Closure laws for interfacial pressure and interfacial velocity are proposed within the frame of two-pressure two-phase flow models. These enable to ensure positivity of void fractions, mass fractions and internal energies when investigating field by field waves in the Riemann problem. Lois de fermeture pour un modèleà deux pressions d'écoulement diphasique Résumé. On propose des lois de fermeture de vitesse et de pression d'interface pour un modèle d'écoulement diphasiqueà deux pressions. Celles-ci assurent champ par champ de respecter la positivité des fractions volumiques, des variables densité eténergie interne si on examine le problème de Riemann.
This paper is devoted to the numerical approximation of the solutions of a system of conservation laws arising in the modeling of two-phase flows in pipelines. The PDEs are closed by two highly nonlinear algebraic relations, namely, a pressure law and a hydrodynamic one. The severe nonlinearities encoded in these laws make the classical approximate Riemann solvers virtually intractable at a reasonable cost of evaluation. We propose a strategy for relaxing solely these two nonlinearities. The relaxation system we introduce is of course hyperbolic but all associated eigenfields are linearly degenerate. Such a property not only makes it trivial to solve the Riemann problem but also enables us to enforce some further stability requirements, in addition to those coming from a Chapman-Enskog analysis. The new method turns out to be fairly simple and robust while achieving desirable positivity properties on the density and the mass fractions. Extensive numerical evidences are provided.
In this paper, an L°°(Ll)-error estimate for a class of finite volume methods for the approximation of scalar multidimensional conservation laws is obtained. These methods can be formally high-order accurate and are defined on general triangulations. The error is proven to be of order ft'/4 , where h represents the "size" of the mesh, via an extension of Kuznetsov approximation theory for which no estimate of the total variation and of the modulus of continuity in time are needed. The result is new even for the finite volume method constructed from monotone numerical flux functions.
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