Entropy-stable discretizations for the compressible Euler equations using simple adaptive averages

TL;DR

This paper introduces a novel entropy-stable discretization method for the compressible Euler equations that employs adaptive averaging techniques to enhance robustness and preserve physical invariants.

Contribution

It proposes a new entropy stabilization approach using simple adaptive averages for density and internal energy, maintaining key flow structures.

Findings

Achieves non-linear robustness with simplified symmetric means.

Ensures entropy conservation asymptotically.

Preserves kinetic energy and pressure equilibrium.

Abstract

Entropy stabilization of the compressible Euler system is achieved by adapting the averages that are applied to the density and internal energy variables. The approach achieves non-linear robustness despite the use of simplified symmetric means (e.g., arithmetic, geometric, or harmonic evaluations), including their related expansions for asymptotic entropy conservation. The proposed formulation works via centralized convective terms and can naturally adhere to additional structures of the flow equations such as kinetic-energy- and pressure-equilibrium-preservation.