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Journal of Scientific Computing

, Volume 49, Issue 2, pp 167–179 | Cite as

Implicit Parallel Time Integrators

  • Andrew Christlieb
  • Benjamin Ong
Article

Abstract

In this work, we discuss a family of parallel implicit time integrators for multi-core and potentially multi-node or multi-gpgpu systems. The method is an extension of Revisionist Integral Deferred Correction (RIDC) by Christlieb, Macdonald and Ong (SISC-2010) which constructed parallel explicit time integrators. The key idea is to re-write the defect correction framework so that, after initial startup costs, each correction loop can be lagged behind the previous correction loop in a manner that facilitates running the predictor and correctors in parallel.

In this paper, we show that RIDC provides a framework to use p cores to generate a pth-order implicit solution to an initial value problem (IVP) in approximately the same wall clock time as a single core, backward Euler implementation (p≤12). The construction, convergence and stability of the schemes are presented, along with supporting numerical evidence.

Keywords

Initial value problems Integral deferred correction Parallel computation Multi-core computing 

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References

  1. 1.
    Auzinger, W., Hofstätter, H., Kreuzer, W., Weinmüller, E.: Modified defect correction algorithms for ODEs, part I: general theory. Numer. Algorithms 36, 135–156 (2004) MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Balay, S., Buschelman, K., Gropp, D., Kaushik, W.D., Knepley, M.G., McInnes, L.C., Smith, B.F., Zhang, H.: PETSc Web page (2009). http://www.mcs.anl.gov/petsc
  3. 3.
    Brakkee, E., Segal, A., Kassels, C.G.M.: A parallel domain decomposition algorithm for the incompressible Navier-Stokes equations. Simul. Pract. Theory 3(4–5), 185–205 (1995) CrossRefGoogle Scholar
  4. 4.
    Christlieb, A., Macdonald, C., Ong, B.: Parallel high-order integrators. SIAM J. Sci. Comput. 32(2), 818–835 (2010) MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Christlieb, A., Morton, M., Ong, B., Qiu, J.-M.: Semi-implicit integral deferred correction constructed with high order additive Runge-Kutta integrators. Submitted Google Scholar
  6. 6.
    Christlieb, A., Ong, B., M Qiu, J.-M.: Comments on high order integrators embedded within integral deferred correction methods. Commun. Appl. Math. Comput. Sci. 4(1), 27–56 (2009) MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Christlieb, A., Ong, B., Qiu, J.-M.: Integral deferred correction methods constructed with high order Runge-Kutta integrators. Math. Comput. 79, 761–783 (2010) MathSciNetzbMATHGoogle Scholar
  8. 8.
    Gander, M.J., Hairer, E.: Nonlinear convergence analysis for the parareal algorithm. Lect. Notes Comput. Sci. Eng. 60, 45 (2008) MathSciNetCrossRefGoogle Scholar
  9. 9.
    Gander, M.J., Vandewalle, S.: Analysis of the parareal time-parallel time-integration method. SIAM J. Sci. Comput. 29(2), 556–578 (2007) MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Gander, M.J., Vandewalle, S.: On the superlinear and linear convergence of the parareal algorithm. Lect. Notes Comput. Sci. Eng. 55, 291 (2007) MathSciNetCrossRefGoogle Scholar
  11. 11.
    Greengard, L., Gropp, W.D.: A parallel version of the fast multipole method. Comput. Math. Appl. 20(7), 63–71 (1990) MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Hagstrom, T., Zhou, R.: On the spectral deferred correction of splitting methods for initial value problems. Commun. Appl. Math. Comput. Sci. 1, 169–205 (2006) MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Hairer, E., Wanner, G.: Stiff and differential-algebraic problems. In: Solving Ordinary Differential Equations, II, 2nd edn. Springer Series in Computational Mathematics, vol. 14, pp. 585–604. Springer, Berlin (1996) Google Scholar
  14. 14.
    Horton, G., Vandewalle, S.: A space-time multigrid method for parabolic PDEs. SIAM J. Sci. Comput. 16(4), 848–864 (1995) MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Van Der Houwen, P.J., Sommeijer, B.P., Couzy, W.: Embedded diagonally implicit Runge-Kutta algorithms on parallel computers. Math. Comput. 58(197), 135–159 (1992) zbMATHGoogle Scholar
  16. 16.
    Huang, J., Jia, J., Minion, M.: Accelerating the convergence of spectral deferred correction methods. J. Comput. Phys. 214(2), 633–656 (2006) MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Huang, J., Jia, J., Minion, M.: Arbitrary order Krylov deferred correction methods for differential algebraic equations. J. Comput. Phys. 221(2), 739–760 (2007) MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Layton, A.T., Minion, M.L.: Implications of the choice of quadrature nodes for Picard integral deferred corrections methods for ordinary differential equations. BIT Numer. Math. 45(2), 341–373 (2005) MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Layton, A.T.: On the choice of correctors for semi-implicit Picard deferred correction methods. Appl. Numer. Math. 58(6), 845–858 (2008) MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Layton, A.T., Minion, M.L.: Implications of the choice of predictors for semi-implicit Picard integral deferred corrections methods. Commun. Appl. Math. Comput. Sci. 1(2), 1–34 (2007) MathSciNetCrossRefGoogle Scholar
  21. 21.
    Lions, J.L., Maday, Y., Turinici, G.: A “parareal” in time discretization of PDEs. C. R. Acad. Sci., Sér. 1 Math. 332(7), 661–668 (2001) MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Maday, Y., Turinici, G.: A parareal in time procedure for the control of partial differential equations. C. R. Math. 335(4), 387–392 (2002) MathSciNetzbMATHGoogle Scholar
  23. 23.
    Minion, M.: A hybrid parareal spectral deferred corrections method Google Scholar
  24. 24.
    Minion, Michael L.: Semi-implicit spectral deferred correction methods for ordinary differential equations. Commun. Math. Sci. 1(3), 471–500 (2003) MathSciNetzbMATHGoogle Scholar
  25. 25.
    Rudolf, F., Rupp, K., Weinbub, J.: ViennaCL web page (2010). http://viennacl.sourceforge.net
  26. 26.
    Runge, C.: Über empirische Funktionen und die Interpolation zwischen äquidistanten Ordinaten. Z. Math. Phys. 46, 224–243 (1901) Google Scholar
  27. 27.
    Smith, B.F., Bjorstad, P., Gropp, W.: Domain Decomposition: Parallel Multilevel Methods for Elliptic Partial Differential Equations, Cambridge University Press, Cambridge (2004) Google Scholar
  28. 28.
    Warren, M.S., Salmon, J.K.: A portable parallel particle program. Comput. Phys. Commun. 87(1–2), 266–290 (1995) zbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Department of MathMichigan State UniversityEast LansingUSA

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