A robust numerical method for approximating solutions of a model of two-phase flows and its properties

The objective of the present paper is to extend our earlier works on simpler systems of balance laws in nonconservative form such as the model of fluid flows in a nozzle with variable cross-section to a more complicated system consisting of seven equations which has applications in the modeling of deflagration-to-detonation transition in granular materials. First, we transform the system into an equivalent one which can be regarded as a composition of three subsystems. Then, depending on the characterization of each subsystem, we propose a convenient numerical treatment of the subsystem separately. Precisely, in the first subsystem of the governing equations in the gas phase, stationary waves are used to absorb the nonconservative terms into an underlying numerical scheme. In the second subsystem of conservation laws of the mixture we can take a suitable scheme for conservation laws. For the third subsystem of the compaction dynamics equation, the fact that the velocities remain constant across solid contacts suggests us to employ the technique of Engquist-Osher's scheme. Then, we prove that our method possesses some interesting properties: it preserves the positivity of the volume fractions in both phases, and in the gas phase, our scheme is capable of capturing equilibrium states, preserves the positivity of the density, and satisfies the numerical minimum entropy principle. Numerical tests show that our scheme can provide reasonable approximations for data the supersonic regions, but the results are not satisfactory in the subsonic region. However, the scheme is numerically stable and robust.