  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.
