Optimized Overlapping Schwarz Methods for Parabolic PDEs with Time-Delay

  • Stefan Vandewalle
  • Martin J. Gander
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 40)


We present overlapping Schwarz methods for the numerical solution of two model problems of delay PDEs: the heat equation with a fixed delay term, and the heat equation with a distributed delay in the form of an integral over the past. We first analyze properties of the solutions of these PDEs and find that their dynamics is fundamentally different from that of regular time-dependent PDEs without time delay. We then introduce and study overlapping Schwarz methods of waveform relaxation type for the two model problems. These methods compute the local solution in each subdomain over many time-levels before exchanging interface information to neighboring subdomains. We analyze the effect of the overlap and derive optimized transmission conditions of Robin type. Finally we illustrate the theoretical results and convergence estimates with numerical experiments.


Domain Decomposition Transmission Condition Constant Mode Schwarz Method Waveform Relaxation 
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  1. A. Bellen and M. Zennaro. Numerical Methods for Delay Differential Equations. Oxford University Press, Oxford, U.K., 2003.Google Scholar
  2. M. J. Gander. A waveform relaxation algorithm with overlapping splitting for reaction diffusion equations. Numerical Linear Algebra with Applications, 6:125–145, 1998.MathSciNetCrossRefGoogle Scholar
  3. M. J. Gander, L. Halpern, and F. Nataf. Optimal convergence for overlapping and non-overlapping Schwarz waveform relaxation. In C.-H. Lai, P. Bjørstad, M. Cross, and O. Widlund, editors, Eleventh international Conference of Domain Decomposition Methods., 1999.Google Scholar
  4. M. J. Gander and A. M. Stuart. Space-time continuous analysis of waveform relaxation for the heat equation. SIAM J. Sci. Comput., 19(6):2014–2031, 1998.MathSciNetCrossRefGoogle Scholar
  5. M. J. Gander and H. Zhao. Overlapping Schwarz waveform relaxation for the heat equation in n-dimensions. BIT, 42(4):779–795, 2002.MathSciNetCrossRefGoogle Scholar
  6. E. Giladi and H. B. Keller. Space time domain decomposition for parabolic problems. Numerische Mathematik, 93(2):279–313, 2002.MathSciNetCrossRefGoogle Scholar
  7. C. Huang and S. Vandewalle. An analysis of delay-dependent stability for ordinary and partial differential equations with fixed and distributed delays. SIAM J. Sci. Comput., 2003. To appear.Google Scholar
  8. S. Vandewalle and M. J. Gander. An analysis of Schwarz methods for delay partial differential equations. Technical report. in preparation.Google Scholar
  9. J. Wu. Theory and applications of Partial Functional Differential Equations. Springer-Verlag, New York, 1996.Google Scholar
  10. B. Zubik-Kowal. Stability in the numerical solution of linear parabolic equations with a delay term. BIT, 41:191–206, 2001.zbMATHMathSciNetCrossRefGoogle Scholar
  11. B. Zubik-Kowal and S. Vandewalle. Waveform relaxation for functional differential equations. SIAM J. Sci. Comput., 21:207–226, 1999.MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Stefan Vandewalle
    • 1
  • Martin J. Gander
    • 2
  1. 1.Department of Computer ScienceKatholieke Universiteit LeuvenLeuven
  2. 2.Department of Mathematics and StatisticsMcGill UniversityCanada

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