Digital Object Identifier (DOI) 10.1007/s00211-004-0558-1 Numer. Math. (2005) 99: 411–440 Numerische Mathematik A relaxation method for two-phase flow models with hydrodynamic closure law Micha¨ el Baudin 1 , Christophe Berthon 2 , Fr´ ed´ eric Coquel 3 , Roland Masson 1 , Quang Huy Tran 1 1 IFP, 1 et 4 avenue de Bois-Pr´ eau, 92852 Rueil-Malmaison Cedex, France 2 MAB, Universit´ e de Bordeaux I, 351 cours de la Lib´ eration, 33405 Talence Cedex, France 3 Lab. J.-L. Lions, Universit´ e Pierre et Marie Curie, Boˆ ıte courrier 187, 75252 Paris Cedex 5 Received September 26, 2002 / Revised version received October 31, 2003 Published online: November 26, 2004 – c  Springer-Verlag 2004 Summary. This paper is devoted to the numerical approximation of the solu- tions 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 non- linearities 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 evi- dences are provided. Mathematics Subject Classification (1991): 76T10, 76N15, 35L65, 65M06 Introduction The purpose of petroleum pipelines is to convey a mixing, made up essen- tially of gas and liquid, over a long distance. In the realm of two-phase flows, there is a wide variety of mathematical models [2, 19, 23, 25]. Those we shall Correspondence to: Quang Huy Tran