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Discrete equations for physical and numerical compressible multiphase mixtures. (English) Zbl 1072.76594
Summary: We have recently proposed, in [J. Comput. Phys. 150, No. 2, 425–467 (1999; Zbl 0937.76053)], 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 the above cited paper, 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 [R. Saurel and O. Lemetayer, J. Fluid Mech. 431, 239–271 (2001; Zbl 1039.76069)]. 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.

76M20 Finite difference methods applied to problems in fluid mechanics
76L05 Shock waves and blast waves in fluid mechanics
76N15 Gas dynamics (general theory)
76T99 Multiphase and multicomponent flows
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
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