Added FineRKN5 Algorithm

[HenryLangner]

Added the 5th order Nyström-Method presented by J. M. Fine in the article "Low Order Practical Runge-Kutta-Nyström Methods" (Also linked here: #677). The embedded method is not yet implemented however I am working on it.

The method passes the convergence tests. Using the harmonic oscillator, the error is plotted below for the solution u at t=3.0 for different timesteps.

JuliaFormatter has been used.

CC @ranocha

[ranocha]

Great, thanks a lot! This looks already quite good to me. Interpolants and error estimators can be added later in another PR.

Could you please add FineRKN5 to docs/src/dynamical/nystrom.md to include it in the docs.

[ranocha]

Thanks! The PR looks good so far but convergence tests are failing?

[HenryLangner]

The convergence tests failing is strange. The only things changed or added in this PR is related to the algorithm FineRKN5. Testing the convergence manually with the (in place and out of place) examples used in tests/algconvergence/partitioned_methods_tests.jl, using the code

using OrdinaryDiffEq, Test, RecursiveArrayTools, DiffEqDevTools, Statistics

u0 = fill(0.0, 2);
v0 = ones(2);
function f1_harmonic(dv, v, u, p, t)
    dv .= -u
end;
function f2_harmonic(du, v, u, p, t)
    du .= v
end;
function harmonic_analytic(y0, p, x)
    v0, u0 = y0.x
    ArrayPartition(-u0 * sin(x) + v0 * cos(x), u0 * cos(x) + v0 * sin(x))
end;
ff_harmonic = DynamicalODEFunction(f1_harmonic, f2_harmonic; analytic = harmonic_analytic);
prob = DynamicalODEProblem(ff_harmonic, v0, u0, (0.0, 5.0));

dts = 1.0 ./ 2.0 .^ (5:-1:0);
println("In Place")
sim = test_convergence(dts, prob, FineRKN5(), dense_errors = true);
println(@test sim.𝒪est[:l2]≈6 rtol=1e-1)
println(@test sim.𝒪est[:L2]≈4 rtol=1e-1)

println("Out of Place")

u0 = 0.0;
v0 = 1.0;
function f1_harmonic_nip(v, u, p, t)
    -u
end;
function f2_harmonic_nip(v, u, p, t)
    v
end;

ff_harmonic_nip = DynamicalODEFunction(f1_harmonic_nip, f2_harmonic_nip;
                                       analytic = harmonic_analytic);
prob = DynamicalODEProblem(ff_harmonic_nip, v0, u0, (0.0, 5.0));

sim = test_convergence(dts, prob, FineRKN5(), dense_errors = true);
println(@test sim.𝒪est[:l2]≈6 rtol=1e-1)
println(@test sim.𝒪est[:L2]≈4 rtol=1e-1)

grants
.
Calling the Test results I get

for the In-Place test and

for the Out-of-Place test. Which shows that FineRKN5 converges fine. Am I missing something here?

I just checked the PR #1949 and it looks like it is failing the same tests with the same error message.

[ranocha]

@ChrisRackauckas Is this a known issue? I can't debug it locally right now...

[ranocha]

Thanks! This looks good to me. The test failures are not related to this PR. May be fixed by merging current master into this PR?

[ChrisRackauckas]

We shouldn't call out this specific method because it's not guaranteed to not change, so this information is non-local. Instead just say that this method requires that the acceleration equation needs to be independent of the velocity.

[ChrisRackauckas]

@ChrisRackauckas Is this a known issue? I can't debug it locally right now...

yes it looks like it was an unrelated regression #1956

[ChrisRackauckas]

Thanks!