# A Cure for numerical shock instability in HLLC Riemann solver using antidiffusion control

## TL;DR

This paper introduces HLLC-ADC, a simple and cost-effective modification to the HLLC Riemann solver that effectively prevents shock instabilities like the Carbuncle phenomenon while maintaining accuracy in high-speed flow simulations.

## Contribution

The paper proposes a novel shock stability enhancement for the HLLC scheme using a pressure-based shock sensor to control antidiffusive terms, improving robustness without sacrificing accuracy.

## Key findings

- HLLC-ADC suppresses shock instabilities effectively.
- The scheme maintains accuracy on shear flows.
- It prevents the Carbuncle phenomenon in tests.

## Abstract

Various forms of numerical shock instabilities are known to plague many contact and shear preserving approximate Riemann solvers, including the popular Harten-Lax-van Leer with Contact (HLLC) scheme, during high speed flow simulations. In this paper we propose a simple and inexpensive novel strategy to prevent the HLLC scheme from developing such spurious solutions without compromising on its linear wave resolution ability. The cure is primarily based on a reinterpretation of the HLLC scheme as a combination of its well-known diffusive counterpart, the HLL scheme, and an antidiffusive term responsible for its accuracy on linear wavefields. In our study, a linear analysis of this alternate form indicates that shock instability in the HLLC scheme could be triggered due to the unwanted activation of the antidiffusive terms of its mass and interface-normal flux components on interfaces that are not aligned with the shock front. This inadvertent activation results in weakening of the favourable dissipation provided by its inherent HLL scheme and causes unphysical mass flux variations along the shock front. To mitigate this, we propose a modified HLLC scheme that employs a simple differentiable pressure based multidimensional shock sensor to achieve smooth control of these critical antidiffusive terms near shocks. Using a linear perturbation analysis and a matrix based stability analysis, we establish that the resulting scheme, called HLLC-ADC (Anti-Diffusion Control), is shock stable over a wide range of freestream Mach numbers. Results from standard numerical test cases demonstrate that the HLLC-ADC scheme is indeed free from the most common manifestations of shock instability including the Carbuncle phenomenon without significant loss of accuracy on shear dominated viscous flows.