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

TL;DR

This paper develops an entropy stable discontinuous Galerkin method for non-conservative Euler equations with gravity, ensuring stability and accuracy for atmospheric flow simulations on complex meshes.

Contribution

It introduces a semi-discrete entropy stable DG method using flux differencing and skew-hybridized formulation for non-conservative balance laws on curvilinear meshes.

Findings

Method achieves entropy stability on complex meshes.

High-order accuracy demonstrated in atmospheric flow tests.

Robustness confirmed across multiple dimensions.

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.