General Procedure for Riemann Solver to Eliminate Carbuncle and Shock Instability

A novel procedure for the Riemann solver flux calculation is proposed in this paper. With this simple normal velocity reconstruction procedure, all of the commonly used flux solvers (such as Godunov, Roe, Harten-Lax-van Leer-Contact, Advection-Upstream-Splitting-Method, etc.) turn out to be carbuncle-free and shock-stable. The normal velocity reconstruction procedure is done by a linear reconstruction of the cell interface normal velocity with the transverse neighbor cells, in consideration of the information transport in the transverse direction, which is neglected in the conventional finite volume/difference method. Some typical cases are performed to show that, when the normal velocity reconstruction procedure is used, various schemes (e.g., Roe) become carbuncle-free and shock-stable. In addition, the normal velocity reconstruction procedure has no influence on the contact-preserving property of the original flux solvers, and it adds very little computational cost. The mechanism of the normal velocity reconstruction procedure also is analyzed by a matrix-based stability method, and the results indicate that the normal velocity reconstruction procedure effectively reduces the system’s positive eigenvalues that are leading to shock instability.