Pressure correction staggered schemes for barotropic one-phase and two-phase flows

We assess in this paper the capability of a pressure correction scheme to compute shock solutions of the homogeneous model for barotropic two-phase flows. This scheme is designed to inherit the stability properties of the continuous problem: the unknowns (in particular the density and the dispersed phase mass fraction y) are kept within their physical bounds, and the entropy of the system is conserved, thus providing an unconditional stability property. In addition, the scheme keeps the velocity and pressure constant through contact discontinuities. These properties are obtained by coupling the mass balance and the transport equation for y in an original pressure correction step. The space discretization is staggered; the numerical schemes which are considered are the Marker-And Cell (MAC) finite volume scheme and the nonconforming low-order Rannacher-Turek and Crouzeix-Raviart finite element approximation. In either case, a finite volume technique is used for all convection terms. Numerical experiments performed here show that, provided that a sufficient dissipation is introduced in the scheme, it converges to the (weak) solution of the continuous hyperbolic system. Observed orders of convergence for 1D Riemann problems as a function of the mesh and time step at constant CFL number vary with the studied case, and the CFL number and on the regularity of the solution. They range from 0.5 to greater than 1 for the velocity and the pressure; in most cases, the density and mass fraction converge with a 0.5 order. Finally, the scheme shows a satisfactory behaviour up to large CFL numbers.