Hostname: page-component-586b7cd67f-2brh9 Total loading time: 0 Render date: 2024-11-22T19:07:35.681Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Published online by Cambridge University Press:  03 June 2015

Shu-Lin Wu  and
Ting-Zhu Huang
Shu-Lin Wu*
Affiliation:
School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, Sichuan 610054, China School of Science, Sichuan University of Science and Engineering, Zigong, Sichuan 643000, China
Ting-Zhu Huang*
Affiliation:
School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, Sichuan 610054, China
*
Corresponding author.Email:[email protected]
Email:[email protected]
Get access

Abstract

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.

Keywords

30E1065M1265M55Schwarz methodwaveform relaxationtime delaymin-max problem
Type
Research Article
Information
Communications in Computational Physics , Volume 14 , Issue 3 , September 2013 , pp. 780 - 800
Copyright
Copyright © Global Science Press Limited 2013

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Bennequin, D., Gander, M. J., Halpern, L., A homographic best approximation problem with application to optimized Schwarz waveform relaxation, Math. Comp. 78 (2009), pp. 185–223.CrossRefGoogle Scholar
[2]Cai, X. C., Additive Schwarz algorithms for parabolic convection-diffusion equations, Numer. Math. 60 (1991), pp. 41–61.Google Scholar
[3]Cai, X. C., Multiplicative Schwarz methods for parabolic problems, SIAM J. Sci. Comput. 15 (1994), pp. 587–603.CrossRefGoogle Scholar
[4]Caetano, F., Gander, M.J., Halpern, L., Szeftel, J., Schwarz waveform relaxation algorithms for semilinear reaction-diffusion equations, Networks and Heterogeneous Media 5 (2010), pp. 487–505.CrossRefGoogle Scholar
[5]Dolean, V., Gander, M. J., Gerardo-Giorda, L., Optimized Schwarz methods for Maxwell’s equations, SIAM J. Sci. Comput. 31 (2009), pp. 2193–2213.Google Scholar
[6]Daoud, D. S., Gander, M.J., Overlapping Schwarz waveform relaxation for advection diffusion equations, Bol. Soc. Esp. Mat. Apl. 46 (2009), pp. 75–90.Google Scholar
[7]Gander, M. J., Schwarz methods over the course of time, Electron. Trans. Numer. Anal. 31 (2008), pp. 228–255.Google Scholar
[8]Gander, M. J., Zhao, H., Overlapping Schwarz waveform relaxation for the heat equation in n-dimensions, BIT Numer. Math. 42 (2002), pp. 779–795.CrossRefGoogle Scholar
[9]Gander, M. J., Stuart, A. M., Space-Time continuous analysis of waveform relaxation for the heat equation, SIAM J. Sci. Comput. 19 (1998), pp. 2014–2031.CrossRefGoogle Scholar
[10]Gander, M. J., Halpern, L., Labbe, S., Santugini-Repiquet, K., An optimized Schwarz waveform relaxation algorithm for Micro-Magnetics, in: Proceedings of the 17th International Conference on Domain Decomposition Methods, 2007.Google Scholar
[11]Gander, M. J., Rohde, C., Overlapping Schwarz waveform relaxation for convection-dominated nonlinear conservation laws, SIAM J. Sci. Comput. 27 (2005), pp. 415–439.Google Scholar
[12]Giladi, E.,Keller, H. B., Space-time domain decomposition for parabolic problems, Numer. Math. 93 (2002), pp. 279–313.Google Scholar
[13]Gander, M. J., Zhao, H., Overlapping Schwarz waveform relaxation for parabolic problems in higher dimension, Proceedings of Algoritmy 97 (1997), pp. 42–51.Google Scholar
[14]Gander, M. J., Overlapping Schwarz for parabolic problems, in: Proceedings of the 9th International Conference on Domain Decomposition Methods, 1997.Google Scholar
[15]Gander, M. J., A waveform relaxation algorithm with overlapping splitting for reaction diffusion equations, Numer. Linear Algebra Appl. 6 (1998), pp. 125–145.Google Scholar
[16]Gander, M. J., Halpern, L., Optimized Schwarz waveform relaxation for advection reaction diffusion problems, SIAM J. Numer. Anal. 45 (2007), pp. 666–697.CrossRefGoogle Scholar
[17]Gander, M. J., Halpern, L., Nataf, F., Optimal Schwarz waveform relaxation for the one dimensional wave equation, SIAM J. Numer. Anal. 41 (2003), pp. 1643–1681.Google Scholar
[18]Gander, M. J., Magoules, F., Nataf, F., Optimized Schwarz methods without overlap for the Helmholtz equation, SIAM J. Sci. Comput. 24 (2002), pp. 38–60.Google Scholar
[19]Gander, M. J., Halpern, L., Methodes de relaxation d’ondes pour l’equation de la chaleur en dimension 1, C. R. Acad. Sci. Paris Ser.I 336, pp. 519–524 (2003).Google Scholar
[20]Halpern, L., Szeftel, J., Nonlinear non-overlapping Schwarz waveform relation for semilinear wave propagation, Math. Comput. 78 (2009), pp. 865–889.Google Scholar
[21]Halpern, L., Absorbing boundary conditions and optimized Schwarz waveform relaxation, BIT Numer. Math. 46 (2006), pp. 21–34.CrossRefGoogle Scholar
[22]Halpern, L., Szeftel, J., Optimized and quasi-optimal Schwarz waveform relaxation for the one dimensional Schrödinger equation, Math. Models Methods Appl. Sci. 20 (2010), pp. 2167–2199.CrossRefGoogle Scholar
[23]Jiang, Y. L., On time-domain simulation of lossless transmission lines with nonlinear terminations, SIAM J. Numer. Anal. 42 (2004), 1018–1031.Google Scholar
[24]Jiang, Y. L., Chen, R. M. M., Computing periodic solutions of linear differential-algebraic equations by waveform relaxation, Math. Comput. 74(2005), 781–804.Google Scholar
[25]Lelarasmee, E., Ruehli, A. E., Sangiovanni-Vincentelli, A. L., The waveform relaxation methods for time-domain analysis of large scale integrated circuits, IEEE Trans. Computer-Aided Design 1 (1982), pp. 131–145.Google Scholar
[26]Martin, V., An optimized Schwarz waveform relaxation method for the unsteady convection diffusion equation in two dimensions, Appl. Numer. Math. 52 (2005), pp. 401–428.Google Scholar
[27]Martin, V., Schwarz waveform relaxation algorithms for the linear viscous equatorial shallow water equations, SIAM J. Sci. Comput. 31 (2009), pp. 3595–3625.Google Scholar
[28]MacMullen, H., O’Riordan, E., Shishkin, G. I., The convergence of classical Schwarz methods applied to convection-diffusion problems with regular boundary layers, Appl. Numer. Math. 43 (2002), pp. 297–313.Google Scholar
[29]Qaddouri, A., Laayouni, L., Loisel, S., Cote, J., Gander, M. J., Optimized Schwarz methods with an overset grid for the shallow-water equations: preliminary results, Appl. Numer. Math. 58 (2008), pp. 459–471.Google Scholar
[30]Qin, L. Z., Xu, X. J., Optimized Schwarz methods with Robin transmission conditions for parabolic problems, SIAM J. Sci. Comput. 31 (2008), pp. 608–623.Google Scholar
[31]Vandewalle, S., Parallel multigrid waveform relaxation for parabolic problems, B. G. Teubner, Stuttgart, 1993.Google Scholar
[32]Vandewalle, S., Gander, M. J., Optimized overlapping Schwarz methods for parabolic PDEs with time-delay, Lecture Notes in Computational Science and Engineering 40 (2005), pp. 291–298.Google Scholar
[33]Wu, J., Theory and applications of partial functional differential equations, Springer-Verlag, New York, 1996.Google Scholar
[34]Wu, S. L., Huang, C. M., Huang, Ting-Zhu, Convergence analysis of overlapping Schwarz waveform relaxation algorithm for reaction diffusion equations with time-delay, IMA J. Numer. Anal. 32 (2012), pp. 632–671.Google Scholar
[35]Wu, S. L., Huang, C. M., Quasi-optimized Schwarz methods for reaction diffusion equations with time delay, J. Math. Anal. Appl. 385 (2012), pp. 354–370.Google Scholar