ECCOMAS 2024

Entropy-stable hydrostatic reconstructions for shallow water systems

  • Ersing, Patrick (Linköping University)
  • Winters, Andrew Ross (Linköping University)

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Hydrostatic reconstructions are an established method in the design of well-balanced schemes for the shallow water equations (SWE). While this method has been successfully extended to the two-layer SWE in a number of works, it is not clear whether these reconstructions satisfy an entropy inequality. In this talk we present a novel hydrostatic reconstruction method for shallow water systems. The new reconstruction method is exactly well-balanced and provably entropy-stable even for wet-dry transitions. Starting from the subcell-reconstruction interpretation in [1], the novel reconstruction incorporates discontinuous bottom topographies as well as a specific discretization of the pressure term. In combination with entropy conserving fluxes we then obtain a reconstruction method that exactly preserves the entropy for the SWE. We extend the method to the two-layer shallow water system and introduce dissipation, to obtain conditional entropy-stability. Furthermore, we show how this technique can be combined with a nodal discontinuous Galerkin spectral element method. In this way we create high-order methods for shallow water flows that are entropy-stable and well-balanced even for wet-dry transitions on two-dimensional curvilinear meshes. To demonstrate the stability and well-balanced properties of our proposed hydrostatic reconstruction technique, we present a number of challenging test cases in one and two spatial dimensions.