Advertisement

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

  • Ronald D. Haynes
  • Benjamin W. Ong
Conference paper
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.

Notes

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.

References

  1. 1.
    Böhmer, K., Stetter, H.: Defect Correction Methods. Theory and Applications. Computing Supplementum, vol. 5. Springer (1984)Google Scholar
  2. 2.
    Cai, X.C.: Additive Schwarz algorithms for parabolic convection-diffusion equations. Numer. Math. 60(1), 41–61 (1991)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Cai, X.C.: Multiplicative Schwarz methods for parabolic problems. SIAM J. Sci. Comput. 15(3), 587–603 (1994)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Christlieb, A., Ong, B.: Implicit parallel time integrators. J. Sci. Comput. 49(2), 167–179 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 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)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Christlieb, A., Macdonald, C., Ong, B.: Parallel high-order integrators. SIAM J. Sci. Comput. 32(2), 818–835 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Christlieb, A., Haynes, R., Ong, B.: A parallel space-time algorithm. SIAM J. Sci. Comput. 34(5), 233–248 (2012)CrossRefMathSciNetGoogle Scholar
  8. 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)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Dutt, A., Greengard, L., Rokhlin, V.: Spectral deferred correction methods for ordinary differential equations. BIT 40(2), 241–266 (2000)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Gander, M.J.: Optimized Schwarz methods. SIAM J. Numer. Anal. 44(2), 699–731 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 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)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 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. 13.
    Koehler, S., Curreri, J., George, A.: Performance analysis challenges and framework for high-performance reconfigurable computing. Parallel Comput. 34(4), 217–230 (2008)CrossRefGoogle Scholar
  14. 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)CrossRefGoogle Scholar
  15. 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)zbMATHMathSciNetGoogle Scholar
  16. 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)CrossRefGoogle Scholar
  17. 17.
    Mpi 3.0 standardization effort. http://meetings.mpi-forum.org/MPI_3.0_main_page.php. Accessed 25 Oct 2012
  18. 18.
    Nievergelt, J.: Parallel methods for integrating ordinary differential equations. Commun. ACM 7(12), 731–733 (1964)CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    Toselli, A., Widlund, O.: Domain Decomposition Methods—Algorithms and Theory. Springer Series in Computational Mathematics, vol. 34. Springer, Berlin (2005)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.Memorial University of NewfoundlandSt. John’sCanada
  2. 2.Michigan State University, Institute for Cyber-Enabled ResearchEast LansingUSA

Personalised recommendations