Overcoming shock instability of the HLLE-type Riemann solvers.

• A new explanation to shock instability is proposed. • Some theoretical research results for viscous shock structure with Davis' wave velocity algorithm are provided. • A simple but effective cure strategy which is based on the insight of shock instability is suggested. This paper concentrates on numerical shock instabilities of some approximate Riemann solvers which stem from the HLLE method, including HLLC-type and HLLEM-type flux functions. Carbuncle phenomena are illustrated for these numerical schemes, especially for those believed to be carbuncle-free, such as the HLLE scheme. A linear matrix stability analysis shows that there exists a threshold independent of shock strength to trigger instability for a given numerical scheme. In this instability mechanism, a specific cell interface in the numerical structure of shock layer is regarded as the source of instability if the density ratio between both sides of this interface exceeds the threshold. The modification by adjusting nonlinear signal velocities in these numerical schemes can change solution behaviors in the shock layer, and reduce the degree of density transition on the specific interface. A very simple cure strategy for the HLLE-type Riemann solvers is presented based on the in-depth understanding of the carbuncle onset of viscous shock. [ABSTRACT FROM AUTHOR]