A well-balanced all-Mach scheme for compressible two-phase flow

We present an implicit-explicit finite volume scheme for the compressible two-phase model in all-Mach number regimes. In order to solve model equations efficiently and accurately in the low Mach regime, the convective term is split in a stiff part associated to fast acoustic waves, and a non-stiff part corresponding to mean flow advection. A Implicit-Explicit Runge-Kutta (IMEX-RK) method is used to integrate the stiff part implicitly, while non-stiff terms are treated explicitly. This leads to a predictor-corrector scheme, where the contribution of pressure waves is accounted for by solving a system of non-linear elliptic equations for the phasic pressures. The resulting numerical scheme is well-balanced and stable under a CFL condition based on the macroscopic velocity only and is capable of simulating two-phase flows in both the incompressible limit and in the highly compressible regime. The asymptotic preserving property of the scheme is also proven.