Stability evaluation of approximate Riemann solvers using the direct Lyapunov method

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 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 is 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.