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Implicit Parallel Time Integrators

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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.

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References

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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. 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)

    Article  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  5. Christlieb, A., Morton, M., Ong, B., Qiu, J.-M.: Semi-implicit integral deferred correction constructed with high order additive Runge-Kutta integrators. Submitted

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  8. Gander, M.J., Hairer, E.: Nonlinear convergence analysis for the parareal algorithm. Lect. Notes Comput. Sci. Eng. 60, 45 (2008)

    Article  MathSciNet  Google Scholar 

  9. Gander, M.J., Vandewalle, S.: Analysis of the parareal time-parallel time-integration method. SIAM J. Sci. Comput. 29(2), 556–578 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  10. Gander, M.J., Vandewalle, S.: On the superlinear and linear convergence of the parareal algorithm. Lect. Notes Comput. Sci. Eng. 55, 291 (2007)

    Article  MathSciNet  Google Scholar 

  11. Greengard, L., Gropp, W.D.: A parallel version of the fast multipole method. Comput. Math. Appl. 20(7), 63–71 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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. Horton, G., Vandewalle, S.: A space-time multigrid method for parabolic PDEs. SIAM J. Sci. Comput. 16(4), 848–864 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    MATH  Google Scholar 

  16. Huang, J., Jia, J., Minion, M.: Accelerating the convergence of spectral deferred correction methods. J. Comput. Phys. 214(2), 633–656 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  19. Layton, A.T.: On the choice of correctors for semi-implicit Picard deferred correction methods. Appl. Numer. Math. 58(6), 845–858 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  23. Minion, M.: A hybrid parareal spectral deferred corrections method

  24. Minion, Michael L.: Semi-implicit spectral deferred correction methods for ordinary differential equations. Commun. Math. Sci. 1(3), 471–500 (2003)

    MathSciNet  MATH  Google Scholar 

  25. Rudolf, F., Rupp, K., Weinbub, J.: ViennaCL web page (2010). http://viennacl.sourceforge.net

  26. Runge, C.: Über empirische Funktionen und die Interpolation zwischen äquidistanten Ordinaten. Z. Math. Phys. 46, 224–243 (1901)

    Google Scholar 

  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. Warren, M.S., Salmon, J.K.: A portable parallel particle program. Comput. Phys. Commun. 87(1–2), 266–290 (1995)

    Article  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Math, Michigan State University, East Lansing, MI, 48823, USA

    Andrew Christlieb & Benjamin Ong

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  1. Andrew Christlieb
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  2. Benjamin Ong
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Correspondence to Andrew Christlieb.

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Christlieb, A., Ong, B. Implicit Parallel Time Integrators. J Sci Comput 49, 167–179 (2011). https://doi.org/10.1007/s10915-010-9452-4

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  • DOI: https://doi.org/10.1007/s10915-010-9452-4

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