Abstract
We propose and analyze non-overlapping Schwarz algorithms for the domain decomposition of the unsteady incompressible Navier–Stokes problem with Discrete Duality Finite Volume (DDFV) discretization. The design of suitable transmission conditions for the velocity and the pressure is a crucial issue. We establish the well-posedness of the method and the convergence of the iterative process, pointing out how the numerical fluxes influence the asymptotic problem which is intended to be a discretization of the Navier–Stokes equations on the entire computational domain. Finally we numerically illustrate the behavior and performances of the algorithm. We discuss on numerical grounds the impact of the parameters for several mesh geometries and we perform simulations of the flow past an obstacle with several domains.
Highlights
The aim of this paper is to develop a non-overlapping iterative Schwarz algorithm for the incompressible Navier–Stokes problem with Discrete Duality Finite Volume (DDFV) schemes
We want the interface conditions to be local and we wish the method to remain free of any restrictive condition on the Reynolds number. We address these issues in the framework of finite volume methods, and by using Discrete Duality Finite Volume discretizations
We start by defining the meshes, and we analyse the scheme on each subdomain, denoted by (Pj), and we introduce the Schwarz algorithm for the domain decomposition
Summary
The aim of this paper is to develop a non-overlapping iterative Schwarz algorithm for the incompressible Navier–Stokes problem with DDFV schemes. Our objective is to decompose the domain Ω of problem (1.1) into smaller subdomains, to solve the incompressible Navier–Stokes problem on those subdomains by imposing some transmission conditions on the interfaces, and to recover by an iterative Schwarz algorithm the discrete solution of (1.1) on the entire computational domain Ω. As a starting point of this study, we refer the reader to [20, 26]: they both build a non-overlapping Schwarz algorithm in a finite volume framework with Fourier-like transmission conditions between subdomains, respectively for anisotropic diffusion with a DDFV discretization, see [4], and for advection-diffusion-reaction in a. We discuss the influence of the parameters λ, α of (1.2) and we apply the method to the simulation of flows past an obstacle, with a multi-domain approach
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