A new all-speed flux scheme for the Euler equations

Abstract

Since being proposed, the HLLEM-type schemes have been widely used because they are with high discontinuity resolutions and can be easily applied to the other system of hyperbolic conservation law. In this paper, we conduct theoretical analyses on the HLLE-type schemes’ performances at low speeds. By realizing that the excessive numerical dissipations corresponding to the velocity-difference terms of the momentum equations make these schemes incapable of obtaining physical solutions at low speeds, we adopt the function g to control such dissipation. Also, we borrow the HLLEMS scheme’s construction and damp the shear waves in the vicinity of the shock to avoid the shock anomaly’s appearance. The moving contact discontinuity case and the Sod shock tube case show that the HLLEMS-AS scheme we propose in this paper can capture contact discontinuities and shocks as sharply as HLLEMS scheme. The Quirk’s odd–even test case and the hypersonic inviscid flow over a cylinder case demonstrate that HLLEMS-AS is robust against the shock anomaly. The inviscid low-speed flow around the NACA0012 airfoil case indicates that HLLEMS-AS is with a high resolution at low speeds. The turbulent flow past a backward facing step case demonstrates the shear wave capturing ability of the HLLEMS-AS scheme. These properties suggest that HLLEMS-AS is promising to be widely used in both cases of low speed and high speed.