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Civil-Comp Proceedings
ISSN 1759-3433
CCP: 101
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Paper 9

Nonlinear Transmission Conditions for Schwarz and Dual Schur Complement Time Domain Decomposition

P. Linel1 and D. Tromeur-Dervout2

1Department of Biostatistics and Computational Biology, University of Rochester Medical Center, Rochester, United States of America
2University of Lyon, University Lyon1, CNRS, ICJ UMR5208, Villeurbanne, France

Full Bibliographic Reference for this paper
P. Linel, D. Tromeur-Dervout, "Nonlinear Transmission Conditions for Schwarz and Dual Schur Complement Time Domain Decomposition", in , (Editors), "Proceedings of the Third International Conference on Parallel, Distributed, Grid and Cloud Computing for Engineering", Civil-Comp Press, Stirlingshire, UK, Paper 9, 2013. doi:10.4203/ccp.101.9
Keywords: Schwarz domain decomposition, time domain decomposition, Aitken's acceleration of the convergence, parabolic equations, boundary values problem.

In this paper, we propose a right transmission condition for the time decomposition that consists to transform the initial boundary value problem into a time boundary values problem. This allows us to use the classical multiplicative Schwarz algorithm using non-overlapping time slices. It also avoids the symmetrizing of the time interval needed to set the unknown value of the solution at the end time boundary of the last time slice. We show that, for nonlinear scalar problems, we must imposed some invariant of the problem as transmission conditions between time slices. We derive a Robin transmission condition in order to break the sequentiality of the propagating of the exact solution from the first time slice to the time slices that follow. We show the purely linear behaviour of this multiplicative Schwarz and its extrapolation to the right transmission conditions using Aitken's technique to accelerate convergence. Then a dual Schur complement technique is used on the nonlinear problem. We derive a method where the nonlinear transmission conditions are used to solve on each time slice while imposing the constraint of the solution continuity between time slices.

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