L²Roe: A low-dissipation version of Roe's approximate Riemann solver for low Mach numbers

Abstract

A modification of the Roe scheme aimed at low Mach number flows is discussed. It improves the dissipation of kinetic energy at the highest resolved wave numbers in a low Mach number test case of decaying isotropic turbulence. This is done by scaling the jumps in all discrete velocity components within the numerical flux function. An asymptotic analysis is used to show the correct pressure scaling at low Mach numbers and to identify the reduced numerical dissipation in that regime, both of the new method $L^2$Roe, as well as a method previously suggested by other authors. Furthermore, the analysis allows comparisons of these methods with a further low Mach scheme, LMRoe. No conflict is observed between the reduced dissipation and the accuracy or stability of the scheme in any of the investigated test cases ranging from low Mach number potential to hypersonic viscous flow. Furthermore, a comparison with the two other methods shows advantages of the new approach.