Simulations of non homogeneous viscous flows with incompressibility constraints

This presentation is an overview on the development of numerical methods for the simulation of non homogeneous flows with incompressibility constraints. We are particularly interested in systems of partial differential equations describing certain mixture flows, like the Kazhikhov-Smagulov system which can be used to model powder-snow avalanches. It turns out that the Incompressible Navier-Stokes system with variable density is a relevant step towards the treatment of such models, and it allows us to bring out some interesting numerical difficulties. We should handle equations of different types, roughly speaking transport and diffusion equations. We present two strategies based on time-splitting. The former relies on a hybrid approach, coupling finite volume and finite element methods. The latter extends discrete duality finite volume schemes for such non homogeneous flows. The methods are confronted to exact solutions and to the simulation of Rayleigh-Taylor instabilities.