Skip to main content

MPI–OpenMP Algorithms for the Parallel Space–Time Solution of Time Dependent PDEs

Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE,volume 98)

Abstract

Recently a space–time algorithm (RIDC–DD) for time dependent partial differential equations, combining an integral deferred correction approach in time with a classical domain decomposition method in space, was proposed and studied. The algorithm enables a multiplicative increase in the number of cores utilized in the spatial solver equal to the order of accuracy in time. This paper reviews the RIDC–DD method and presents in detail two hybrid MPI–OpenMP implementations.

Keywords

  • Time Dependent PDEs
  • Partial Differential Equations (PDEs)
  • Space-time Analogy
  • Classical Domain Decomposition Methods
  • Compliance Correction

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.

This is a preview of subscription content, access via your institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   149.00
Price excludes VAT (Canada)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD   199.99
Price excludes VAT (Canada)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions
Fig. 1
Fig. 2
Fig. 3
Fig. 4

References

  1. Böhmer, K., Stetter, H.: Defect Correction Methods. Theory and Applications. Computing Supplementum, vol. 5. Springer (1984)

    Google Scholar 

  2. Cai, X.C.: Additive Schwarz algorithms for parabolic convection-diffusion equations. Numer. Math. 60(1), 41–61 (1991)

    CrossRef  MATH  MathSciNet  Google Scholar 

  3. Cai, X.C.: Multiplicative Schwarz methods for parabolic problems. SIAM J. Sci. Comput. 15(3), 587–603 (1994)

    CrossRef  MATH  MathSciNet  Google Scholar 

  4. Christlieb, A., Ong, B.: Implicit parallel time integrators. J. Sci. Comput. 49(2), 167–179 (2011)

    CrossRef  MATH  MathSciNet  Google Scholar 

  5. Christlieb, A., Ong, B., Qiu, J.M.: Comments on high order integrators embedded within integral deferred correction methods. Commun. Appl. Math. Comput. Sci. 4(1), 27–56 (2009)

    CrossRef  MATH  MathSciNet  Google Scholar 

  6. Christlieb, A., Macdonald, C., Ong, B.: Parallel high-order integrators. SIAM J. Sci. Comput. 32(2), 818–835 (2010)

    CrossRef  MATH  MathSciNet  Google Scholar 

  7. Christlieb, A., Haynes, R., Ong, B.: A parallel space-time algorithm. SIAM J. Sci. Comput. 34(5), 233–248 (2012)

    CrossRef  MathSciNet  Google Scholar 

  8. Dubois, O., Gander, M., Loisel, S., St-Cyr, A., Szyld, D.: The optimized Schwarz method with a coarse grid correction. SIAM J. Sci. Comput. 34(1), A421–A458 (2012)

    CrossRef  MATH  MathSciNet  Google Scholar 

  9. Dutt, A., Greengard, L., Rokhlin, V.: Spectral deferred correction methods for ordinary differential equations. BIT 40(2), 241–266 (2000)

    CrossRef  MATH  MathSciNet  Google Scholar 

  10. Gander, M.J.: Optimized Schwarz methods. SIAM J. Numer. Anal. 44(2), 699–731 (2006)

    CrossRef  MATH  MathSciNet  Google Scholar 

  11. Gander, M.J., Halpern, L.: Optimized Schwarz waveform relaxation methods for advection reaction diffusion problems. SIAM J. Numer. Anal. 45(2), 666–697 (2007)

    CrossRef  MATH  MathSciNet  Google Scholar 

  12. Gander, M., Vandewalle, S.: On the superlinear and linear convergence of the parareal algorithm. Domain decomposition methods in science and engineering XVI. Lecture Notes in Computational Science and Engineering, vol. 55. Springer p. 291 (2007)

    Google Scholar 

  13. Koehler, S., Curreri, J., George, A.: Performance analysis challenges and framework for high-performance reconfigurable computing. Parallel Comput. 34(4), 217–230 (2008)

    CrossRef  Google Scholar 

  14. Lelarasmee, E., Ruehli, A.E., Sangiovanni-Vincentelli, A.L.: The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Trans. CAD IC Syst. 1, 131–145 (1982)

    CrossRef  Google Scholar 

  15. Lions, J., Maday, Y., Turinici, G.: A “parareal” in time discretization of PDEs. C. R. Acad. Sci. Ser. I Math. 332(7), 661–668 (2001)

    MATH  MathSciNet  Google Scholar 

  16. Minion, M., Williams, S.: Parareal and spectral deferred corrections. In: Numerical Analysis and Applied Mathematics: International Conference on Numerical Analysis and Applied Mathematics 2008. AIP Conference Proceedings, vol. 1048, pp. 388–391 (2008)

    CrossRef  Google Scholar 

  17. Mpi 3.0 standardization effort. http://meetings.mpi-forum.org/MPI_3.0_main_page.php. Accessed 25 Oct 2012

  18. Nievergelt, J.: Parallel methods for integrating ordinary differential equations. Commun. ACM 7(12), 731–733 (1964)

    CrossRef  MATH  MathSciNet  Google Scholar 

  19. Toselli, A., Widlund, O.: Domain Decomposition Methods—Algorithms and Theory. Springer Series in Computational Mathematics, vol. 34. Springer, Berlin (2005)

    Google Scholar 

Download references

Acknowledgements

This work was supported by the Institute for Cyber-Enabled Research (iCER) at MSU, NSERC Discovery Grant 311796, and AFOSR Grant FA9550-12-1-0455.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Ronald D. Haynes .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and Permissions

Copyright information

© 2014 Springer International Publishing Switzerland

About this paper

Cite this paper

Haynes, R.D., Ong, B.W. (2014). MPI–OpenMP Algorithms for the Parallel Space–Time Solution of Time Dependent PDEs. In: Erhel, J., Gander, M., Halpern, L., Pichot, G., Sassi, T., Widlund, O. (eds) Domain Decomposition Methods in Science and Engineering XXI. Lecture Notes in Computational Science and Engineering, vol 98. Springer, Cham. https://doi.org/10.1007/978-3-319-05789-7_14

Download citation