A Genuinely Multidimensional HLL Riemann Solver Based on AUSM Splitting

A genuinely multidimensional HLL Riemann solver was given. The flux vector of the Euler equations was split into convection and pressure parts. The convection part was evaluated by using the upwind scheme and the pressure part was evaluated by using a modified HLL scheme. In the modified HLL scheme, the pressure difference was replaced by density difference in the dissipative term in order to capture the contact accurately. In order to obtain the genuinely multidimensional property, the numerical fluxes at the midpoint and the two corners of the cell interface were evaluated respectively, and the Simpson rule was used to obtain the final numerical flux through the interface. The linear reconstruction using SDWLS gradients was used for second order spatial accuracy, and the time derivative was discretized using the second order strong stability preserving Runge-Kutta method. Compared with the traditional one dimensional HLL scheme, the genuinely multidimensional HLL scheme can capture the contact discontinuity, and can use larger time step. Unlike other schemes which can capture the contact discontinuity accurately such as the HLLC scheme, the genuinely multidimensional HLL scheme eliminates the phenomena of numerical shock instability in 2D cases.