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.
Keywords: Schwarz methods, Waveform relaxation, Semi-discrete
Simon Clement 1; Florian Lemarié 1; Eric Blayo 1
@article{SMAI-JCM_2022__8__99_0, author = {Simon Clement and Florian Lemari\'e and Eric Blayo}, title = {Discrete analysis of {Schwarz} waveform relaxation for a diffusion reaction problem with discontinuous coefficients}, journal = {The SMAI Journal of computational mathematics}, pages = {99--124}, publisher = {Soci\'et\'e de Math\'ematiques Appliqu\'ees et Industrielles}, volume = {8}, year = {2022}, doi = {10.5802/smai-jcm.81}, language = {en}, url = {https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.81/} }
TY - JOUR AU - Simon Clement AU - Florian Lemarié AU - Eric Blayo TI - Discrete analysis of Schwarz waveform relaxation for a diffusion reaction problem with discontinuous coefficients JO - The SMAI Journal of computational mathematics PY - 2022 SP - 99 EP - 124 VL - 8 PB - Société de Mathématiques Appliquées et Industrielles UR - https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.81/ DO - 10.5802/smai-jcm.81 LA - en ID - SMAI-JCM_2022__8__99_0 ER -
%0 Journal Article %A Simon Clement %A Florian Lemarié %A Eric Blayo %T Discrete analysis of Schwarz waveform relaxation for a diffusion reaction problem with discontinuous coefficients %J The SMAI Journal of computational mathematics %D 2022 %P 99-124 %V 8 %I Société de Mathématiques Appliquées et Industrielles %U https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.81/ %R 10.5802/smai-jcm.81 %G en %F SMAI-JCM_2022__8__99_0
Simon Clement; Florian Lemarié; Eric Blayo. Discrete analysis of Schwarz waveform relaxation for a diffusion reaction problem with discontinuous coefficients. The SMAI Journal of computational mathematics, Volume 8 (2022), pp. 99-124. doi : 10.5802/smai-jcm.81. https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.81/
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