We are interested in the numerical approximation of the solutions of the compressible seven-equation two-phase flow model. We propose a numerical srategy based on the derivation of a simple, accurate and explicit approximate Riemann solver. The source terms associated with the external forces and the drag force are included in the definition of the Riemann problem, and thus receive an upwind treatment. The objective is to try to preserve, at the numerical level, the asymptotic property of the solutions of the model to behave like the solutions of a drift-flux model with an algebraic closure law when the source terms are stiff. Numerical simulations and comparisons with other strategies are proposed.

Abstract. This paper is concerned with the numerical approximation of the solutions of a two-fluid two-pressure model used in the modelling of two-phase flows. We present a relaxation strategy for easily dealing with both the nonlinearities associated with the pressure laws and the nonconservative terms that are inherently present in the set of convective equations and that couple the two phases. In particular, the proposed approximate Riemann solver is given by explicit formulas, preserves the natural phase space, and exactly captures the coupling waves between the two phases. Numerical evidences are given to corroborate the validity of our approach.Mathematics Subject Classification. 76T10, 35L60, 76M12.

Abstract. This work is devoted to the coupling of two fluid models, such as two Euler systems in Lagrangian coordinates, at a fixed interface. We define coupling conditions which can be expressed in terms of continuity of some well chosen variables and then solve the coupled Riemann problem. In the present setting where the interface is characteristic, a particular choice of dependent variables which are transmitted ensures the overall conservativity.