A robust entropy−satisfying finite volume scheme for the isentropic Baer−Nunziato model

¹ CMAP, École Polytechnique CNRS, UMR 7641, Route de Saclay, 91128 Palaiseau Cedex, France.
² EDF-R&D, Département MFEE, 6 Quai Watier, 78401 Chatou Cedex, France.
³ UPMC Univ Paris 06, UMR 7598, Laboratoire Jacques-Louis Lions, 75005, Paris, France.
saleh@ann.jussieu.fr
⁴ CNRS, UMR 7598, Laboratoire Jacques-Louis Lions, 75005, Paris, France.
⁵ Inria Paris-Rocquencourt, BP 105, 78153 Le Chesnay Cedex, France.

Received: 25 June 2012
Revised: 1 March 2013

Abstract

We construct an approximate Riemann solver for the isentropic Baer−Nunziato two-phase flow model, that is able to cope with arbitrarily small values of the statistical phase fractions. The solver relies on a relaxation approximation of the model for which the Riemann problem is exactly solved for subsonic relative speeds. In an original manner, the Riemann solutions to the linearly degenerate relaxation system are allowed to dissipate the total energy in the vanishing phase regimes, thereby enforcing the robustness and stability of the method in the limits of small phase fractions. The scheme is proved to satisfy a discrete entropy inequality and to preserve positive values of the statistical fractions and densities. The numerical simulations show a much higher precision and a more reduced computational cost (for comparable accuracy) than standard numerical schemes used in the nuclear industry. Finally, two test-cases assess the good behavior of the scheme when approximating vanishing phase solutions.