Well-balanced mortars

[andrewwinters5000]

This seeks to rewrite the mortar projection routines specifically for the shallow water equations to preserve well-balancedness with non-conforming approximations. That is, the shallow water solver could utilize the AMR capabilities of TreeMesh or P4estMesh. The implementation might become delicate at wet/dry transition regions as one needs to take care in what solution values and/or fluxes are projected to and from the mortar while maintaining positivity.

• Decide on file names and structure in the src/solvers folder
• Implement and test on TreeMesh in 2D Save for a future PR
• Implement and test on P4estMesh in 2D
  • Wet well balanced
  • Wet/dry well balanced
  • Wet perturbation with AMR
  • Wet/dry perturbation with AMR (runs but conservation errors for water height are ≈1e-5 and robustness is sensitive to velocity desingularization)

Further explorations and improvements are discussed in #77

[codecov]

Codecov Report

Attention: Patch coverage is 99.36709% with 2 lines in your changes missing coverage. Please review.

Project coverage is 99.33%. Comparing base (2267af4) to head (8a6a2fb).
Report is 1 commits behind head on main.

Files with missing lines	Patch %	Lines
src/solvers/dgsem_p4est/dg_2d.jl	98.67%	2 Missing ⚠️

Additional details and impacted files

@@           Coverage Diff            @@
##             main      #45    +/-   ##
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  Coverage   99.32%   99.33%            
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  Files          60       66     +6     
  Lines        2801     3115   +314     
========================================
+ Hits         2782     3094   +312     
- Misses         19       21     +2     

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[andrewwinters5000]

The current version provides a well-balanced approximation for a fully wet configuration. The examples/p4est_2d_dgsem/elixir_shallowwater_well_balanced_wet_dry.jl crashes in the first time step. So this requires further investigation. The main issues with the fully wet version are:

1. There are still significant allocations in the computation of the mortar fluxes.
2. More pressing is that we require a slight modification to the P4estMortarContainer within Trixi.jl to add additional storage of the unprojected large element solution in order to have this information available to subtract off a superfluous contribution of the local physical flux in the large element. I want to avoid this because this specailized mortar treatment should only need to live within the new package.

The modification below in Trixi's src/solvers/dgsem_p4est/containers.jl is slight, but incredibly hacky. I added a third "slot" to store the large element information that I needed as given in the code snippet below. There is probably a better way to precompute and store the parent flux contributions on each mortar. I am just not familiar enough with the P4estMesh implementations and its various caches to see where one could store such information.

# Create mortar container and initialize mortar data.
function init_mortars(mesh::Union{P4estMesh, T8codeMesh}, equations, basis, elements)
    NDIMS = ndims(elements)
    uEltype = eltype(elements)

    # Initialize container
    n_mortars = count_required_surfaces(mesh).mortars

    _u = Vector{uEltype}(undef,
                         3 * nvariables(equations) * 2^(NDIMS - 1) *
                         nnodes(basis)^(NDIMS - 1) * n_mortars)
    u = unsafe_wrap(Array, pointer(_u),
                    (3, nvariables(equations), 2^(NDIMS - 1),
                     ntuple(_ -> nnodes(basis), NDIMS - 1)..., n_mortars))

    _neighbor_ids = Vector{Int}(undef, (2^(NDIMS - 1) + 1) * n_mortars)
    neighbor_ids = unsafe_wrap(Array, pointer(_neighbor_ids),
                               (2^(NDIMS - 1) + 1, n_mortars))

    _node_indices = Vector{NTuple{NDIMS, Symbol}}(undef, 2 * n_mortars)
    node_indices = unsafe_wrap(Array, pointer(_node_indices), (2, n_mortars))

    mortars = P4estMortarContainer{NDIMS, uEltype, NDIMS + 1, NDIMS + 3}(u,
                                                                         neighbor_ids,
                                                                         node_indices,
                                                                         _u,
                                                                         _neighbor_ids,
                                                                         _node_indices)

    if n_mortars > 0
        init_mortars!(mortars, mesh)
    end

    return mortars
end

UPDATE: This can be ignored as a new P4estShallowWaterMortarContainer with the required additional storage was introduced in 4821434 that avoids the need to modify the Trixi base code.

[andrewwinters5000]

I wonder if all the coveralls "noise" in the CI means we should update the reporting strategy as mentioned here

[patrickersing]

From my end this looks almost ready to merge. I just have a question and one comment that needs to be changed.

[patrickersing]

I wonder if all the coveralls "noise" in the CI means we should update the reporting strategy as mentioned here

Since the coverage issue has been solved by setting a higher maxiters in the coverage run, I would say we can move the discussion about this to a separate issue.

[ranocha]

Do you want me to review the PR again? If so, please let me know (and tell me whether there is anything in particular you where you would like to have feedback). Otherwise, I am fine merging this PR when the remaining comments of Patrick are resolved.

[andrewwinters5000]

Thanks a lot @ranocha and @patrickersing for your help getting this experimental feature over the finish line!