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Recent Advances in Schwarz Waveform Moving Mesh Methods – A New Moving Subdomain Method

  • Ronald D. Haynes
Conference paper
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 78)

Abstract

It is well accepted that the efficient solution of complex partial differential equations (PDEs) often requires methods which are adaptive in both space and time. In this paper we are interested in a class of spatially adaptive moving mesh (r-refinement) methods introduced in [9, 10, 12]. Our purpose is to introduce and explore a natural coupling of domain decomposition, Schwarz waveform relaxation (SWR) [4], and spatially adaptive moving mesh PDE (MMPDE) methods for time dependent PDEs. SWR allows the focus of computational energy to evolve to the changing behaviour of the solution locally in regions or subdomains of the space-time domain. In particular, this will enable different time steps and indeed integration methods in each subdomain. The spatial mesh, provided by the MMPDE, will react to the local solution dynamics, providing distinct advantages for problems with evolving regions of interesting features.

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Notes

Acknowledgments

The author would like to acknowledge the support of NSERC (Canada) under discovery grant 311796.

Bibliography

  1. 1.
    U.M. Ascher. DAEs that should not be solved. In Dynamics of algorithms (Minneapolis, MN, 1997), volume 118 of IMA Volumes in Mathematics and Its Applications, pp. 55–67. Springer, New York, NY 2000.Google Scholar
  2. 2.
    C.J. Budd, W. Huang, and R.D. Russell. Adaptivity with moving grids. Acta Numer., 18:111–241, 2009. URL http://dx.doi.org/10.1017/S0962492906400015.Google Scholar
  3. 3.
    M.J. Gander and L. Halpern. Optimized Schwarz waveform relaxation methods for advection reaction diffusion problems. SIAM J. Numer. Anal., 45(2):666–697 (electronic), 2007. URL http://dx.doi.org/10.1137/050642137.Google Scholar
  4. 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.MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    R.D. Haynes. A domain decomposition approach for the equidistribution princple. In Preparation, May 2010.Google Scholar
  6. 6.
    R.D. Haynes, W. Huang, and R.D. Russell. A moving mesh method for time-dependent problems based on Schwarz waveform relaxation. In Domain Decomposition Methods in Science and Engineering XVII, volume 60 of Lecture Notes in Computational Science and Engineering, pp. 229–236. Springer, Berlin, 2008.Google Scholar
  7. 7.
    R.D. Haynes and R.D. Russell. A Schwarz waveform moving mesh method. SIAM J. Sci. Comput., 29(2):656–673 (electronic), 2007.MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    W. Huang. Practical aspects of formulation and solution of moving mesh partial differential equations. J. Comput. Phys., 171(2):753–775, 2001.MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    W. Huang, Y. Ren, and R.D. Russell. Moving mesh methods based on moving mesh partial differential equations. J. Comput. Phys., 113(2):279–290, 1994.MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    W. Huang, Y. Ren, and R.D. Russell. Moving mesh partial differential equations (MMPDES) based on the equidistribution principle. SIAM J. Numer. Anal., 31(3):709–730, 1994.MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    L.R. Petzold. A description of DASSL: a differential/algebraic system solver. In Scientific computing (Montreal, Que., 1982), IMACS Trans. Sci. Comput., I, pp. 65–68. IMACS, New Brunswick, NJ, 1983.Google Scholar
  12. 12.
    Y. Ren and R.D. Russell. Moving mesh techniques based upon equidistribution, and their stability. SIAM J. Sci. Statist. Comput., 13(6):1265–1286, 1992.MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsMemorial University of NewfoundlandSt. John’sCanada

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