The paper presents a new approach of stability evaluation of the approximate Riemann solvers based on the direct Lyapunov method. The present methodology offers a detailed understanding of the origins of numerical shock instability in the approximate Riemann solvers. The pressure perturbation feeding the density and transverse momentum perturbations is identified as the cause of the numerical shock instabilities in the complete approximate Riemann solvers, while the magnitude of the numerical shock instabilities are found to be proportional to the magnitude of the pressure perturbations. A shock-stable HLLEM scheme is proposed based on the insights obtained from this analysis about the origins of numerical shock instability in the approximate Riemann solvers. A set of numerical test cases are solved to show that the proposed scheme is free from numerical shock instability problems of the original HLLEM scheme at high Mach numbers.

翻译：本文提出了一种基于直接李雅普诺夫方法的近似黎曼求解器稳定性评估新方法。该方法能够深入理解近似黎曼求解器中数值激波不稳定性的来源。研究发现，压力扰动对密度和横向动量扰动的反馈是完整近似黎曼求解器中数值激波不稳定性的诱因，且数值激波不稳定性的大小与压力扰动幅度成正比。基于上述关于近似黎曼求解器数值激波不稳定性起源的分析，本文提出了一种激波稳定的HLLEM格式。通过求解一系列数值测试算例表明，所提格式在高马赫数下能够完全避免原始HLLEM格式的数值激波不稳定性问题。