Development of an accurate and robust numerical scheme for simulating all Mach number flows

Abstract

The HLLEM scheme is a popular numerical method for computing the convective fluxes in the Euler equations and the Navier–Stokes equations due to its accuracy and positivity-preserving. However, it still has two defects. One is the numerical instability, such as the carbuncle phenomenon, in computations involving multidimensional strong shock waves; the other is the non-physical results caused by accuracy issues in computations of low Mach number flows. In the engineering field, high Mach number flows with strong shock waves and low Mach number flows often coexist. Therefore, it is necessary to develop a numerical scheme that is both accurate and robust across all Mach number flows. The shock instabilities of the HLLEM scheme are addressed by simply modifying the wave speeds to balance advection dissipation and acoustic dissipation. Additionally, the performance in computing low-speed flows is improved by controlling the excessive numerical dissipation corresponding to the velocity-difference terms in the momentum equations. Numerical results from high Mach number test cases and low Mach number test cases demonstrate the accuracy and robustness of the proposed scheme for simulating flows across all Mach number.