Nouvelles constructions de méthodes de volumes/éléments finis pour les écoulements transsoniques/supersoniques compressibles

Thèse numérisée par la Direction des bibliothèques de l'Université de Montréal

Entropy Stable Discontinuous Galerkin Finite Element Method with Multi-Dimensional Slope Limitation for Euler Equations

International audienceAbstract We present an entropy stable Discontinuous Galerkin (DG) finite element method to approximate systems of 2-dimensional symmetrizable conservation laws on unstructured grids. The scheme is constructed using a combination of entropy conservative fluxes and entropy-stable numerical dissipation operators. The method is designed to work on structured as well as on unstructured meshes. As solutions of hyperbolic conservation laws can develop discontinuities (shocks) in finite time, we include a multidimensional slope limitation step to suppress spurious oscillations in the vicinity of shocks. The numerical scheme has two steps: the first step is a finite element calculation which includes calculations of fluxes across the edges of the elements using 1-D entropy stable solver. The second step is a procedure of stabilization through a truly multi-dimensional slope limiter. We compared the Entropy Stable Scheme (ESS) versus Roe’s solvers associated with entropy corrections and Osher’s solver. The method is illustrated by computing solution of the two stationary problems: a regular shock reflection problem and a 2-D flow around a double ellipse at high Mach number

THREE-DIMENSIONAL ADAPTIVE CENTRAL SCHEMES ON UNSTRUCTURED STAGGERED GRIDS ∗

Abstract. We present an explicit second-order finite volume generalization of the onedimensional (1D) Nessyahu–Tadmor schemes for hyperbolic equations on adaptive unstructured tetrahedral grids. The nonoscillatory central difference scheme of Nessyahu and Tadmor, in which the resolution of the Riemann problem at the cell interfaces is bypassed thanks to the use of the staggered Lax–Friedrichs scheme, is extended here to a two-steps scheme. In order to reduce artificial viscosity, we start with an adaptively refined primal grid in three dimensions (3D), where the theoretical a posteriori result of the first-order scheme is used to derive appropriate refinement indicators. We apply those methods to solve Euler’s equations. Numerical experimental tests on classical problems are obtained by our method and by the computational fluid dynamics software Fluent. These tests include results for the 3D Euler system (shock tube problem) and flow around an NACA0012 airfoil. Key words. 3D adaptive central schemes, unstructured staggered grid mesh adaptation, finite volume method