Numerical approximation for a Baer–Nunziato model of two-phase flows

We present a well-balanced numerical scheme for approximating the solution of the Baer–Nunziato model of two-phase flows by balancing the source terms and discretizing the compaction dynamics equation. First, the system is transformed into a new one of three subsystems: the first subsystem consists of the balance laws in the gas phase, the second subsystem consists of the conservation law of the mass in the solid phase and the conservation law of the momentum of the mixture, and the compaction dynamic equation is considered as the third subsystem. In the first subsystem, stationary waves are used to build up a well-balanced scheme which can capture equilibrium states. The second subsystem is of conservative form and thus can be numerically treated in a standard way. For the third subsystem, the fact that the solid velocity is constant across the solid contact suggests us to compose the technique of the Engquist–Osher scheme. We show that our scheme is capable of capturing exactly equilibrium states. Moreover, numerical tests show the convergence of approximate solutions to the exact solution.