Add multilayer shallow water equations in 2D

[patrickersing]

This PR extends the multilayer shallow water equations introduced in #30 to 2D.
The PR introduces the new equations file and basic tests for well-balancedness, entropy-conservation and convergence on TreeMesh and UnstructuredMesh

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Codecov Report

Attention: Patch coverage is 99.46950% with 2 lines in your changes are missing coverage. Please review.

Project coverage is 99.26%. Comparing base (b928469) to head (de6251f).

Files	Patch %	Lines
src/equations/shallow_water_multilayer_2d.jl	99.33%	2 Missing ⚠️

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@@            Coverage Diff             @@
##             main      #40      +/-   ##
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+ Coverage   99.22%   99.26%   +0.04%     
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  Lines        1680     2054     +374     
==========================================
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[patrickersing]

Here are some convergence results on TreeMesh2D.
Overall the convergence order is a bit worse than expected.

polydeg = 3, EC flux:

polydeg=3, flux_lax_friedrichs

polydeg=4, EC flux:

polydeg=4, flux_lax_friedrichs:

polydeg=5, EC flux:

polydeg=5, flux_lax_friedrichs`:

[andrewwinters5000]

Great work @patrickersing ! It is nice that the 2D version of the multilayer equations came together quite easily.

[andrewwinters5000]

Here are some convergence results on TreeMesh2D. Overall the convergence order is a bit worse than expected.

I agree that some of the convergence values are strange. The polydeg=5 look pretty okay, but the polydeg=3or 4 are a bit odd. Can you try running the ES convergence with flux_hll? Sometimes the flux_lax_friedrichs give dodgy results compare to the more "sophisticated" Riemann solvers.

[patrickersing]

Here are the convergence results using FluxHLL. The results look a bit more reasonable, but are still a bit strange.
polydeg=3

polydeg=4

polydeg=5

[andrewwinters5000]

Here are the convergence results using FluxHLL. The results look a bit more reasonable, but are still a bit strange.

Agreed, these are still a bit odd. The final EOC for the odd polynomial degrees is now better (around polydeg+2) and the even is now worse (around polydeg) than the expected polydeg + 1 for an ES scheme. Would it be possible to run one more refinement level for each of the polydegs to see if this weird trend continues? Or if the EOCs "settle" in some sense.

[andrewwinters5000]

Everything already looks quite good. Just a suggestion about the explicit inclusion o fthe multiplication symbols. This would likely need done in 1D as well.

[patrickersing]

Here are the convergence results using FluxHLL. The results look a bit more reasonable, but are still a bit strange.

Agreed, these are still a bit odd. The final EOC for the odd polynomial degrees is now better (around polydeg+2) and the even is now worse (around polydeg) than the expected polydeg + 1 for an ES scheme. Would it be possible to run one more refinement level for each of the polydegs to see if this weird trend continues? Or if the EOCs "settle" in some sense.

I followed your suggestion and set the gravity_constant=1.1 to obtain a MMS in supersonic flow conditions. Now the convergence rates look pretty good, which suggests that the weird behavior was caused by the transonic MMS.
The results below are obtained with FluxHLL for t=0.2 and a slightly modified MMS with g=1.1.

polydeg = 3:

polydeg = 4:

polydeg = 5:

Since this fixes the weird convergence, I would suggest to update the MMS for both ShallowWaterMultiLayer1D and ShallowWaterMultiLayer2D.

[andrewwinters5000]

Looks good! I just left a few clean-up comments / questions. One thing we should keep in mind is that the docstrings will need updating once the multilayer preprint is finished.

[andrewwinters5000]

LGTM! We just have to hope to get an Intel chipset for the macOS tests due to the ARM issues.

[patrickersing]

I think the reason why tests are suddenly failing is that macOS-latest now defaults to macOS-14-arm64 instead of macOS-12. This also explains why even with multiple tries I never get Intel runners.
So I would suggest to fix the version to macOS-12 for now to merge the PR (assuming this works) and tackle the problem in a dedicated PR.

[andrewwinters5000]

I think the reason why tests are suddenly failing is that macOS-latest now defaults to macOS-14-arm64 instead of macOS-12. This also explains why even with multiple tries I never get Intel runners. So I would suggest to fix the version to macOS-12 for now to merge the PR (assuming this works) and tackle the problem in a dedicated PR.

Sounds reasonable. We can adjust down to macOS-12 and hopefully grab the Intel runners to fix this problem. We can then open a corresponding issue regarding the CI on Mac-ARM and address it later.

[patrickersing]

With the recent CI change in #42 the issues with macOS have been fixed, so we should be good to merge.