Semi-implicit multiresolution for multiphase flows

In the context of multiphase flows we are faced with vector PDE solutions combining waves whose speeds are several orders of magnitude apart. The wave of interest is the transport one, and is relatively slow. The other fast acoustic waves are not interesting but impose a very restrictive CFL condition if a fully explicit in time scheme is considered. We therefore use a time semi-implicit conservative scheme where the fast waves are handled with a linearized implicit formulation and the slow wave remains explicitly solved. The CFL condition, governed by the explicit wave speed is then optimal. We combine this method with a multiscale analysis of the vector solution which enables to use a time varying adaptive grid based on the relevant smoothness properties of the discrete solution. In this short paper we compare different strategies to evaluate the fluxes at cells interfaces on a non uniform grid.