We propose a numerical method adapted to the modelling of phase transitions in compressible fluid flows. Pressure laws taking into account phase transitions are complex and lead to difficulties such as the non-uniqueness of entropy solutions. In order to avoid these difficulties, we propose a projection finite volume scheme. This scheme is based on a Riemann solver with a simpler pressure law and an entropy maximization procedure that enables us to recover the original complex pressure law. Several numerical experiments are presented that validate this approach.

Abstract.In this work, we propose a general framework for the construction of pressure law for phase transition. These equations of state are particularly suitable for a use in a relaxation finite volume scheme. The approach is based on a constrained convex optimization problem on the mixture entropy. It is valid for both miscible and immiscible mixtures. We also propose a rough pressure law for modelling a super-critical fluid.Mathematics Subject Classification. 76M12, 65M12.

Abstract. In this paper we investigate algorithms based on the Fast Legendre Transform (FLT) in order to compute tabulated Equation Of State (EOS) for fluids with phase transition. The equation of state of a binary mixture is given by an energy minimization principle. According to the miscible or immiscible nature of the mixture, the energy of the system is either a convex envelope or an inf-convolution of the energies of the two phases. Because these operations are closely linked to Legendre Transform, it is possible to construct O(N ) algorithms that compute efficiently these operations, where N is the number of points in the table. In addition, it appears that the natural mathematical tool for studying mixture thermodynamics in the Legendre space is the max-plus algebra theory.