Shallow water exner equations 1D

[patrickersing]

This PR implements the shallow water equations coupled with the exner model for sediment transport in 1D.
The implementation uses a specific formulation with an extra nonconservative term to ensure entropy conservation / stability.

To compute the sediment discharge the Meyer-Peter-Mueller and Grass model are available and the Manning formula is used to compute the shear stress. The implementation of the models is designed to be flexible, so that additional models can be added later on.

For validation there are three elixirs to test for convergence, well-balancedness and entropy conservation.

[codecov]

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[patrickersing]

Here are some convergence results using elixir_shallowwater_exner_source_terms.jl with polydeg=4 for different fluxes and formulas for the sediment discharge:

• EC fluxes / Grass:
  

• Local Lax-Friedrichs / Meyer-Peter-Müller:
  

• Roe / Meyer-Peter-Müller:
  

[andrewwinters5000]

Here are some convergence results using elixir_shallowwater_exner_source_terms.jl with polydeg=4 for different fluxes and formulas for the sediment discharge

Overall, the EOCs look very good. One small question. Do we still see an odd/even effect when using the EC fluxes. That is, can you run with polydeg=3 for the EC + Grass test?

[patrickersing]

Here are some convergence results using elixir_shallowwater_exner_source_terms.jl with polydeg=4 for different fluxes and formulas for the sediment discharge

Overall, the EOCs look very good. One small question. Do we still see an odd/even effect when using the EC fluxes. That is, can you run with polydeg=3 for the EC + Grass test?

Yes, there is some odd-even effect visible. Here are results for EC + Grass with polydeg=2:

and polydeg=3:

[andrewwinters5000]

This already looks very nice! Most of my suggestions are expanding upon the docstrings.

One question: What would be the modifications necessary to handle wetting / drying?

[patrickersing]

@andrewwinters5000 thanks for your review. I like the additional details that you added to the docstrings.

I did not look too much into wetting / drying yet, but I suspect that most changes should be similar to the SWE. So we need to switch to FV near dry states, add a hydrostatic reconstruction and a velocity desingularization.
I expect the additional Exner-equation to less problematic since we do not need to recover the velocity.

[andrewwinters5000]

I did not look too much into wetting / drying yet, but I suspect that most changes should be similar to the SWE. So we need to switch to FV near dry states, add a hydrostatic reconstruction and a velocity desingularization. I expect the additional Exner-equation to less problematic since we do not need to recover the velocity.

Okay, we can explore the wetting/drying possibilities in a separate PR and/or open an issue describing what would need done / modified, e.g., thresholds, desingularizing the velocities in the Manning friction, hydrostatic reconstruction etc. For now, the "wet" version of the SW-Exner is mergable to main.

[andrewwinters5000]

Very nice work @patrickersing ! Everything looks good from my side.