On the carbuncle instability of the HLLE-type solvers

Abstract

The carbuncle phenomenon is a numerical instability that affects the numerical capturing of shock waves when low-dissipative upwind scheme is used. This paper investigates shock instabilities of the HLLE-type methods for the Euler equations under the strong shock interaction, where the HLLE-type methods include the HLLE, HLLC, HLLEM, HLLCM and HLLEC Riemann solvers with specific wavespeed estimates. Based on a matrix stability analysis for two dimensional steady shocks, a new factor to influence carbuncle phenomenon is pointed out and the choice of the signal velocity plays an important role. A numerical flux function with wave velocity estimates which can crisply resolve shocks seems to be vulnerable to the shock anomalies even if the numerical fluxes to be regarded to be free from the carbuncle phenomenon. A suggestion to the choice of the wave speed is proposed when calculating strong shock wave problems.