Generalized Riemann problem method for the Kapila model of compressible multiphase flows

The Kapila model, reduced from the Baer–Nunziato model, is becoming increasingly popular for numerical simulations of compressible multiphase flows. In the Kapila model, the volume fraction equation includes a stiff source term containing velocity divergence, posing significant challenges for numerical schemes. Most existing numerical methods employ Runge–Kutta time stepping strategies and Riemann solvers. The effect of the source term is not fully resolved in Riemann solutions. In this paper, a robust second-order finite volume scheme is developed for the Kapila model of compressible multiphase flows. The scheme is developed using the generalized Riemann problem (GRP) solver as the cornerstone. Besides Riemann solutions, the GRP solver provides time derivatives of flow variables at cell interfaces, achieving second-order accuracy in time within a single stage. The use of the GRP solver enhances the capability of the resulting scheme to handle the stiffness of the Kapila model in two ways. First, Riemann solutions and time derivatives give the cell interface values of flow variables at the new time level, yielding an approximation to the velocity divergence at the new time level in a computational step. This allows a semi-implicit time discretization to the stiff source term in the volume fraction equation. Second, the effect of the source term is included in numerical fluxes by considering time derivatives. The resulting numerical flux is able to capture the interactions between phases. The robustness of the scheme is, therefore, further improved. Several challenging numerical experiments are conducted to demonstrate the good performance of the proposed scheme.