Preventing pressure oscillations does not fix local linear stability issues of entropy-based split-form high-order schemes

TL;DR

This paper investigates whether pressure equilibrium preservation can address local linear stability issues in entropy-based split-form high-order schemes for the Euler equations, revealing that it does not resolve the stability problems caused by anti-diffusion mechanisms.

Contribution

The study provides a theoretical analysis and numerical validation showing that pressure equilibrium preservation does not fix the local linear stability issues in entropy-conserving high-order schemes.

Findings

Pressure equilibrium preservation does not improve local linear stability.

Anti-diffusion in entropy-conserving fluxes causes stability issues.

Numerical fluxes with specific properties are characterized and tested.

Abstract

Recently, it was discovered that the entropy-conserving/dissipative high-order split-form discontinuous Galerkin discretizations have robustness issues when trying to solve the simple density wave propagation example for the compressible Euler equations. The issue is related to missing local linear stability, i.e. the stability of the discretization towards perturbations added to a stable base flow. This is strongly related to an anti-diffusion mechanism, that is inherent in entropy-conserving two-point fluxes, which are a key ingredient for the high-order discontinuous Galerkin extension. In this paper, we investigate if pressure equilibrium preservation is a remedy to these recently found local linear stability issues of entropy-conservative/dissipative high-order split-form discontinuous Galerkin methods for the compressible Euler equations. Pressure equilibrium preservation describes the property of a discretization to keep pressure and velocity constant for pure density wave propagation. We present the full theoretical derivation, analysis, and show corresponding numerical results to underline our findings. In addition, we characterize numerical fluxes for the Euler equations that are entropy-conservative, kinetic-energy-preserving, pressure-equilibrium-preserving, and have a density flux that does not depend on the pressure. The source code to reproduce all numerical experiments presented in this article is available online (DOI: 10.5281/zenodo.4054366).