Discrete equations for physical and numerical compressible multiphase mixtures

We have recently proposed, in [21], a compressible two-phase unconditionally hyperbolic model able to deal with a wide range of applications: interfaces between compressible materials, shock waves in condensed multiphase mixtures, homogeneous two-phase flows (bubbly and droplet flows) and cavitation in liquids. One of the difficulties of the model, as always in this type of physical problems, was the occurrence of non-conservative products. In [21], we have proposed a discretisation technique that was without any ambiguity only in the case of material interfaces, not in the case of shock waves. This model was extended to several space dimensions in [24], In this paper, thanks to a deeper analysis of the model, we propose a class of schemes that are able to converge to the correct solution even when shock waves interact with volume fraction discontinuities. This analysis provides a more accurate estimate of closure terms, but also an accurate resolution method for the conservative fluxes as well as non-conservative terms even for situations involving discontinuous solutions. The accuracy of the model and method is clearly demonstrated on a sequence of difficult test problems.