A Uniform Algorithm for All-Speed Shock-Capturing Schemes

TL;DR

This paper introduces a unified framework for all-speed shock-capturing schemes, enabling better understanding, comparison, and analysis of their mechanisms, especially in low-Mach number flows.

Contribution

A novel uniform algorithm that unifies Roe, HLL, and AUSM schemes within a single framework, offering explicit physical interpretation and aiding further research.

Findings

Unified framework for shock-capturing schemes

Clear explanation of low-Mach number behavior

A simple rule to evaluate low-Mach performance

Abstract

There are many ideas for developing shock-capturing schemes and their extension for all-speed flow. The representatives of them are Roe, HLL and AUSM families. In this paper, a uniform algorithm is proposed, which expresses three families in the same framework. The algorithm has explicit physical meaning, provides a new angel of understanding and comparing the mechanism of schemes, and may play a great role in the further research. As an example of applying the uniform algorithm, the low-Mach number behaviour of the schemes is analyzed. Then, a very clear and simple explanation is given based on the wall boundary, and a concise rule is proposed to judge whether a scheme has satisfied low-Mach number behaviour.