[WIP] Add 1D multilayer shallow water equations

[patrickersing]

This PR will add the 1D multilayer shallow water equations, which are a generalization of the currently implemented two-layer shallow water equations for an arbitrary amount of layers. The implementation employs a specific formulation of the system with a nonconservative pressure term.

The implementation is done similar to the multicomponent equations in Trixi.jl with some modifications, to account for the layer-wise structure.

Additional tests:

• Well-balanced tests
• Convergence testing for more than two layers
• Application tests

Additional functionality:

• Add ES fluxes
• Add additional analysis functions (lake_at_rest_error, energy_total, etc.)

[codecov]

Codecov Report

Attention: Patch coverage is 98.19005% with 4 lines in your changes are missing coverage. Please review.

Project coverage is 99.22%. Comparing base (cbabd75) to head (54b0f18).

Files	Patch %	Lines
src/equations/equations.jl	50.00%	2 Missing ⚠️
src/equations/shallow_water_multilayer_1d.jl	98.96%	2 Missing ⚠️

Additional details and impacted files

@@            Coverage Diff             @@
##             main      #30      +/-   ##
==========================================
- Coverage   99.38%   99.22%   -0.16%     
==========================================
  Files          40       45       +5     
  Lines        1459     1680     +221     
==========================================
+ Hits         1450     1667     +217     
- Misses          9       13       +4     

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[patrickersing]

Here are some convergence results for elixir_shallowwater_multilayer_convergence.jl with
polydeg=3:

polydeg=4:

polydeg=5:

[andrewwinters5000]

Here are some convergence results for elixir_shallowwater_multilayer_convergence.jl

These results all seem reasonable. Are these all with the EC scheme? It is a bit strange that the h1 and h1v tend to converge with polydeg+1/2 whereas the other layers tend more toward polydeg+1.

[patrickersing]

Yes, all the results were obtained with the EC flux. The trend remains roughly the same when I test with flux_lax_friedrichs, here e.g. for polydeg=3

In most cases it seems that convergence for layer 2 is better than for layers 1 & 3 (even above the expected convergence order), but I don't have an explanation why that is.

[andrewwinters5000]

It is quite strange that the "sandwiched" layer converges with slightly higher order than the other layers. It is also weird that the convergence of h2_v and h3_v are both higher than expected because you are setting the velocities to be constants. So I would assume that the convergence behavior of h3_v would match that of h (for instance)

[andrewwinters5000]

Nice work @patrickersing ! It is very cool that we can have an entropy stable scheme on an arbitrary number of layers. This abstraction should also make it easier to do the mullti-layer wetting and drying. I left a few comments and questions.

[andrewwinters5000]

LGTM! Although I still cannot work out why some functions are not reported as covered when they should be.