Entropy stable hydrostatic reconstruction schemes for shallow water systems

TL;DR

This paper introduces a new entropy stable hydrostatic reconstruction scheme for shallow water systems, capable of handling multilayer flows, wetting/drying, and high-order accuracy on complex meshes, with proven stability and robustness.

Contribution

It develops a novel hydrostatic reconstruction method combined with entropy conservative fluxes and high-order DG schemes, ensuring stability, well-balancedness, and positivity in shallow water simulations.

Findings

Proves entropy stability and well-balancedness of the scheme.

Demonstrates robustness through numerical validation.

Extends to high-order and curvilinear meshes.

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.