A new type of approximate Riemann solver with adaptive anti-diffusion and its application

Constructing numerical fluxes using Riemann solvers is one of the most important steps for solving hyperbolic systems of conservation laws. However, it is still of great challenge for existing Riemann solvers to simulate high-speed flows with shock waves. The Riemann solver based on the HLL approximation has a high dissipation, while those based on the Roe, HLLEM, and HLLC approximation may result in non-physical solutions and numerical instability when simulating complex flows with strong shock waves. Because of this, a new type of Riemann solver with adaptive anti-diffusion is developed by a dedicated design of the anti-diffusion matrix. Due to its low dissipation, the proposed Riemann solver yields high-accuracy solutions when applied to high-order weighted compact finite difference schemes. The standard numerical experiments verify computational accuracy and stability for the new method, which show that the new Riemann solver overcomes the shortcomings of the traditional approximate Riemann solver. The new Riemann solver can accurately identify the fine structures such as shear layers, and maintain the numerical stability for the computation of shock wave.