Strategies to cure numerical shock instability in HLLEM Riemann solver

TL;DR

This paper proposes strategies to prevent shock instability in the HLLEM Riemann solver by modifying flux discretizations, ensuring stable shock solutions without sacrificing accuracy in shear flows.

Contribution

It introduces two novel approaches to modify flux components in HLLEM, effectively mitigating shock instability while maintaining accuracy.

Findings

Both strategies stabilize shock solutions in test cases.

Modified schemes retain accuracy in shear-dominated flows.

Stability bounds on CFL number are established.

Abstract

The HLLEM scheme is a popular contact and shear preserving approximate Riemann solver for cheap and accurate computation of high speed gasdynamical flows. Unfortunately this scheme is known to be plagued by various forms of numerical shock instability. In this paper we present various strategies to save the HLLEM scheme from developing such spurious solutions. A linear scale analysis of its mass and interface-normal momentum flux discretizations reveal that its antidiffusive terms, which are primarily responsible for resolution of linear wavefields, are inadvertently activated along a normal shock front due to numerical perturbations. These erroneously activated terms counteract the favourable damping mechanism provided by its inherent HLL-type diffusive terms and trigger the shock instability. To avoid this, two different strategies are proposed for discretization of these critical flux components in the vicinity of a shock: one that deals with increasing the magnitude of inherent HLL-type dissipation through careful manipulation of specific non-linear wave speed estimates while the other deals with reducing the magnitude of these critical antidiffusive terms. A linear perturbation analysis is performed to gauge the effectiveness of these cures and estimate von-Neumann type stability bounds on the CFL number arising from their use. Results from classic numerical test cases show that both types of modified HLLEM schemes are able to provide excellent shock stable solutions while retaining commendable accuracy on shear dominated viscous flows.