A Mach-sensitive implicit–explicit scheme adapted to compressible multi-scale flows

The method presented below focuses on the numerical approximation of the Euler compressible system. It pursues a two-fold objective: being able to accurately follow slow material waves as well as strong shock waves in the context of low Mach number flows. The resulting implicit–explicit(IMEX) fractional step approach leans on a dynamic splitting designed to react to the time fluctuations of the maximal flow Mach number. When the latter rises suddenly, the IMEX scheme, so far driven by a material-wave Courant number, turn into a time-explicit approximate Riemann solver constrained by an acoustic-wave Courant number. It is also possible to enrich the dynamic splitting in order to capture high pressure jumps even when the flow Mach number is low. One-dimensional low Mach number test cases involving single or multiple waves confirm that the present approach is as accurate and efficient as an IMEX Lagrange-Projection method. Besides, numerical results suggest that the stability of the present method holds for any Mach number if the Courant number related to the convective subsystem arising from the splitting is of order unity.