Commun. Comput. Phys., 14 (2013), pp. 780-800.

Quasi-Optimized Overlapping Schwarz Waveform Relaxation Algorithm for PDEs with Time-Delay

Shu-Lin Wu 1*, Ting-Zhu Huang 2

1 School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, Sichuan 610054, China.
2 School of Science, Sichuan University of Science and Engineering, Zigong, Sichuan 643000, China.

Received 10 March 2012; Accepted (in revised version) 7 November 2012
Available online 28 February 2013


Schwarz waveform relaxation (SWR) algorithm has been investigated deeply and widely for regular time dependent problems. But for time delay problems, complete analysis of the algorithm is rare. In this paper, by using the reaction diffusion equations with a constant discrete delay as the underlying model problem, we investigate the convergence behavior of the overlapping SWR algorithm with Robin transmission condition. The key point of using this transmission condition is to determine a free parameter as better as possible and it is shown that the best choice of the parameter is determined by the solution of a min-max problem, which is more complex than the one arising for regular problems without delay. We propose new notion to solve the min-max problem and obtain a quasi-optimized choice of the parameter, which is shown efficient to accelerate the convergence of the SWR algorithm. Numerical results are provided to validate the theoretical conclusions.

AMS subject classifications: 30E10, 65M12, 65M55

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Key words: Schwarz method, waveform relaxation, time delay, min-max problem.

*Corresponding author.
Email: (S.-L. Wu), (T.-Z. Huang)

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