Discretely entropy stable weight-adjusted discontinuous Galerkin methods on curvilinear meshes

Abstract

We construct entropy conservative and entropy stable high order accurate discontinuous Galerkin (DG) discretizations for time-dependent nonlinear hyperbolic conservation laws on curvilinear meshes. The resulting schemes preserve a semi-discrete quadrature approximation of a continuous global entropy inequality. The proof requires the satisfaction of a discrete geometric conservation law, which we enforce through an appropriate polynomial approximation. We extend the construction of entropy conservative and entropy stable DG schemes to the case when high order accurate curvilinear mass matrices are approximated using low-storage weight-adjusted approximations, and describe how to retain global conservation properties under such an approximation. The theoretical results are verified through numerical experiments for the compressible Euler equations on triangular and tetrahedral meshes.