A Note on Numerical Fluxes Conserving a Member of Harten's One-Parameter Family of Entropies for the Compressible Euler Equations

TL;DR

This paper proves that for the compressible Euler equations, no numerical flux can simultaneously conserve a member of Harten's entropy family, preserve pressure equilibrium, and have a pressure-independent density flux, highlighting limitations in entropy-conserving schemes.

Contribution

It establishes a fundamental impossibility result for certain entropy-conserving fluxes in the Euler equations, contrasting with fluxes based on physical entropy.

Findings

No flux conserving Harten's entropy can preserve pressure equilibrium.

Such fluxes cannot have a pressure-independent density flux.

Fluxes based on physical entropy can preserve kinetic energy.

Abstract

Entropy-conserving numerical fluxes are a cornerstone of modern high-order entropy-dissipative discretizations of conservation laws. In addition to entropy conservation, other structural properties mimicking the continuous level such as pressure equilibrium and kinetic energy preservation are important. This note proves that there are no numerical fluxes conserving (one of) Harten's entropies for the compressible Euler equations that also preserve pressure equilibria and have a density flux independent of the pressure. This is in contrast to fluxes based on the physical entropy, where even kinetic energy preservation can be achieved in addition.