A linearly implicit shock capturing scheme for compressible two-phase flows at all Mach numbers

Abstract

We present a semi-implicit solver for the solution of compressible two-phase flows governed by the Baer–Nunziato model. A novel linearly implicit discretization is proposed for the pressure fluxes as well as for the relaxation source terms, whereas an explicit scheme is retained for the nonlinear convective contributions. Consequently, the CFL-type stability condition on the maximum admissible time step is based only on the mean flow velocity and not on the sound speed of each phase, so that the novel scheme works uniformly for all Mach numbers. Central finite difference operators on Cartesian grids are adopted for the implicit terms, thus avoiding any need of numerical diffusion that might destroy accuracy in the low Mach number regime. To comply with high Mach number flows, shock capturing finite volume schemes are employed for the approximation of the convective fluxes. The discretization of the non-conservative terms ensures the preservation of moving equilibrium solutions, making the new method well-balanced. The new scheme is also proven to be asymptotic preserving in the low Mach limit of the mixture model. Second order of accuracy is achieved by means of an implicit-explicit (IMEX) time stepping algorithm combined with a total variation diminishing (TVD) reconstruction technique. The novel method is benchmarked against a set of test cases involving different Mach number regimes, permitting to validate both accuracy and robustness.