Explicit staggered schemes for the compressible euler equations

ESAIM: PROCEEDINGS, Juillet 2013, Vol. 40, p. 83-102 C. Bourdarias, S. Gerbi, Editors EXPLICIT STAGGERED SCHEMES FOR THE COMPRESSIBLE EULER EQUATIONS R. Herbin 1 , J.-C. Latché 2 and T.T. Nguyen 3 Abstract. We review in this paper an explicit scheme for the numerical simulation of inviscid compressible flows; we analyze it for both the barotropic Euler equations and the full Euler equations for an ideal gas. 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. 1. Introduction The objective pursued in this work is to develop and study, from a theoretical point of view, an explicit scheme for the simulation of non viscous compressible flows, modeled either by the barotropic Euler equations or by the full Euler equations for an ideal gas. Our basic choice is to use an explicit variant of implicit and semi-implicit schemes that were developed and studied recently in the framework of the simulation of compressible flows at all speeds [6, 10, 11, 13]; in these latter works, the implicit scheme is studied as a first step in the mathematical analysis of pressure correction schemes, which extend algorithms that are classical in the incompressible framework; these are based on (inf-sup stable) staggered discretizations. In our approach, the upwinding techniques which are implemented for stability reasons are performed for each equation separately and with respect to the material velocity only. This is in contradiction with the most common strategy adopted for hyperbolic systems, where upwinding is built from the wave structure of the system (see e.g. [2, 22] for surveys). However, it yields algorithms which are used in practice (see e.g. the so-called AUSM family of schemes [18, 19]), because of their generality (a closed-form solution of Riemann problems is not needed), their implementation simplicity and their efficiency, thanks to an easy construction of the fluxes at the cell faces. But these schemes are scarcely studied from a theoretical point of view; one of our main concerns here will thus be to bring, as far as possible, theoretical arguments supporting our numerical developments. We give in this paper a review of the results obtained for the explicit version of the schemes in the case of the (inviscid) Euler equations, and refer to [12] for a review of the results of the implicit and semi–implicit versions, to [10, 11, 15, 16, 20] for the detailed proofs of the results, and to [7] for the implementation of the pressure correction scheme in the case of a drift-diffusion model for two phase flows. The paper is organized as follows. We start by the description of the staggered meshes which are used for the discretization in space, using either a finite volume – non-conforming finite element or a full “MAC-type” 1 Université d’Aix-Marseille, 2 Institut de Radioprotection et de Sûreté Nucléaire, jean-claude.latche]@irsn.fr 3 Institut de Radioprotection et de Sûreté Nucléaire, c EDP Sciences, SMAI 2013 Article published online by EDP Sciences and available at http://www.esaim-proc.org or http://dx.doi.org/10.1051/proc/201340006 84 ESAIM: PROCEEDINGS finite volume scheme. We then study the scheme for the barotropic Euler equations in Section 3, and for the full Euler equations in Section 4, and give some numerical results in Section 5. 2. Meshes and unknowns Let Ω, the computational domain, be an open bounded connected subset of Rd , 1 ≤ d ≤ 3, of boundary ∂Ω. In this section, we focus on the definition of the space discretization in the multidimensional cases, the adaptation to the 1D case being straightforward (see Section 3.3). Let M be a decomposition of Ω, supposed to be regular in the usual sense of the finite element literature (e.g. [3]). The cells may be: - for a general domain Ω, either convex quadrilaterals (d = 2) or hexahedra (d = 3) or simplices, both type of cells being possibly combined in a same mesh, - for a domain the boundaries of which are hyperplanes normal to a coordinate axis, rectangles (d = 2) or rectangular parallelepipeds (d = 3) (the faces of which, of course, are then also necessarily normal to a coordinate axis). By E and E(K) we denote the set of all (d − 1)-faces σ of the mesh and of the element K ∈ M respectively. The set of faces included in the boundary of Ω is denoted by Eext and the set of internal edges (i.e. E \ Eext ) is denoted by Eint ; a face σ ∈ Eint separating the cells K and L is denoted by σ = K|L. The outward normal vector to a face σ of K is denoted by nK,σ . For K ∈ M and σ ∈ E, we denote by |K| the measure of K and (i) by |σ| the (d − 1)-measure of the face σ. For 1 ≤ i ≤ d, we denote by E (i) ⊂ E and Eext ⊂ Eext the subset of the th faces of E and Eext respectively which are perpendicular to the i unit vector of the canonical basis of Rd . The space discretization is staggered, using either the Marker-And Cell (MAC) scheme [8,9], or nonconforming low-order finite element approximations, namely the Rannacher and Turek (RT) element [21] for quadrilateral or hexahedric meshes, or the lowest degree Crouzeix-Raviart (CR) element [4] for simplicial meshes. For all these space discretizations, the degrees of freedom for the scalar variables (i.e. the discrete pressure and density unknowns in the barotropic case and the discrete pressure, density and internal  energy unknowns for the Euler equations) are associated to the cells of the mesh M. They are denoted by pK , ρK , K ∈ M  (barotropic case) or pK , ρK , eK , K ∈ M (Euler equations). Let us then turn to the degrees of freedom for the velocity (i.e. the discrete velocity unknowns). - Rannacher-Turek or Crouzeix-Raviart discretizations – The degrees of freedom for the velocity components are located at the center of the faces of the mesh, and we choose the version of the element where they represent the average of the velocity through a face. The set of degrees of freedom reads: {uσ,i , σ ∈ E, 1 ≤ i ≤ d}. - MAC discretization – The degrees of freedom for the ith component of the velocity are defined at the centre of the faces of E (i) , so the whole set of discrete velocity unknowns reads:  uσ,i , σ ∈ E (i) , 1 ≤ i ≤ d . We now introduce a dual mesh, which will be used for the finite volume approximation of the time derivative and convection terms in the momentum balance equation. - Rannacher-Turek or Crouzeix-Raviart discretizations – For the RT or CR discretizations, the dual mesh is the same for all the velocity components. When K ∈ M is a simplex, a rectangle or a cuboid, for σ ∈ E(K), we define DK,σ as the cone with basis σ and with vertex the mass center of K (see Figure 1). We thus obtain a partition of K in m sub-volumes, where m is the number of faces of the mesh, each sub-volume having the same measure |DK,σ | = |K|/m. We extend this definition to general quadrangles 85 EXPLICIT STAGGERE (...truncated)