# A simple cure for numerical shock instability in HLLC Riemann solver

## TL;DR

This paper introduces the HLLC-SWM framework, a modification to the HLLC Riemann solver that enhances shock stability by selectively increasing dissipation near shocks, while preserving accuracy on other wavefields.

## Contribution

The paper proposes a novel framework, HLLC-SWM, that improves shock stability of the HLLC scheme by adding targeted dissipation without sacrificing accuracy on contact and shear waves.

## Key findings

- HLLC-SWM schemes are shock stable over a wide Mach number range.
- They retain positivity and exact contact resolution.
- HLLC-SWM-P is highly accurate for viscous flows.

## Abstract

The Harten-Lax-van Leer with contact (HLLC) scheme is known to be plagued by various forms of numerical shock instabilities. In this paper, we propose a new framework for developing shock stable, contact and shear preserving approximate Riemann solvers based on the HLLC scheme for the Euler system of equations. The proposed framework termed as HLLC-SWM (\textbf{S}elective \textbf{W}ave \textbf{M}odified) scheme identifies and increases the magnitude of the inherent diffusive HLL component within the HLLC scheme in the vicinity of a shock wave while leaving its antidiffusive component unmodified to retain accuracy on linearly degenerate wavefields. We present two strategies to compute the requisite supplementary dissipation which results in HLLC-SWM-E and HLLC-SWM-P variants. Through a linear perturbation analysis of the HLLC-SWM framework, we clarify how the additional dissipation introduced in this way helps in damping of unphysical perturbations in primitive quantities under a derived CFL constraint. A matrix based stability analysis of a steady two-dimensional normal shock is used to show that both variants of the HLLC-SWM scheme are shock stable over a wide range of inlet Mach numbers. Results from standard test cases demonstrate that the HLLC-SWM schemes are capable of computing shock stable solutions on a variety of problems while retaining positivity and exact inviscid contact ability. On viscous flows, while the HLLC-SWM-P variant is quite accurate, the HLLC-SWM-E variant introduces slight inaccuracy which can be corrected through a simple Mach number based switching function.