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Schwarz waveform relaxation methods for systems of semilinear reaction-diffusion equations. (English) Zbl 1217.65187

Huang, Yunqing (ed.) et al., Domain decomposition methods in science and engineering XIX. Selected papers based on the presentations at the 19th international conference on domain decoposition (DD19), Zhanjiajie, China, August 17–22, 2009. Berlin: Springer (ISBN 978-3-642-11303-1/hbk; 978-3-642-11304-8/ebook). Lecture Notes in Computational Science and Engineering 78, 423-430 (2011).
Summary: Schwarz waveform relaxation methods have been studied for a wide range of scalar linear partial differential equations (PDEs) of parabolic and hyperbolic type. They are based on a space-time decomposition of the computational domain and the subdomain iteration uses an overlapping decomposition in space. There are only few convergence studies for nonlinear PDEs.
We analyze in this paper the convergence of Schwarz waveform relaxation applied to systems of semi-linear reaction-diffusion equations. We show that the algorithm converges linearly under certain conditions over long time intervals. We illustrate our results, and further possible convergence behavior, with numerical experiments.
For the entire collection see [Zbl 1204.65002].

MSC:

65M55 Multigrid methods; domain decomposition for initial value and initial-boundary value problems involving PDEs
35K57 Reaction-diffusion equations
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
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