Classic LWR traffic flow

[DanielDoehring]

As we integrate Trixi more and more into our teaching I plan to add classic conservation laws to Trixi.
The first one is the Lighthill-Witham-Richards (LWR) 1D traffic model that is referenced in every lecture notes I have seen on hyperbolic PDEs.

---

Convergence test:

####################################################################################################
l2
car-density         
error     EOC       
8.46e-04  -         
4.68e-05  4.17      
2.87e-06  4.03      
1.79e-07  4.00      
1.12e-08  4.00      
6.98e-10  4.00      

mean      4.04      
----------------------------------------------------------------------------------------------------
linf
car-density         
error     EOC       
4.59e-03  -         
3.27e-04  3.81      
2.05e-05  3.99      
1.29e-06  3.99      
8.06e-08  4.00      
5.04e-09  4.00      

mean      3.96      
----------------------------------------------------------------------------------------------------

initial_condition_constant:

────────────────────────────────────────────────────────────────────────────────────────────────────
 Simulation running 'TrafficFlowLWREquations1D' with DGSEM(polydeg=3)
────────────────────────────────────────────────────────────────────────────────────────────────────
 #timesteps:               2001                run time:       1.71623150e-02 s
 Δt:             3.53850282e-12                └── GC time:    0.00000000e+00 s (0.000%)
 sim. time:      1.00000000e+02 (100.000%)     time/DOF/rhs!:  2.32239583e-08 s
                                               PID:            2.75562500e-08 s
 #DOFs per field:            64                alloc'd memory:        161.149 MiB
 #elements:                  16

 Variable:       car-density   
 L2 error:       4.13046273e-17
 Linf error:     1.11022302e-16
 ∑∂S/∂U ⋅ Uₜ :   0.00000000e+00
────────────────────────────────────────────────────────────────────────────────────────────────────

---

Other than the LWR equations I plan to add 1D LinearizedEuler and 1D Wave equation (as a first order hyperbolic system) whenever I have time and am in the mood to code something up.

[github-actions]

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[codecov]

Codecov Report

Attention: 6 lines in your changes are missing coverage. Please review.

Comparison is base (29e173e) 96.38% compared to head (42eaf42) 96.36%.

Files	Patch %	Lines
src/equations/traffic_flow_lwr_1d.jl	85.00%	6 Missing ⚠️

Additional details and impacted files

@@            Coverage Diff             @@
##             main    #1840      +/-   ##
==========================================
- Coverage   96.38%   96.36%   -0.01%     
==========================================
  Files         455      458       +3     
  Lines       36336    36392      +56     
==========================================
+ Hits        35019    35069      +50     
- Misses       1317     1323       +6     

Flag	Coverage Δ
unittests	96.36% <89.29%> (-0.01%)	⬇️

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[andrewwinters5000]

This is a nice equation to add to Trixi, especially to expand the use Trixi for educational purposes. I also like that it would be easy for students to add a more complicated flux function (like those given in Whitham's book) for traffic modelling and compare.

[DanielDoehring]

Thanks for the quick review @andrewwinters5000 !

[ranocha]

Nice idea!

[ranocha]

Thanks!

[andrewwinters5000]

LGTM! I approved this because my only note is a small fix in the NEWS.md