On the Mechanism of Roe-type Schemes for All-Speed Flows

TL;DR

This paper analyzes the mechanisms behind Roe-type schemes for all-speed flows, identifying causes of accuracy and checkerboard issues, and proposes new schemes with improved stability and accuracy based on sound speed coefficient adjustments.

Contribution

It reveals the root causes of accuracy and checkerboard problems in Roe-type schemes and introduces novel schemes with optimized sound speed coefficients for better all-speed flow simulation.

Findings

Problems caused by sound speed coefficient choices are resolved by zero-order sound speed coefficients.

Local variable-based sound speed terms improve accuracy without sacrificing stability.

Numerical experiments validate the theoretical analysis and proposed rules.

Abstract

In recent years, Roe-type schemes based on different ideas have been developed for all-speed flows, such as the preconditioned Roe, the All-Speed Roe, Thornber's modified Roe and the LM-Roe schemes. This work explores why these schemes succeed or fail with the accuracy and checkerboard problems. Comparison and analysis show that the accuracy and checkerboard problems are caused by the order of the sound speed being too large and too small in the coefficients of the velocity-derivative and pressure-derivative dissipation terms, respectively. These problems can be resolved by choosing coefficients with zero-order sound speed. In addition, to avoid the negative effects of the global cut-off strategy on accuracy while maintaining computational stability, the sound speed terms in the numerator of the coefficients can be determined by local variables, while those in the denominator remain the global cut-off. Two novel schemes are proposed as examples to demonstrate how these ideas can be applied to construct more satisfactory schemes for all-speed flows. Asymptotic analysis and numerical experiments support the theoretical analysis and the rules obtained in the work.