A relaxation-projection method for compressible flows. Part I: The numerical equation of state for the Euler equations Richard Saurel * , Erwin Franquet, Eric Daniel, Olivier Le Metayer Polytech’Marseille, University Institute of France, Universite ´ de Provence and SMASH Project UMR CNRS 6595 – IUSTI-INRIA, 5 rue E. Fermi, 13453 Marseille Cedex 13, France Received 30 September 2005; received in revised form 27 September 2006; accepted 4 October 2006 Available online 27 November 2006 Abstract A new projection method is developed for the Euler equations to determine the thermodynamic state in computational cells. It consists in the resolution of a mechanical relaxation problem between the various sub-volumes present in a com- putational cell. These sub-volumes correspond to the ones traveled by the various waves that produce states with different pressures, velocities, densities and temperatures. Contrarily to Godunov type schemes the relaxed state corresponds to mechanical equilibrium only and remains out of thermal equilibrium. The pressure computation with this relaxation pro- cess replaces the use of the conventional equation of state (EOS). A simplified relaxation method is also derived and pro- vides a specific EOS (named the Numerical EOS). The use of the Numerical EOS gives a cure to spurious pressure oscillations that appear at contact discontinuities for fluids governed by real gas EOS. It is then extended to the compu- tation of interface problems separating fluids with different EOS (liquid–gas interface for example) with the Euler equa- tions. The resulting method is very robust, accurate, oscillation free and conservative. For the sake of simplicity and efficiency the method is developed in a Lagrange-projection context and is validated over exact solutions. In a companion paper [F. Petitpas, E. Franquet, R. Saurel, A relaxation-projection method for compressible flows. Part II: computation of interfaces and multiphase mixtures with stiff mechanical relaxation. J. Comput. Phys. (submitted for publication)], the method is extended to the numerical approximation of a non-conservative hyperbolic multiphase flow model for interface computation and shock propagation into mixtures. Ó 2006 Elsevier Inc. All rights reserved. Keywords: Primitive variables computation; Interfaces; Real gases; Multiphase flows; Riemann problem 0. Introduction The Godunov method, its extensions and approximate versions, is the most popular method to solve hyperbolic systems of conservation laws. However, inaccuracies and even computational failure appear 0021-9991/$ - see front matter Ó 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.jcp.2006.10.004 * Corresponding author. Tel.: +339 128 8511; fax: +339 128 8322. E-mail address: Richard.Saurel@polytech.univ-mrs.fr (R. Saurel). Journal of Computational Physics 223 (2007) 822–845 www.elsevier.com/locate/jcp