Comparison of some Entropy Conservative Numerical Fluxes for the Euler Equations

TL;DR

This paper compares various entropy conservative numerical fluxes for the Euler equations, extending flux differencing theory, constructing new fluxes, and analyzing their stability and positivity properties through theoretical and numerical methods.

Contribution

It extends flux differencing theory to high-order schemes, develops a procedure for constructing entropy conservative fluxes, and investigates their robustness and positivity preservation.

Findings

Extended flux differencing theory for high-order schemes

Derived new entropy conservative fluxes

Proven positivity preservation with dissipation operators

Abstract

Entropy conservation and stability of numerical methods in gas dynamics have received much interest. Entropy conservative numerical fluxes can be used as ingredients in two kinds of schemes: Firstly, as building blocks in the subcell flux differencing form of Fisher and Carpenter (2013) and secondly (enhanced by dissipation) as numerical surface fluxes in finite volume like schemes. The purpose of this article is threefold. Firstly, the flux differencing theory is extended, guaranteeing high-order for general symmetric and consistent numerical fluxes and investigating entropy stability in a generalised framework of summation-by-parts operators applicable to multiple dimensions and simplex elements. Secondly, a general procedure to construct affordable entropy conservative fluxes is described explicitly and used to derive several new fluxes. Finally, robustness properties of entropy stable numerical fluxes are investigated and positivity preservation is proven for several entropy conservative fluxes enhanced with local Lax-Friedrichs type dissipation operators. All these theoretical investigations are supplemented with numerical experiments.