An Entropy Stable Discontinuous Galerkin Method for the Two-Layer Shallow Water Equations on Curvilinear Meshes

Abstract

We present an entropy stable nodal discontinuous Galerkin spectral element method (DGSEM) for the two-layer shallow water equations on two dimensional curvilinear meshes. We mimic the continuous entropy analysis on the semi-discrete level with the DGSEM constructed on Legendre–Gauss–Lobatto (LGL) nodes. The use of LGL nodes endows the collocated nodal DGSEM with the summation-by-parts property that is key in the discrete analysis. The approximation exploits an equivalent flux differencing formulation for the volume contributions, which generate an entropy conservative split-form of the governing equations. A specific combination of a numerical surface flux and discretization of the nonconservative terms is then applied to obtain a high-order path-conservative scheme that is entropy conservative. Furthermore, we find that this combination yields an analogous discretization for the pressure and nonconservative terms such that the numerical method is well-balanced for discontinuous bathymetry on curvilinear domains. Dissipation is added at the interfaces to create an entropy stable approximation that satisfies the second law of thermodynamics in the discrete case, while maintaining the well-balanced property. We conclude with verification of the theoretical findings through numerical tests and demonstrate results about convergence, entropy stability and well-balancedness of the scheme.