Large time-step positivity-preserving method for multiphase flows F. Coquel 1 , Q. L. Nguyen 2 , M. Postel 1 and Q. H. Tran 2 1 Universit´ e Pierre et Marie Curie-Paris6, UMR 7598 LJLL, F-75005 France; CNRS, UMR 7598 LJLL, Paris, F-75005 France coquel@ann.jussieu.fr, postel@ann.jussieu.fr 2 D´ epartement Math´ ematiques Appliqu´ ees, Institut Fran¸cais du P´ etrole, 1 et 4 avenue de Bois-Pr´ eau, 92852 Rueil-Malmaison Cedex, France q-long.nguyen@ifp.fr, q-huy.tran@ifp.fr Summary. Using a relaxation strategy in a Lagrangian-Eulerian formulation, we propose a scheme in local conservation form for approximating weak solutions of complex compressible flows involving wave speeds of different orders of magnitude. Explicit time integration is performed on slow transport waves for the sake of accu- racy while fast acoustic waves are dealt with implicitly to enable large time stepping. A CFL condition based on the slow waves is derived ensuring positivity properties on the density and the mass fraction. Numerical benchmarks validate the method. 1 Statement of the problem The present work treats the numerical approximation of the discontinuous solutions of the following PDE system       ∂ t (ρ) + ∂ x (ρu) =0, ∂ t (ρY )+ ∂ x (ρY u) =0, ∂ t (ρu)+ ∂ x (ρu 2 + P )=0. (1) Here and with classical notations, ρ stands for the total density of a com- pressible material, u its velocity and Y a gas mass fraction. The pressure P is a given nonlinear function of the unknown u =(ρ, ρY, ρu), in the form P (τ,Y ) with τ =1/ρ, which in the present setting (oil production) turns to be highly nonlinear. For the closure laws P (u) to be dealt with, the system of conservation laws (1) is typically hyperbolic over Ω =  u ∈ IR 3 ; ρ> 0, 0 ≤ Y ≤ 1,u ∈ IR  . (2) The PDE system (1) governs solutions made up of distinct waves propagating with the next three distinct eigenvalues : u-c(u) <u<u+c(u) where the def- inition of the sound speed c(u) follows from the prescribed pressure law P (u).