On discretely entropy stable weight-adjusted discontinuous Galerkin methods: curvilinear meshes

Abstract

We construct entropy conservative and entropy stable 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 curvilinear mass matrices are approximated using low-storage weight-adjusted approximations, and describe how to retain global conservation properties under such an approximation. For certain types of curvilinear meshes, these weight-adjusted approximations deliver optimal rates of convergence. Finally, the high order accuracy, local conservation, and discrete conservation or dissipation of entropy for under-resolved solutions are verified through numerical experiments for the compressible Euler equations on triangular and tetrahedral meshes.