A low-Mach number fix for Roe’s approximate Riemann solver

We present a low-Mach number fix for Roe’s approximate Riemann solver (LMRoe). As the Mach number Ma tends to zero, solutions to the Euler equations converge to solutions of the incompressible equations. Yet, standard upwind schemes do not reproduce this convergence: the artificial viscosity grows like 1/Ma, leading to a loss of accuracy as Ma → 0. With a discrete asymptotic analysis of the Roe scheme we identify the responsible term: the jump in the normal velocity component ΔU of the Riemann problem. The remedy consists of reducing this term by one order of magnitude in terms of the Mach number. This is achieved by simply multiplying ΔU with the local Mach number. With an asymptotic analysis it is shown that all discrepancies between continuous and discrete asymptotics disappear, while, at the same time, checkerboard modes are suppressed. Low Mach number test cases show, first, that the accuracy of LMRoe is independent of the Mach number, second, that the solution converges to the incompressible limit for Ma → 0 on a fixed mesh and, finally, that the new scheme does not produce pressure checkerboard modes. High speed test cases demonstrate the fall back of the new scheme to the classical Roe scheme at moderate and high Mach numbers.