A discussion on numerical shock stability of unstructured finite volume method: Riemann solvers and limiters

TL;DR

This paper analyzes the causes of numerical shock instability in unstructured finite volume methods, focusing on Riemann solvers and slope limiters, and proposes a pressure-based shock indicator to improve stability without sacrificing accuracy.

Contribution

It provides an in-depth discussion of shock instability mechanisms and introduces a pressure-based shock indicator to balance dissipation and accuracy in shock computations.

Findings

Pressure-based shock indicator effectively locates shock waves.

Extra dissipation reduces instability but may decrease accuracy.

Proper dissipation tuning improves stability and accuracy.

Abstract

Numerical shock instability is a complexity which may occur in supersonic simulations. Riemann solver is usually the crucial factor that affects both the computation accuracy and numerical shock stability. In this paper, several classical Riemann solvers are discussed, and the intrinsic mechanism of shock instability is especially concerned. It can be found that the momentum perturbation traversing shock wave is a major reason that invokes instability. Furthermore, slope limiters used to depress oscillation across shock wave is also a key factor for computation stability. Several slope limiters can cause significant numerical errors near shock waves, and make the computation fail to converge. Extra dissipation of Riemann solvers and slope limiters can be helpful to eliminate instability, but reduces the computation accuracy. Therefore, to properly introduce numerical dissipation is critical for numerical computations. Here, pressure based shock indicator is used to show the position of shock wave and tunes the numerical dissipation. Overall, the presented methods are showing satisfactory results in both the accuracy and stability.