Add Shallow water moment equation models in 1D

[patrickersing]

This PR adds two new equation types for the one-dimensional Shallow water moment equations and Shallow water linearized moment equations:

• ShallowWaterMomentEquations1D
• ShallowWaterLinearizedMomentEquations1D

Details about the implementation and models can be found in the preprint:

• Julio Careaga, Patrick Ersing, Julian Koellermeier, Andrew R. Winters (2026)
  Entropy analysis and entropy stable DG methods for the shallow water moment equations
  DOI: 10.48550/arXiv.2602.06513

[codecov]

Codecov Report

❌ Patch coverage is 98.81188% with 6 lines in your changes missing coverage. Please review.
✅ Project coverage is 99.27%. Comparing base (729b1a8) to head (f70060a).
⚠️ Report is 1 commits behind head on main.

Files with missing lines	Patch %	Lines
src/equations/equations.jl	50.00%	2 Missing ⚠️
...c/equations/shallow_water_linearized_moments_1d.jl	98.81%	2 Missing ⚠️
src/equations/shallow_water_moments_1d.jl	99.11%	2 Missing ⚠️

Additional details and impacted files

@@            Coverage Diff             @@
##             main     #128      +/-   ##
==========================================
- Coverage   99.32%   99.27%   -0.05%     
==========================================
  Files          87       94       +7     
  Lines        4587     5092     +505     
==========================================
+ Hits         4556     5055     +499     
- Misses         31       37       +6     

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[patrickersing]

Convergence results:

---

SWME

elixir_shallowwater_moments_convergence with polydeg=3:

---

SWLME

elixir_shallowwater_linearized_moments_convergence with polydeg=3:

[andrewwinters5000]

Thanks for adding the new equation systems and functionality! I only left a few comments.

[andrewwinters5000]

Everything looks good to me. The small drop in coverage is acceptable as noise.

[andrewwinters5000]

It all looks good to me! Just one small thing for the sake of consistency; in the Monai tutorial we could maybe change source_terms_manning_friction to be source_term_manning_friction on lines 265 and 319 in the file docs/src/tutorials/elixir_shallowwater_monai_tsunami.jl.

[patrickersing]

It all looks good to me! Just one small thing for the sake of consistency; in the Monai tutorial we could maybe change source_terms_manning_friction to be source_term_manning_friction on lines 265 and 319 in the file docs/src/tutorials/elixir_shallowwater_monai_tsunami.jl.

@andrewwinters5000 Right, we should make that consistent. However, we currently have not implemented source_term_manning_friction for the standard ShallowWaterEquations1D, so I think we would need to extend the existing TrixiSW function inside the tutorial. I don't think that is optimal for an introductory example.

Should we just open an issue for now and then properly implement the source_term_manning_friction for the SWE in a separate PR?

[andrewwinters5000]

Should we just open an issue for now and then properly implement the source_term_manning_friction for the SWE in a separate PR?

Yeah, let's just open an issue about naming consistency of the friction source terms. This should be done separate from this feature PR.

[patrickersing]

I have created issue #129 to document the inconsistency and will start working on a respective PR soon. With that I think everything should be addressed in this PR.

[andrewwinters5000]

With that I think everything should be addressed in this PR.

I agree, we are good to merge.