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This article is the first of two in which we develop a relaxation finitevolume scheme for the convective part of the multiphase flow models introducedin the series of papers [10, 9, 4]. In the present article we focus onbarotropic flows where in each phase the pressure is a given function of thedensity. The case of general equations of state will be the purpose of thesecond article. We show how it is possible to extend the relaxation schemedesigned in [8] for the barotropic Baer-Nunziato two-phase flow model to themultiphase flow model with N-where N is arbitrarily large-phases. The obtainedscheme inherits the main properties of the relaxation scheme designed for theBaer-Nunziato two phase flow model. The approximated phase fractions and phasedensities are proven to remain positive and a discrete energy inequality isalso proven under a classical CFL condition. For the same level of refinement,the relaxation scheme is shown to be much more accurate than Rusanov's scheme,and for a given level of approximation error, the relaxation scheme is shown toperform much better in terms of computational cost than Rusanov's scheme.Moreover, contrary to Rusanov's scheme which develops strong oscillations whenapproximating vanishing phase solutions, the numerical results show that therelaxation scheme remains stable in such regimes.

A relaxation scheme for a hyperbolic multiphase flow model. Part I: barotropic eos