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[WIP] Shallow-water exner equations with wetting/drying #56

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This PR aims to add wetting and drying for the ShallowWaterExnerEquations1D, following the strategy used for the SWE.
For the Exner equation drying is less problematic, as there is no separate velocity computation. For now there is no mechanism to ensure positive sediment height, so we need to assume that there is always some sediment present.
An open problem is that the hydrostatic reconstruction leads to continuous sediment height at interfaces and therefore removes any dissipation in the sediment layer, which leads to oscillations.
Furthermore, the sediment transport is very sensitive to velocity oscillations that might be present at/near the dry states.

Open problems:

  • Find a way to introduce viscosity in the sediment layer when hydrostatic reconstruction is used
  • Mechanism to prevent sediment transport for vanishing sediment height (optional)
  • Improve the velocity computation near dry cells (optional)

@patrickersing patrickersing added the enhancement New feature or request label Aug 23, 2024
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codecov bot commented Aug 23, 2024

Codecov Report

Attention: Patch coverage is 7.86517% with 82 lines in your changes missing coverage. Please review.

Project coverage is 96.28%. Comparing base (7e1d491) to head (68148a7).

Files Patch % Lines
...c/callbacks_stage/positivity_shallow_water_dg1d.jl 0.00% 37 Missing ⚠️
src/equations/shallow_water_exner_1d.jl 23.33% 23 Missing ⚠️
...s/tree_1d_dgsem/elixir_shallowwater_exner_beach.jl 0.00% 14 Missing ⚠️
...sem/elixir_shallowwater_exner_well_balanced_dry.jl 0.00% 8 Missing ⚠️
Additional details and impacted files
@@            Coverage Diff             @@
##             main      #56      +/-   ##
==========================================
- Coverage   99.27%   96.28%   -3.00%     
==========================================
  Files          61       63       +2     
  Lines        2630     2719      +89     
==========================================
+ Hits         2611     2618       +7     
- Misses         19      101      +82     

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