A Low Cost Semi-implicit Low-Mach Relaxation Scheme for the Full Euler Equations

We introduce a semi-implicit two-speed relaxation scheme to solve the compressible Euler equations in the low Mach regime. The scheme involves a relaxation system with two speeds, already introduced by Bouchut et al. (Numer Math, 2020. https://doi.org/10.1007/s00211-020-01111-5) in the barotropic case. It is entropy satisfying and has a numerical viscosity well-adapted to low Mach flows. This relaxation system is solved via a dynamical Mach number dependent splitting, similar to the one proposed by Iampietro et al. (J Comput Appl Math 340:122–150, 2018). Stability conditions are derived, they limit the range of admissible relaxation and splitting parameters. We resolve separately the advection part of the splitting by an explicit method, and the acoustic part by an implicit method. The relaxation speeds are chosen so that the implicit system fully linearizes the acoustics and requires just to invert an elliptic operator with constant coefficients. The scheme is shown to well capture with low cost the incompressible slow scale dynamics with a timestep adapted to the velocity field scale, and rather well the fast acoustic waves.