Geodesic
Discrete analysis of Schwarz waveform relaxation for a diffusion reaction problem with discontinuous coefficients
Clement, Simon1 ; Lemarié, Florian1 ; Blayo, Eric1
1 Univ. Grenoble Alpes, Inria, CNRS, Grenoble INP, LJK, Grenoble, France
The SMAI Journal of computational mathematics, Tome 8 (2022), pp. 99-124.

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In this paper, we investigate the effect of the space and time discretisation on the convergence properties of Schwarz Waveform Relaxation (SWR) algorithms. We consider a reaction-diffusion problem with discontinuous coefficients discretised on two non-overlapping domains with several numerical schemes (in space and time). A methodology to determine the rate of convergence of the classical SWR method with standard interface conditions (Dirichlet-Neumann or Robin-Robin) accounting for discretisation errors is presented. We discuss how such convergence rates differ from the ones derived at a continuous level (i.e. assuming an exact discrete representation of the continuous problem). In this work we consider a second-order finite difference scheme and a finite volume scheme based on quadratic spline reconstruction in space, combined with either a simple backward Euler scheme or a two-step “Padé” scheme (resembling a Diagonally Implicit Runge Kutta scheme) in time. We prove those combinations of space-time schemes to be unconditionally stable on bounded domains. We illustrate the relevance of our analysis with specifically designed numerical experiments.

Publié le :
DOI : 10.5802/smai-jcm.81
Classification : 65B99, 65L12, 65M12
Keywords: Schwarz methods, Waveform relaxation, Semi-discrete

Clement, Simon 1 ; Lemarié, Florian 1 ; Blayo, Eric 1

1 Univ. Grenoble Alpes, Inria, CNRS, Grenoble INP, LJK, Grenoble, France 
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     title = {Discrete analysis of {Schwarz} waveform relaxation for a diffusion reaction problem with discontinuous coefficients},
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Clement, Simon; Lemarié, Florian; Blayo, Eric. Discrete analysis of Schwarz waveform relaxation for a diffusion reaction problem with discontinuous coefficients. The SMAI Journal of computational mathematics, Tome 8 (2022), pp. 99-124. doi : 10.5802/smai-jcm.81. http://geodesic.mathdoc.fr/articles/10.5802/smai-jcm.81/

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