DDFV method : applications to fluid mechanics and domain decomposition

Giulia Lissoni 1
1 COFFEE - COmplex Flows For Energy and Environment
CRISAM - Inria Sophia Antipolis - Méditerranée , JAD - Laboratoire Jean Alexandre Dieudonné : UMR7351
Abstract : The goal of this thesis is to study and develop numerical schemes of finite volume type for problems arising in fluid mechanics, namely Stokes and Navier-Stokes problems. The schemes we choosed are of discrete duality type, denoted by DDFV; this method works on staggered grids, where the velocity unknowns are located at the centers of control volumes and at the vertices of the mesh, and the pressure unknowns are on the edges of the mesh. This kind of construction has two main advantages: it allows to consider general meshes (that do not necessarily verify the classical orthogonality condition required by finite volume meshes) and to reconstruct and mimic at the discrete level the dual properties of the continuous differential operators. We start by the study of the discretization of Stokes problem with mixed boundary conditions of Dirichlet/Neumann type; the well-posed character of this problem is strictly relied to Inf-sup inequality, that has to be verified. In the DDFV setting, this inequality has been proven for particular meshes; we can avoid this hypothesis, by adding some stabilization terms in the equation of conservation of mass. In the first place, we study a stabilized scheme for Stokes problem in Laplace form, by showing its well-posedness, some error estimates and numerical tests. We study the same problem in divergence form, where the strain rate tensor replaces the gradient; here, we suppose that the Inf-sup inequality is verified, and we design a well-posed scheme followed by some numerical tests. We consider then the incompressible Navier-Stokes problem. At first, we study this problem coupled with « open » boundary conditions on the outflow; this kind of conditions arises when an artificial boundary is introduced, to save computational ressources or for physical reasons. We write a well-posed scheme and some energy estimates, validated by numerical simulations. Secondly, we address the domain decomposition method without overlap for the incompressible Navier-Stokes problem, by writing a discrete Schwarz algorithm. We discretize the problem with a semi-implicit Euler scheme in time, and at each time iteration we apply Schwarz algorithm to the resulting linear system. We show the convergence of this algorithm and we end by some numerical experiments. This thesis ends with a last chapter concerning the work done during CEMRACS 2019, where the goal is to extend DPIR (a recent technique for interface reconstruction between two materials) to the case of curved interfaces and of three materials. Some numerical simulations show the results.
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Giulia Lissoni. DDFV method : applications to fluid mechanics and domain decomposition. Mathematics [math]. Université de Nice - Sophia Antipolis; UCA, LJAD, 2019. English. ⟨tel-02309356⟩

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