Will be set by the publisher STAGGERED SCHEMES FOR ALL SPEED FLOWS

We present in this paper a class of schemes for the numerical simulation of compressible flows. In order to ensure the stability of the discretizations in a wide range of Mach numbers and introduce sufficient decoupling for the numerical resolution, we choose to implement and study pressure correction schemes on staggered meshes. The implicit version of the schemes is also considered for the theoretical study. We give both algorithms for the barotropic Navier Stokes equations, for the full Navier-Stokes equations and for the Euler equations. In each case, we summarize the theoretical results that were recently obtained concerning the stability and consistency of the schemes and present some numerical results which confirm their good performance. Introduction The aim of this paper is to review some recent contributions to the development of a class of schemes for the simulation of industrial compressible flows. The considered systems of governing equations are coupled and strongly nonlinear, and the applications in view involve complex geometry and flows, possibly combining quasi-steady states with quick transient phases, with strong physical properties contrasts (in particular, for the viscosity, the compressibility and the density). Accordingly, the algorithms are developed so as to realize a compromise between two main requirements: preserve the stability in a wide range of Mach numbers and introduce sufficient decoupling to facilitate the resolution of discrete algebraic systems. Pressure correction methods seem to be a good choice to address these requirements. This class of schemes was first introduced in the framework of incompressible flows a long time ago [6, 54], and such algorithms are now quite widespread and well understood in this context (see, for example, [44] for an introduction and [20] for a review of most of the variants). Pressure correction schemes are less popular in the context of compressible flows, even though their application to compressible Navier-Stokes equations may also be traced back to the late sixties, with the seminal work of Harlow and Amsden [25,26], who developed an iterative algorithm (the so-called ICE method) including an elliptic corrector step for the pressure. Later on, pressure correction equations appeared in numerical schemes proposed by several researchers, essentially in the finite-volume framework, using either a collocated [11, 35, 38, 45, 50, 52] or a staggered arrangement [3, 5, 8, 33, 34,36,56,57,59–61] of unknowns; in the first case, some corrective actions are to be foreseen to avoid the usual odd-even decoupling of the pressure in the low Mach number regime. Some of these algorithms are essentially implicit, the end-of-step solution being then obtained by SIMPLE-like iterative processes [11,35,36,38,45,52,58]. The other schemes [3,8,33,34,50,56,57,59,61,62] are predictor-corrector methods, where basically two steps are performed sequentially: first a semi-explicit decoupled prediction of the momentum or velocity (and possibly 1 Université d’Aix-Marseille, herbin@cmi.univ-mrs.fr 2 Institut de Radioprotection et de Sûreté Nucléaire, kheriji.walid@gmail.com 3 Institut de Radioprotection et de Sûreté Nucléaire, jean-claude.latche]@irsn.fr c © EDP Sciences, SMAI 2011 ha l-0 06 36 17 3, v er si on 1 26 O ct 2 01 1