,7 @@ <h2>Part C: Time-parallel pySDC with space-parallel PETSc<a class="headerlink" h
    <span class="n">Range</span> <span class="n">of</span> <span class="n">values</span> <span class="k">for</span> <span class="n">number</span> <span class="n">of</span> <span class="n">iterations</span><span class="p">:</span>  <span class="mi">0</span> 
    <span class="n">Position</span> <span class="n">of</span> <span class="nb">max</span><span class="o">/</span><span class="nb">min</span> <span class="n">number</span> <span class="n">of</span> <span class="n">iterations</span><span class="p">:</span>  <span class="mi">0</span> <span class="o">--</span>  <span class="mi">0</span>
    <span class="n">Std</span> <span class="ow">and</span> <span class="n">var</span> <span class="k">for</span> <span class="n">number</span> <span class="n">of</span> <span class="n">iterations</span><span class="p">:</span> <span class="mf">0.00</span> <span class="o">--</span> <span class="mf">0.00</span>
-<span class="n">Time</span> <span class="n">to</span> <span class="n">solution</span><span class="p">:</span> <span class="mf">2.8030</span> <span class="n">sec</span><span class="o">.</span>
+<span class="n">Time</span> <span class="n">to</span> <span class="n">solution</span><span class="p">:</span> <span class="mf">2.0782</span> <span class="n">sec</span><span class="o">.</span>
 <span class="n">Error</span> <span class="n">vs</span><span class="o">.</span> <span class="n">PDE</span> <span class="n">solution</span><span class="p">:</span> <span class="mf">1.9479e-04</span>
 </pre></div>
 </div>
@@ -726,26 +726,26 @@ <h2>Part C: Time-parallel pySDC with space-parallel PETSc<a class="headerlink" h
    <span class="n">Range</span> <span class="n">of</span> <span class="n">values</span> <span class="k">for</span> <span class="n">number</span> <span class="n">of</span> <span class="n">iterations</span><span class="p">:</span>  <span class="mi">0</span> 
    <span class="n">Position</span> <span class="n">of</span> <span class="nb">max</span><span class="o">/</span><span class="nb">min</span> <span class="n">number</span> <span class="n">of</span> <span class="n">iterations</span><span class="p">:</span>  <span class="mi">0</span> <span class="o">--</span>  <span class="mi">0</span>
    <span class="n">Std</span> <span class="ow">and</span> <span class="n">var</span> <span class="k">for</span> <span class="n">number</span> <span class="n">of</span> <span class="n">iterations</span><span class="p">:</span> <span class="mf">0.00</span> <span class="o">--</span> <span class="mf">0.00</span>
-<span class="n">Time</span> <span class="n">to</span> <span class="n">solution</span><span class="p">:</span> <span class="mf">0.9242</span> <span class="n">sec</span><span class="o">.</span>
+<span class="n">Time</span> <span class="n">to</span> <span class="n">solution</span><span class="p">:</span> <span class="mf">0.9128</span> <span class="n">sec</span><span class="o">.</span>
 <span class="n">Error</span> <span class="n">vs</span><span class="o">.</span> <span class="n">PDE</span> <span class="n">solution</span><span class="p">:</span> <span class="mf">1.9479e-04</span>
 </pre></div>
 </div>
 <p>2 processor in time, 2 processors in space</p>
-<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">This</span> <span class="ow">is</span> <span class="n">time</span><span class="o">-</span><span class="n">rank</span> <span class="mf">0.</span><span class="o">..</span>
-<span class="n">Number</span> <span class="n">of</span> <span class="n">iterations</span> <span class="k">for</span> <span class="n">time</span> <span class="mf">0.00</span><span class="p">:</span> <span class="mi">12</span>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">This</span> <span class="ow">is</span> <span class="n">time</span><span class="o">-</span><span class="n">rank</span> <span class="mf">1.</span><span class="o">..</span>
+<span class="n">Number</span> <span class="n">of</span> <span class="n">iterations</span> <span class="k">for</span> <span class="n">time</span> <span class="mf">0.12</span><span class="p">:</span> <span class="mi">12</span>
    <span class="n">Mean</span> <span class="n">number</span> <span class="n">of</span> <span class="n">iterations</span><span class="p">:</span> <span class="mf">12.00</span>
    <span class="n">Range</span> <span class="n">of</span> <span class="n">values</span> <span class="k">for</span> <span class="n">number</span> <span class="n">of</span> <span class="n">iterations</span><span class="p">:</span>  <span class="mi">0</span> 
    <span class="n">Position</span> <span class="n">of</span> <span class="nb">max</span><span class="o">/</span><span class="nb">min</span> <span class="n">number</span> <span class="n">of</span> <span class="n">iterations</span><span class="p">:</span>  <span class="mi">0</span> <span class="o">--</span>  <span class="mi">0</span>
    <span class="n">Std</span> <span class="ow">and</span> <span class="n">var</span> <span class="k">for</span> <span class="n">number</span> <span class="n">of</span> <span class="n">iterations</span><span class="p">:</span> <span class="mf">0.00</span> <span class="o">--</span> <span class="mf">0.00</span>
-<span class="n">Time</span> <span class="n">to</span> <span class="n">solution</span><span class="p">:</span> <span class="mf">0.8208</span> <span class="n">sec</span><span class="o">.</span>
+<span class="n">Time</span> <span class="n">to</span> <span class="n">solution</span><span class="p">:</span> <span class="mf">0.7904</span> <span class="n">sec</span><span class="o">.</span>
 <span class="n">Error</span> <span class="n">vs</span><span class="o">.</span> <span class="n">PDE</span> <span class="n">solution</span><span class="p">:</span> <span class="mf">1.9479e-04</span>
-<span class="n">This</span> <span class="ow">is</span> <span class="n">time</span><span class="o">-</span><span class="n">rank</span> <span class="mf">1.</span><span class="o">..</span>
-<span class="n">Number</span> <span class="n">of</span> <span class="n">iterations</span> <span class="k">for</span> <span class="n">time</span> <span class="mf">0.12</span><span class="p">:</span> <span class="mi">12</span>
+<span class="n">This</span> <span class="ow">is</span> <span class="n">time</span><span class="o">-</span><span class="n">rank</span> <span class="mf">0.</span><span class="o">..</span>
+<span class="n">Number</span> <span class="n">of</span> <span class="n">iterations</span> <span class="k">for</span> <span class="n">time</span> <span class="mf">0.00</span><span class="p">:</span> <span class="mi">12</span>
    <span class="n">Mean</span> <span class="n">number</span> <span class="n">of</span> <span class="n">iterations</span><span class="p">:</span> <span class="mf">12.00</span>
    <span class="n">Range</span> <span class="n">of</span> <span class="n">values</span> <span class="k">for</span> <span class="n">number</span> <span class="n">of</span> <span class="n">iterations</span><span class="p">:</span>  <span class="mi">0</span> 
    <span class="n">Position</span> <span class="n">of</span> <span class="nb">max</span><span class="o">/</span><span class="nb">min</span> <span class="n">number</span> <span class="n">of</span> <span class="n">iterations</span><span class="p">:</span>  <span class="mi">0</span> <span class="o">--</span>  <span class="mi">0</span>
    <span class="n">Std</span> <span class="ow">and</span> <span class="n">var</span> <span class="k">for</span> <span class="n">number</span> <span class="n">of</span> <span class="n">iterations</span><span class="p">:</span> <span class="mf">0.00</span> <span class="o">--</span> <span class="mf">0.00</span>
-<span class="n">Time</span> <span class="n">to</span> <span class="n">solution</span><span class="p">:</span> <span class="mf">0.8205</span> <span class="n">sec</span><span class="o">.</span>
+<span class="n">Time</span> <span class="n">to</span> <span class="n">solution</span><span class="p">:</span> <span class="mf">0.7905</span> <span class="n">sec</span><span class="o">.</span>
 <span class="n">Error</span> <span class="n">vs</span><span class="o">.</span> <span class="n">PDE</span> <span class="n">solution</span><span class="p">:</span> <span class="mf">1.9479e-04</span>
 </pre></div>
 </div>
@@ -850,27 +850,27 @@ <h2>Part D: pySDC and PyTorch<a class="headerlink" href="#part-d-pysdc-and-pytor
 </pre></div>
 </div>
 <p>Results:</p>
-<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">Training</span> <span class="k">for</span> <span class="n">dt</span><span class="o">=</span><span class="mf">1.55e-03</span><span class="p">:</span> <span class="n">Epoch</span> <span class="p">[</span>  <span class="mi">50</span><span class="o">/</span> <span class="mi">250</span><span class="p">],</span> <span class="n">Loss</span><span class="p">:</span> <span class="mf">1.7391e-02</span>
-<span class="n">Training</span> <span class="k">for</span> <span class="n">dt</span><span class="o">=</span><span class="mf">1.55e-03</span><span class="p">:</span> <span class="n">Epoch</span> <span class="p">[</span> <span class="mi">100</span><span class="o">/</span> <span class="mi">250</span><span class="p">],</span> <span class="n">Loss</span><span class="p">:</span> <span class="mf">8.3421e-05</span>
-<span class="n">Training</span> <span class="k">for</span> <span class="n">dt</span><span class="o">=</span><span class="mf">1.55e-03</span><span class="p">:</span> <span class="n">Epoch</span> <span class="p">[</span> <span class="mi">150</span><span class="o">/</span> <span class="mi">250</span><span class="p">],</span> <span class="n">Loss</span><span class="p">:</span> <span class="mf">4.8431e-07</span>
-<span class="n">Training</span> <span class="k">for</span> <span class="n">dt</span><span class="o">=</span><span class="mf">1.55e-03</span><span class="p">:</span> <span class="n">Epoch</span> <span class="p">[</span> <span class="mi">200</span><span class="o">/</span> <span class="mi">250</span><span class="p">],</span> <span class="n">Loss</span><span class="p">:</span> <span class="mf">2.0377e-09</span>
-<span class="n">Training</span> <span class="k">for</span> <span class="n">dt</span><span class="o">=</span><span class="mf">1.55e-03</span><span class="p">:</span> <span class="n">Epoch</span> <span class="p">[</span> <span class="mi">250</span><span class="o">/</span> <span class="mi">250</span><span class="p">],</span> <span class="n">Loss</span><span class="p">:</span> <span class="mf">1.1031e-11</span>
-<span class="n">Error</span> <span class="n">of</span> <span class="n">prediction</span> <span class="n">at</span> <span class="mf">1.55e-03</span> <span class="n">during</span> <span class="n">training</span><span class="p">:</span> <span class="mf">1.82e-05</span>
-<span class="n">Training</span> <span class="k">for</span> <span class="n">dt</span><span class="o">=</span><span class="mf">6.45e-03</span><span class="p">:</span> <span class="n">Epoch</span> <span class="p">[</span>  <span class="mi">50</span><span class="o">/</span> <span class="mi">250</span><span class="p">],</span> <span class="n">Loss</span><span class="p">:</span> <span class="mf">6.7525e-05</span>
-<span class="n">Training</span> <span class="k">for</span> <span class="n">dt</span><span class="o">=</span><span class="mf">6.45e-03</span><span class="p">:</span> <span class="n">Epoch</span> <span class="p">[</span> <span class="mi">100</span><span class="o">/</span> <span class="mi">250</span><span class="p">],</span> <span class="n">Loss</span><span class="p">:</span> <span class="mf">1.0774e-06</span>
-<span class="n">Training</span> <span class="k">for</span> <span class="n">dt</span><span class="o">=</span><span class="mf">6.45e-03</span><span class="p">:</span> <span class="n">Epoch</span> <span class="p">[</span> <span class="mi">150</span><span class="o">/</span> <span class="mi">250</span><span class="p">],</span> <span class="n">Loss</span><span class="p">:</span> <span class="mf">4.2838e-09</span>
-<span class="n">Training</span> <span class="k">for</span> <span class="n">dt</span><span class="o">=</span><span class="mf">6.45e-03</span><span class="p">:</span> <span class="n">Epoch</span> <span class="p">[</span> <span class="mi">200</span><span class="o">/</span> <span class="mi">250</span><span class="p">],</span> <span class="n">Loss</span><span class="p">:</span> <span class="mf">2.6739e-11</span>
-<span class="n">Training</span> <span class="k">for</span> <span class="n">dt</span><span class="o">=</span><span class="mf">6.45e-03</span><span class="p">:</span> <span class="n">Epoch</span> <span class="p">[</span> <span class="mi">250</span><span class="o">/</span> <span class="mi">250</span><span class="p">],</span> <span class="n">Loss</span><span class="p">:</span> <span class="mf">1.1213e-13</span>
-<span class="n">Error</span> <span class="n">of</span> <span class="n">prediction</span> <span class="n">at</span> <span class="mf">6.45e-03</span> <span class="n">during</span> <span class="n">training</span><span class="p">:</span> <span class="mf">5.23e-07</span>
-<span class="n">Training</span> <span class="k">for</span> <span class="n">dt</span><span class="o">=</span><span class="mf">1.00e-02</span><span class="p">:</span> <span class="n">Epoch</span> <span class="p">[</span>  <span class="mi">50</span><span class="o">/</span> <span class="mi">250</span><span class="p">],</span> <span class="n">Loss</span><span class="p">:</span> <span class="mf">3.1718e-05</span>
-<span class="n">Training</span> <span class="k">for</span> <span class="n">dt</span><span class="o">=</span><span class="mf">1.00e-02</span><span class="p">:</span> <span class="n">Epoch</span> <span class="p">[</span> <span class="mi">100</span><span class="o">/</span> <span class="mi">250</span><span class="p">],</span> <span class="n">Loss</span><span class="p">:</span> <span class="mf">8.5910e-08</span>
-<span class="n">Training</span> <span class="k">for</span> <span class="n">dt</span><span class="o">=</span><span class="mf">1.00e-02</span><span class="p">:</span> <span class="n">Epoch</span> <span class="p">[</span> <span class="mi">150</span><span class="o">/</span> <span class="mi">250</span><span class="p">],</span> <span class="n">Loss</span><span class="p">:</span> <span class="mf">4.2362e-10</span>
-<span class="n">Training</span> <span class="k">for</span> <span class="n">dt</span><span class="o">=</span><span class="mf">1.00e-02</span><span class="p">:</span> <span class="n">Epoch</span> <span class="p">[</span> <span class="mi">200</span><span class="o">/</span> <span class="mi">250</span><span class="p">],</span> <span class="n">Loss</span><span class="p">:</span> <span class="mf">1.6118e-12</span>
-<span class="n">Training</span> <span class="k">for</span> <span class="n">dt</span><span class="o">=</span><span class="mf">1.00e-02</span><span class="p">:</span> <span class="n">Epoch</span> <span class="p">[</span> <span class="mi">250</span><span class="o">/</span> <span class="mi">250</span><span class="p">],</span> <span class="n">Loss</span><span class="p">:</span> <span class="mf">2.6642e-14</span>
-<span class="n">Error</span> <span class="n">of</span> <span class="n">prediction</span> <span class="n">at</span> <span class="mf">1.00e-02</span> <span class="n">during</span> <span class="n">training</span><span class="p">:</span> <span class="mf">4.06e-07</span>
-<span class="n">Error</span> <span class="n">of</span> <span class="n">prediction</span> <span class="n">at</span> <span class="mf">1.55e-03</span> <span class="n">after</span> <span class="n">training</span><span class="p">:</span> <span class="mf">4.25e-01</span>
-<span class="n">Error</span> <span class="n">of</span> <span class="n">prediction</span> <span class="n">at</span> <span class="mf">6.45e-03</span> <span class="n">after</span> <span class="n">training</span><span class="p">:</span> <span class="mf">1.12e-01</span>
-<span class="n">Error</span> <span class="n">of</span> <span class="n">prediction</span> <span class="n">at</span> <span class="mf">1.00e-02</span> <span class="n">after</span> <span class="n">training</span><span class="p">:</span> <span class="mf">4.06e-07</span>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">Training</span> <span class="k">for</span> <span class="n">dt</span><span class="o">=</span><span class="mf">1.55e-03</span><span class="p">:</span> <span class="n">Epoch</span> <span class="p">[</span>  <span class="mi">50</span><span class="o">/</span> <span class="mi">250</span><span class="p">],</span> <span class="n">Loss</span><span class="p">:</span> <span class="mf">8.1769e-03</span>
+<span class="n">Training</span> <span class="k">for</span> <span class="n">dt</span><span class="o">=</span><span class="mf">1.55e-03</span><span class="p">:</span> <span class="n">Epoch</span> <span class="p">[</span> <span class="mi">100</span><span class="o">/</span> <span class="mi">250</span><span class="p">],</span> <span class="n">Loss</span><span class="p">:</span> <span class="mf">3.0408e-05</span>
+<span class="n">Training</span> <span class="k">for</span> <span class="n">dt</span><span class="o">=</span><span class="mf">1.55e-03</span><span class="p">:</span> <span class="n">Epoch</span> <span class="p">[</span> <span class="mi">150</span><span class="o">/</span> <span class="mi">250</span><span class="p">],</span> <span class="n">Loss</span><span class="p">:</span> <span class="mf">1.7127e-07</span>
+<span class="n">Training</span> <span class="k">for</span> <span class="n">dt</span><span class="o">=</span><span class="mf">1.55e-03</span><span class="p">:</span> <span class="n">Epoch</span> <span class="p">[</span> <span class="mi">200</span><span class="o">/</span> <span class="mi">250</span><span class="p">],</span> <span class="n">Loss</span><span class="p">:</span> <span class="mf">9.1137e-10</span>
+<span class="n">Training</span> <span class="k">for</span> <span class="n">dt</span><span class="o">=</span><span class="mf">1.55e-03</span><span class="p">:</span> <span class="n">Epoch</span> <span class="p">[</span> <span class="mi">250</span><span class="o">/</span> <span class="mi">250</span><span class="p">],</span> <span class="n">Loss</span><span class="p">:</span> <span class="mf">4.6576e-12</span>
+<span class="n">Error</span> <span class="n">of</span> <span class="n">prediction</span> <span class="n">at</span> <span class="mf">1.55e-03</span> <span class="n">during</span> <span class="n">training</span><span class="p">:</span> <span class="mf">1.30e-05</span>
+<span class="n">Training</span> <span class="k">for</span> <span class="n">dt</span><span class="o">=</span><span class="mf">6.45e-03</span><span class="p">:</span> <span class="n">Epoch</span> <span class="p">[</span>  <span class="mi">50</span><span class="o">/</span> <span class="mi">250</span><span class="p">],</span> <span class="n">Loss</span><span class="p">:</span> <span class="mf">6.7436e-05</span>
+<span class="n">Training</span> <span class="k">for</span> <span class="n">dt</span><span class="o">=</span><span class="mf">6.45e-03</span><span class="p">:</span> <span class="n">Epoch</span> <span class="p">[</span> <span class="mi">100</span><span class="o">/</span> <span class="mi">250</span><span class="p">],</span> <span class="n">Loss</span><span class="p">:</span> <span class="mf">6.8079e-07</span>
+<span class="n">Training</span> <span class="k">for</span> <span class="n">dt</span><span class="o">=</span><span class="mf">6.45e-03</span><span class="p">:</span> <span class="n">Epoch</span> <span class="p">[</span> <span class="mi">150</span><span class="o">/</span> <span class="mi">250</span><span class="p">],</span> <span class="n">Loss</span><span class="p">:</span> <span class="mf">1.7740e-09</span>
+<span class="n">Training</span> <span class="k">for</span> <span class="n">dt</span><span class="o">=</span><span class="mf">6.45e-03</span><span class="p">:</span> <span class="n">Epoch</span> <span class="p">[</span> <span class="mi">200</span><span class="o">/</span> <span class="mi">250</span><span class="p">],</span> <span class="n">Loss</span><span class="p">:</span> <span class="mf">1.4685e-11</span>
+<span class="n">Training</span> <span class="k">for</span> <span class="n">dt</span><span class="o">=</span><span class="mf">6.45e-03</span><span class="p">:</span> <span class="n">Epoch</span> <span class="p">[</span> <span class="mi">250</span><span class="o">/</span> <span class="mi">250</span><span class="p">],</span> <span class="n">Loss</span><span class="p">:</span> <span class="mf">9.2464e-14</span>
+<span class="n">Error</span> <span class="n">of</span> <span class="n">prediction</span> <span class="n">at</span> <span class="mf">6.45e-03</span> <span class="n">during</span> <span class="n">training</span><span class="p">:</span> <span class="mf">8.52e-07</span>
+<span class="n">Training</span> <span class="k">for</span> <span class="n">dt</span><span class="o">=</span><span class="mf">1.00e-02</span><span class="p">:</span> <span class="n">Epoch</span> <span class="p">[</span>  <span class="mi">50</span><span class="o">/</span> <span class="mi">250</span><span class="p">],</span> <span class="n">Loss</span><span class="p">:</span> <span class="mf">2.0351e-05</span>
+<span class="n">Training</span> <span class="k">for</span> <span class="n">dt</span><span class="o">=</span><span class="mf">1.00e-02</span><span class="p">:</span> <span class="n">Epoch</span> <span class="p">[</span> <span class="mi">100</span><span class="o">/</span> <span class="mi">250</span><span class="p">],</span> <span class="n">Loss</span><span class="p">:</span> <span class="mf">3.8560e-08</span>
+<span class="n">Training</span> <span class="k">for</span> <span class="n">dt</span><span class="o">=</span><span class="mf">1.00e-02</span><span class="p">:</span> <span class="n">Epoch</span> <span class="p">[</span> <span class="mi">150</span><span class="o">/</span> <span class="mi">250</span><span class="p">],</span> <span class="n">Loss</span><span class="p">:</span> <span class="mf">5.8901e-10</span>
+<span class="n">Training</span> <span class="k">for</span> <span class="n">dt</span><span class="o">=</span><span class="mf">1.00e-02</span><span class="p">:</span> <span class="n">Epoch</span> <span class="p">[</span> <span class="mi">200</span><span class="o">/</span> <span class="mi">250</span><span class="p">],</span> <span class="n">Loss</span><span class="p">:</span> <span class="mf">2.8962e-12</span>
+<span class="n">Training</span> <span class="k">for</span> <span class="n">dt</span><span class="o">=</span><span class="mf">1.00e-02</span><span class="p">:</span> <span class="n">Epoch</span> <span class="p">[</span> <span class="mi">250</span><span class="o">/</span> <span class="mi">250</span><span class="p">],</span> <span class="n">Loss</span><span class="p">:</span> <span class="mf">1.5985e-14</span>
+<span class="n">Error</span> <span class="n">of</span> <span class="n">prediction</span> <span class="n">at</span> <span class="mf">1.00e-02</span> <span class="n">during</span> <span class="n">training</span><span class="p">:</span> <span class="mf">3.44e-07</span>
+<span class="n">Error</span> <span class="n">of</span> <span class="n">prediction</span> <span class="n">at</span> <span class="mf">1.55e-03</span> <span class="n">after</span> <span class="n">training</span><span class="p">:</span> <span class="mf">4.28e-01</span>
+<span class="n">Error</span> <span class="n">of</span> <span class="n">prediction</span> <span class="n">at</span> <span class="mf">6.45e-03</span> <span class="n">after</span> <span class="n">training</span><span class="p">:</span> <span class="mf">1.13e-01</span>
+<span class="n">Error</span> <span class="n">of</span> <span class="n">prediction</span> <span class="n">at</span> <span class="mf">1.00e-02</span> <span class="n">after</span> <span class="n">training</span><span class="p">:</span> <span class="mf">3.44e-07</span>
 </pre></div>
 </div>
 </section>
diff --git a/tutorial/step_8.html b/tutorial/step_8.html
index c4e9ece79f..bd30204419 100644
--- a/tutorial/step_8.html
+++ b/tutorial/step_8.html
@@ -136,7 +136,7 @@ <h2>Part A: Visualizing Residuals<a class="headerlink" href="#part-a-visualizing
 <span class="n">Number</span> <span class="n">of</span> <span class="n">iterations</span> <span class="k">for</span> <span class="n">time</span> <span class="mf">0.88</span><span class="p">:</span> <span class="mi">7</span>
 </pre></div>
 </div>
-<a class="reference internal image-reference" href="../_images/step_8_residuals.png"><img alt="../_images/step_8_residuals.png" src="../_images/step_8_residuals.png" style="width: 614.0px; height: 433.5px;" />
+<a class="reference internal image-reference" href="../_images/step_8_residuals.png"><img alt="../_images/step_8_residuals.png" src="../_images/step_8_residuals.png" style="width: 754.0px; height: 505.5px;" />
 </a>
 </section>
 <section id="part-b-multi-step-sdc">
@@ -342,9 +342,9 @@ <h2>Part B: Multi-step SDC<a class="headerlink" href="#part-b-multi-step-sdc" ti
 <span class="n">Number</span> <span class="n">of</span> <span class="n">iterations</span> <span class="k">for</span> <span class="n">time</span> <span class="mf">0.88</span> <span class="p">(</span><span class="n">PFASST</span><span class="o">/</span><span class="n">parMSSDC</span><span class="o">/</span><span class="n">serMSSDC</span><span class="p">):</span>  <span class="mi">8</span> <span class="o">/</span> <span class="mi">18</span> <span class="o">/</span> <span class="mi">11</span>
 </pre></div>
 </div>
-<a class="reference internal image-reference" href="../_images/step_8_residuals_mssdc_jac.png"><img alt="../_images/step_8_residuals_mssdc_jac.png" src="../_images/step_8_residuals_mssdc_jac.png" style="width: 614.0px; height: 433.5px;" />
+<a class="reference internal image-reference" href="../_images/step_8_residuals_mssdc_jac.png"><img alt="../_images/step_8_residuals_mssdc_jac.png" src="../_images/step_8_residuals_mssdc_jac.png" style="width: 754.0px; height: 505.5px;" />
 </a>
-<a class="reference internal image-reference" href="../_images/step_8_residuals_mssdc_gs.png"><img alt="../_images/step_8_residuals_mssdc_gs.png" src="../_images/step_8_residuals_mssdc_gs.png" style="width: 614.0px; height: 433.5px;" />
+<a class="reference internal image-reference" href="../_images/step_8_residuals_mssdc_gs.png"><img alt="../_images/step_8_residuals_mssdc_gs.png" src="../_images/step_8_residuals_mssdc_gs.png" style="width: 754.0px; height: 505.5px;" />
 </a>
 </section>
 <section id="part-c-iteration-estimator">
@@ -678,7 +678,7 @@ <h2>Part C: Iteration estimator<a class="headerlink" href="#part-c-iteration-est
    Errors after step   0.8750 with 6 iterations: 3.6530e-08 / 2.0645e-13
    Errors after step   1.0000 with 6 iterations: 3.6478e-08 / 1.2759e-13
 
-...done, took 0.3308976680000342 seconds!
+...done, took 0.33569771999998466 seconds!
 -----------------------------------------------------------------------------
 Running diffusion in 1 dimensions with time-step size 0.125...
 
@@ -692,7 +692,7 @@ <h2>Part C: Iteration estimator<a class="headerlink" href="#part-c-iteration-est
    Errors after step   0.8750 with 3 iterations: 3.5501e-08 / 3.9957e-13
    Errors after step   1.0000 with 3 iterations: 3.6813e-08 / 1.6975e-13
 
-...done, took 0.7408457070000054 seconds!
+...done, took 0.7528066140000078 seconds!
 -----------------------------------------------------------------------------
 Running advection in 1 dimensions with time-step size 0.125...
 
@@ -706,7 +706,7 @@ <h2>Part C: Iteration estimator<a class="headerlink" href="#part-c-iteration-est
    Errors after step   0.8750 with 4 iterations: 2.5517e-08 / 3.3307e-16
    Errors after step   1.0000 with 4 iterations: 2.9147e-08 / 4.4409e-16
 
-...done, took 0.21992689800003973 seconds!
+...done, took 0.22154832900000088 seconds!
 -----------------------------------------------------------------------------
 Running advection in 1 dimensions with time-step size 0.125...
 
@@ -720,7 +720,7 @@ <h2>Part C: Iteration estimator<a class="headerlink" href="#part-c-iteration-est
    Errors after step   0.8750 with 2 iterations: 2.5987e-08 / 3.1841e-13
    Errors after step   1.0000 with 2 iterations: 2.9666e-08 / 3.1819e-13
 
-...done, took 0.6080810290000045 seconds!
+...done, took 0.6121239379999963 seconds!
 -----------------------------------------------------------------------------
 Running auzinger in 1 dimensions with time-step size 0.125...
 
@@ -734,7 +734,7 @@ <h2>Part C: Iteration estimator<a class="headerlink" href="#part-c-iteration-est
    Errors after step   0.8750 with 6 iterations: 2.8266e-08 / 0.0000e+00
    Errors after step   1.0000 with 6 iterations: 2.8338e-08 / 0.0000e+00
 
-...done, took 0.13996988500002772 seconds!
+...done, took 0.14012889900001824 seconds!
 -----------------------------------------------------------------------------
 Running auzinger in 1 dimensions with time-step size 0.125...
 
@@ -748,7 +748,7 @@ <h2>Part C: Iteration estimator<a class="headerlink" href="#part-c-iteration-est
    Errors after step   0.8750 with 4 iterations: 3.1486e-08 / 0.0000e+00
    Errors after step   1.0000 with 4 iterations: 3.3802e-08 / 0.0000e+00
 
-...done, took 0.1660917489999747 seconds!
+...done, took 0.16614925400000402 seconds!
 -----------------------------------------------------------------------------
 </pre></div>
 </div>