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Parareal Schwarz Waveform Relaxation Methods

  • Martin J. Gander
  • Yao-Lin Jiang
  • Rong-Jian Li
Conference paper
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 91)

Abstract

Solving an evolution problem in parallel is naturally undertaken by trying to parallelize the algorithm in space, and then still follow a time stepping method from the initial time t = 0 to the final time t = T. This is especially easy to do when an explicit time stepping method is used, because in that case the time step for each component is only based on past, known data, and the time stepping can be performed in an embarrassingly parallel way.

Keywords

Transmission Condition Domain Decomposition Method Superlinear Convergence Waveform Relaxation Parareal Algorithm 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Bibliography

  1. [1].
    Daniel Bennequin, Martin J. Gander, and Laurence Halpern. A homographic best approximation problem with application to optimized Schwarz waveform relaxation. Math. of Comp., 78(265):185–232, 2009.Google Scholar
  2. [2].
    Philippe Chartier and Bernard Philippe. A parallel shooting technique for solving dissipative ODEs. Computing, 51:209–236, 1993.MathSciNetzbMATHCrossRefGoogle Scholar
  3. [3].
    Ashish Deshpande, Sachit Malhotra, Craig C. Douglas, and Martin H. Schultz. A rigorous analysis of time domain parallelism. Parallel Algorithms and Applications, 6:53–62, 1995.zbMATHCrossRefGoogle Scholar
  4. [4].
    Martin J. Gander and Ernst Hairer. Nonlinear convergence analysis for the parareal algorithm. In U. Langer, M. Discacciati, D.E. Keyes, O.B. Widlund, and W. Zulehner, editors, Domain Decomposition Methods in Science and Engineering XVII, volume 60, pages 45–56. Springer-Verlag, 2007.Google Scholar
  5. [5].
    Martin J. Gander and Laurence Halpern. Optimized Schwarz waveform relaxation methods for advection reaction diffusion problems. SIAM J. Numer. Anal., 45(2):666–697, 2007.Google Scholar
  6. [6].
    Martin J. Gander, Laurence Halpern, and Frédéric Nataf. Optimal Schwarz waveform relaxation for the one dimensional wave equation. SIAM J. Numer. Anal., 41(5):1643–1681, 2003.Google Scholar
  7. [7].
    Martin J. Gander, Yao-Lin Jiang, Rong-Jian Li, and Bo Song. A family of parareal Schwarz waveform relaxation algorithms. In preparation, 2012.Google Scholar
  8. [8].
    Martin J. Gander and Andrew M. Stuart. Space-time continuous analysis of waveform relaxation for the heat equation. SIAM J. Sci. Comput., 19(6):2014–2031, 1998.Google Scholar
  9. [9].
    Martin J. Gander and Stefan Vandewalle. Analysis of the parareal time-parallel time-integration method. SIAM J. Sci. Comput., 29(2):556–578, 2007.Google Scholar
  10. [10].
    Martin J. Gander and Hongkai Zhao. Overlapping Schwarz waveform relaxation for the heat equation in n-dimensions. BIT, 42(4):779–795, 2002.MathSciNetzbMATHCrossRefGoogle Scholar
  11. [11].
    Wolfgang Hackbusch. Parabolic multi-grid methods. In Roland Glowinski and Jacques-Louis Lions, editors, Computing Methods in Applied Sciences and Engineering, VI, pages 189–197. North-Holland, 1984.Google Scholar
  12. [12].
    Yao-Lin Jiang. A general approach to waveform relaxation solutions of differential-algebraic equations: the continuous-time and discrete-time cases. IEEE Trans. Circuits and Systems - Part I, 51(9):1770–1780, 2004.Google Scholar
  13. [13].
    Yao-Lin Jiang. Waveform Relaxation Methods. Scientific Press, Beijing, 2010.zbMATHGoogle Scholar
  14. [14].
    Yao-Lin Jiang and Hui Zhang. Schwarz waveform relaxation methods for parabolic equations in space-frequency domain. Computers and Mathematics with Applications, 55(12):2924–2933, 2008.MathSciNetzbMATHCrossRefGoogle Scholar
  15. [15].
    Ekachai Lelarasmee, Albert E. Ruehli, and Alberto L. Sangiovanni-Vincentelli. The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Trans. on CAD of IC and Syst., 1:131–145, 1982.Google Scholar
  16. [16].
    Jacques-Louis Lions, Yvon Maday, and Gabriel Turinici. A parareal in time discretization of pde’s. C.R. Acad. Sci. Paris, Serie I, 332:661–668, 2001.Google Scholar
  17. [17].
    C. Lubich and A. Ostermann. Multi-grid dynamic iteration for parabolic equations. BIT, 27(2):216–234, 1987.MathSciNetzbMATHCrossRefGoogle Scholar
  18. [18].
    Yvon Maday and Gabriel Turinici. The parareal in time iterative solver: a further direction to parallel implementation. In U. Langer, M. Discacciati, D.E. Keyes, O.B. Widlund, and W. Zulehner, editors, Domain Decomposition Methods in Science and Engineering XVII, volume 60, pages 441–448. Springer-Verlag, 2007.Google Scholar
  19. [19].
    Willard L. Miranker and Werner Liniger. Parallel methods for the numerical integration of ordinary differential equations. Math. Comp., 91:303–320, 1967.Google Scholar
  20. [20].
    Jörg Nievergelt. Parallel methods for integrating ordinary differential equations. Comm. ACM, 7:731–733, 1964.Google Scholar
  21. [21].
    Stefan Vandewalle and Eric Van de Velde. Space-time concurrent multigrid waveform relaxation. Ann. Numer. Math., 1(1–4):347–363, 1994.Google Scholar
  22. [22].
    David E. Womble. A time-stepping algorithm for parallel computers. SIAM J. Sci. Stat. Comput., 11(5):824–837, 1990.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Martin J. Gander
    • 1
  • Yao-Lin Jiang
    • 2
  • Rong-Jian Li
    • 2
  1. 1.Mathematics SectionUniversity of GenevaGenevaSwitzerland
  2. 2.Department of Mathematics SciencesXi’an Jiaotong UniversityXi’anChina

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