Cures for the Expansion Shock and the Shock Instability of the Roe Scheme

TL;DR

This paper introduces a new Roe scheme that effectively suppresses both shock instability and expansion shock, overcoming limitations of previous methods and maintaining simplicity and low computational cost.

Contribution

A novel Roe scheme is developed that simultaneously cures shock instability and expansion shock without additional computational costs.

Findings

The new scheme is concise and easy to implement.

It effectively suppresses shock-related issues.

The scheme maintains low computational cost.

Abstract

A common defect of the Roe scheme is the production of non-physical expansion shock and shock instability. An improved method with several advantages was presented to suppress the shock instability. However, this method cannot prevent expansion shock and is incompatible with the traditional curing method for expansion shock. Therefore, the traditional curing mechanism is analyzed. The discussion explains the effectiveness of the traditional curing method and identifies several defects, one of which leads to incompatibility between curing the shock instability and expansion shock. Consequently, a new improved Roe scheme is proposed in this study. This scheme is concise, easy to implement, low computational cost, and robust. More importantly, the scheme can simultaneously cure the shock instability and expansion shock without additional costs.