Entropy Stable Discontinuous Galerkin Methods for Balance Laws in Non-Conservative Form: Applications to the Euler Equations with Gravity

Abstract

In this work a non-conservative balance law formulation is considered that encompasses the rotating, compressible Euler equations for dry atmospheric flows. We develop a semi-discretely entropy stable discontinuous Galerkin method on curvilinear meshes using a generalization of flux differencing for numerical fluxes in fluctuation form. The method uses the skew-hybridized formulation of the element operators to ensure that, even in the presence of under-integration on curvilinear meshes, the resulting discretization is entropy stable. Several atmospheric flow test cases in one, two, and three dimensions confirm the theoretical entropy stability results as well as show the high-order accuracy and robustness of the method.

31 Pages, 13 figures

References in corpus (2)

• Efficient Exascale Discretizations: High-Order Finite Element Methods
• Entropy stable discontinuous Galerkin methods for ten-moment Gaussian closure equations

Cited by in corpus (4)

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• Structure-preserving approximations of the Serre-Green-Naghdi equations in standard and hyperbolic form
• An entropy stable discontinuous Galerkin method for the two-layer shallow water equations on curvilinear meshes
• Multiderivative time integration methods preserving nonlinear functionals via relaxation