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arXiv · 2406.14119

Entropy stable hydrostatic reconstruction schemes for shallow water systems

Patrick Ersing·Sven Goldberg·Andrew R. Winters
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Abstract

In this work, we develop a new hydrostatic reconstruction procedure to construct well-balanced schemes for one and multilayer shallow water flows, including wetting and drying. Initially, we derive the method for a path-conservative finite volume scheme and combine it with entropy conservative fluxes and suitable numerical dissipation to preserve an entropy inequality in the semi-discrete case. We then combine the novel hydrostatic reconstruction with a collocated nodal split-form discontinuous Galerkin spectral element method, extending the method to high-order and curvilinear meshes. The high-order method incorporates an additional positivity-limiter and is blended with a compatible subcell finite volume method to maintain well-balancedness at wet/dry fronts. We prove entropy stability, well-balancedness, and positivity-preservation for both methods. Numerical results for the high-order method validate the theoretical findings and demonstrate the robustness of the scheme.

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Patrick Ersing, Sven Goldberg, Andrew R. Winters. 2024-06-20. Entropy stable hydrostatic reconstruction schemes for shallow water systems. https://arxiv.org/abs/2406.14119

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