On discretely entropy conservative and entropy stable discontinuous Galerkin methods

TL;DR

This paper develops a framework for constructing high-order entropy conservative and stable discontinuous Galerkin methods for hyperbolic PDEs, allowing more flexible quadrature choices while maintaining stability and accuracy.

Contribution

It introduces flux differencing and SBP-like operators to create entropy conservative schemes with arbitrary quadrature rules in DG methods.

Findings

Numerical experiments confirm stability for Euler equations.

Methods achieve high order accuracy.

Schemes are entropy stable with flexible quadrature choices.

Abstract

High order methods based on diagonal-norm summation by parts operators can be shown to satisfy a discrete conservation or dissipation of entropy for nonlinear systems of hyperbolic PDEs. These methods can also be interpreted as nodal discontinuous Galerkin methods with diagonal mass matrices. In this work, we describe how use flux differencing, quadrature-based projections, and SBP-like operators to construct discretely entropy conservative schemes for DG methods under more arbitrary choices of volume and surface quadrature rules. The resulting methods are semi-discretely entropy conservative or entropy stable with respect to the volume quadrature rule used. Numerical experiments confirm the stability and high order accuracy of the proposed methods for the compressible Euler equations in one and two dimensions.