A positive and entropy-satisfying finite volume scheme for the Baer-Nunziato model

We present a relaxation scheme for approximating the entropy dissipating weak solutions of the BaerNunziato two-phase flow model. This relaxation scheme is straightforwardly obtained as an extension of the relaxation scheme designed in [16] for the isentropic Baer-Nunziato model and consequently inherits its main properties. Up to our knowledge, this is the only existing scheme for which the approximated phase fractions, phase densities and phase pressures are proven to remain positive without any restrictive condition other than a classical fully computable CFL condition. It is also the only scheme for which a discrete entropy inequality is proven, under a CFL condition derived from the natural sub-characteristic condition associated with the relaxation approximation. These two properties of the numerical scheme (discrete positivity and entropy inequality) are satisfied for any admissible equation of state. We provide a numerical study for the convergence of the approximate solutions towards some exact Riemann solutions. The numerical simulations show a higher precision and a more reduced computational cost (for comparable accuracy) than standard numerical schemes used in the nuclear industry. We also assess the good behavior of the scheme when approximating vanishing phase solutions. Key-words : Two-phase flows, energy-entropy duality, entropy-satisfying methods, relaxation techniques, Riemann problem. AMS subject classifications : 76T05, 35L60, 35F55.