Location via proxy:   [ UP ]  
[Report a bug]   [Manage cookies]                
Skip to main content
Springer Nature Link
Log in

A multi-level spectral deferred correction method

  • Published:
BIT Numerical Mathematics Aims and scope Submit manuscript

Abstract

The spectral deferred correction (SDC) method is an iterative scheme for computing a higher-order collocation solution to an ODE by performing a series of correction sweeps using a low-order timestepping method. This paper examines a variation of SDC for the temporal integration of PDEs called multi-level spectral deferred corrections (MLSDC), where sweeps are performed on a hierarchy of levels and an FAS correction term, as in nonlinear multigrid methods, couples solutions on different levels. Three different strategies to reduce the computational cost of correction sweeps on the coarser levels are examined: reducing the degrees of freedom, reducing the order of the spatial discretization, and reducing the accuracy when solving linear systems arising in implicit temporal integration. Several numerical examples demonstrate the effect of multi-level coarsening on the convergence and cost of SDC integration. In particular, MLSDC can provide significant savings in compute time compared to SDC for a three-dimensional problem.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2

Similar content being viewed by others

Notes

  1. We adopt here and in the upcoming examples the following notation: Solutions of PDEs are denoted with an underline, e.g. \(\underline{u}\), and depend continuously on one or more spatial variables and a time variable. Discretizing a PDE in space by the method of lines results in an IVP with dimension \(N\) equal to the degrees of freedom of the spatial discretization. The solution of such an IVP is a vector-valued function denoted by a lower case letter, e.g. \(u\), and depends continuously on time. The numerical approximation of \(u\) at some point in time \(t_{m}\) is denoted by a capital letter, e.g. \({U}_{m}^{k}\), where \(k\) corresponds to the iteration number.

References

  1. Alam, J.M., Kevlahan, N.K.R., Vasilyev, O.V.: Simultaneous spacetime adaptive wavelet solution of nonlinear parabolic differential equations. J. Comput. Phys. 214(2), 829–857 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  2. Ascher, U.M., Petzold, L.R.: Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations. SIAM, Philadelphia (2000)

    Google Scholar 

  3. Böhmer, K., Hemker, P., Stetter, H.J.: The defect correction approach. In: K. Böhmer, H.J. Stetter (eds.) Defect Correction Methods. Theory and Applications, pp. 1–32. Springer, Berlin (1984)

  4. Bolten, M.: Evaluation of a multigrid solver for 3-level Toeplitz and circulant matrices on Blue Gene/Q. In: K. Binder, Münster, G., Kremer, M. (eds.) NIC Symposium 2014, pp. 345–352. John von Neumann Institute for Computing (2014, to appear)

  5. Bourlioux, A., Layton, A.T., Minion, M.L.: High-order multi-implicit spectral deferred correction methods for problems of reactive flow. J. Comput. Phys. 189(2), 651–675 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  6. Bouzarth, E.L., Minion, M.L.: A multirate time integrator for regularized stokeslets. J. Comput. Phys. 229(11), 4208–4224 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  7. Brandt, A.: Multi-level adaptive solutions to boundary-value problems. Math. Comput. 31(138), 333–390 (1977)

    Article  MATH  Google Scholar 

  8. Briggs, W.L.: A Multigrid Tutorial. SIAM, Philadelphia (1987)

    MATH  Google Scholar 

  9. Chorin, A.J., Marsden, J.E.: A Mathematical Introduction to Fluid Mechanics, 2nd edn. Springer, Berlin (1990)

  10. Chow, E., Falgout, R.D., Hu, J.J., Tuminaro, R.S., Yang, U.M.: A survey of parallelization techniques for multigrid solvers. In: Parallel Processing for Scientific Computing, SIAM Series of Software, Environements and Tools. SIAM (2006)

  11. Christlieb, A., Morton, M., Ong, B., Qiu, J.M.: Semi-implicit integral deferred correction constructed with additive Runge–Kutta methods. Commun. Math. Sci. 9(3), 879–902 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  12. 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)

    Article  MATH  MathSciNet  Google Scholar 

  13. Christlieb, A., Ong, B.W., Qiu, J.M.: Integral deferred correction methods constructed with high order Runge-Kutta integrators. Math. Comput. 79, 761–783 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  14. Dai, X., Le Bris, C., Legoll, F., Maday, Y.: Symmetric parareal algorithms for Hamiltonian systems. ESAIM. Math. Model. Numer. Anal. 47, 717–742 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  15. Daniel, J.W., Pereyra, V., Schumaker, L.L.: Iterated deferred corrections for initial value problems. Acta Cient. Venezolana 19, 128–135 (1968)

    MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  17. Emmett, M., Minion, M.L.: Efficient implementation of a multi-level parallel in time algorithm. In: Proceedings of the 21st International Conference on Domain Decomposition Methods, Lecture Notes in Computational Science and Engineering. Springer (2012)

  18. Emmett, M., Minion, M.L.: Toward an efficient parallel in time method for partial differential equations. Commun. Appl. Math. Comput. Sci. 7, 105–132 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  19. Hagstrom, T., Zhou, R.: On the spectral deferred correction of splitting methods for initial value problems. Commun. Appl. Math. Comput. Sci. 1(1), 169–205 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  20. Hairer, E., Nørsett, S.P., Wanner, G.: Solving Ordinary Differential Equations I, Nonstiff Problems. Springer, Berlin (1987)

  21. Hairer, E., Wanner, G.: Solving Ordinary Differential Equations II. Stiff and Differential-Algebraic Problems. Springer, Berlin (1991)

    Book  MATH  Google Scholar 

  22. Hansen, A.C., Strain, J.: Convergence theory for spectral deferred correction (2006, preprint)

  23. Haut, T., Wingate, B.: An asymptotic parallel-in-time method for highly oscillatory PDEs. SIAM J. Sci. Comput. (2014, in press)

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

    Article  MATH  MathSciNet  Google Scholar 

  25. Hunter, J.D.: Matplotlib: a 2D graphics environment. Comput. Sci. Eng. 9(3), 90–95 (2007)

    Article  Google Scholar 

  26. Jiang, G.S., Shu, C.W.: Efficient implementation of weighted ENO schemes. J. Comput. Phys. 126, 202–228 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  27. Layton, A.T.: On the efficiency of spectral deferred correction methods for time-dependent partial differential equations. Appl. Numer. Math. 59(7), 1629–1643 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  28. Layton, A.T., Minion, M.L.: Conservative multi-implicit spectral deferred correction methods for reacting gas dynamics. J. Comput. Phys. 194(2), 697–715 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  29. 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, 341–373 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  30. Lele, S.K.: Compact finite difference schemes with spectral-like resolution. J. Comput. Phys. 103(1), 16–42 (1992)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  32. Minion, M.L.: Semi-implicit projection methods for incompressible flow based on spectral deferred corrections. Appl. Numer. Math. 48(3), 369–387 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  33. Minion, M.L.: A hybrid parareal spectral deferred corrections method. Commun. Appl. Math. Comput. Sci. 5(2), 265–301 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  34. Pereyra, V.: Iterated deferred corrections for nonlinear operator equations. Numer. Math. 10, 316–323 (1966)

    Article  MathSciNet  Google Scholar 

  35. Pereyra, V.: On improving an approximate solution of a functional equation by deferred corrections. Numer. Math. 8, 376–391 (1966)

    Article  MATH  MathSciNet  Google Scholar 

  36. Ruprecht, D., Krause, R.: Explicit parallel-in-time integration of a linear acoustic-advection system. Comput. Fluids 59, 72–83 (2012)

    Article  MathSciNet  Google Scholar 

  37. South, J.C., Brandt, A.: Application of a multi-level grid method to transonic flow calculations. In: Transonic Flow Problems in Turbomachinery, Hemisphere, pp. 180–206 (1977)

  38. Speck, R., Ruprecht, D., Krause, R., Emmett, M., Minion, M., Winkel, M., Gibbon, P.: A massively space-time parallel N-body solver. In: Proceedings of the International Conference on High Performance Computing, Networking, Storage and Analysis, SC ’12, pp. 92:1–92:11. IEEE Computer Society Press, Los Alamitos (2012)

  39. Speck, R., Ruprecht, D., Minion, M., Emmett, M., Krause, R.: Inexact spectral deferred corrections In: Domain Decomposition Methods in Science and Engineering XXII, Lecture Notes in Computational Science and Engineering (2014, accepted). arXiv:1401.7824

  40. Spotz, W.F., Carey, G.F.: A high-order compact formulation for the 3D Poisson equation. Numer. Methods Partial Differ. Equ. 12(2), 235–243 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  41. Stetter, H.J.: Economical global error estimation. In: Willoughby, R.A. (ed.) Stiff Differential Systems, pp. 245–258 (1974)

  42. Trottenberg, U., Oosterlee, C.W.: Multigrid: Basics. Parallelism and Adaptivity. Academic Press, London (2000)

    Google Scholar 

  43. Weiser, M.: Faster SDC convergence on non-equidistant grids with DIRK sweeps (2013). ZIB Report, pp 13–30

  44. Xia, Y., Xu, Y., Shu, C.W.: Efficient time discretization for local discontinuous Galerkin methods. Discrete Contin. Dyn. Syst. Ser. B 8(3), 677–693 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  45. Zadunaisky, P.: A method for the estimation of errors propagated in the numerical solution of a system of ordinary differential equations. In: Contopoulus, G. (ed.) The Theory of Orbits in the Solar System and in Stellar Systems. Proceedings of International Astronomical Union, Symposium 25 (1964)

Download references

Acknowledgments

The plots were generated with the Python Matplotlib [25] package.

Author information

Authors and Affiliations

  1. Jülich Supercomputing Centre, Forschungszentrum Jülich, Jülich, Germany

    Robert Speck

  2. Institute of Computational Science, Università della Svizzera italiana, Lugano, Switzerland

    Robert Speck, Daniel Ruprecht & Rolf Krause

  3. Center for Computational Sciences and Engineering, Lawrence Berkeley National Laboratory, Berkeley, USA

    Matthew Emmett

  4. Institute for Computational and Mathematical Engineering, Stanford University, Stanford, USA

    Michael Minion

  5. Department of Mathematics and Science, University of Wuppertal, 42097, Wuppertal, Germany

    Matthias Bolten

Authors
  1. Robert Speck
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Daniel Ruprecht
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Matthew Emmett
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Michael Minion
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Matthias Bolten
    View author publications

    You can also search for this author in PubMed Google Scholar

  6. Rolf Krause
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Robert Speck.

Additional information

Communicated by Jan Nordström.

Robert Speck and Daniel Ruprecht acknowledge supported by Swiss National Science Foundation grant 145271 under the lead agency agreement through the project “ExaSolvers” within the Priority Programme 1648 “Software for Exascale Computing” of the Deutsche Forschungsgemeinschaft. Matthias Bolten acknowledges support from DFG through the project “ExaStencils” within SPPEXA. Daniel Ruprecht and Matthew Emmett also thankfully acknowledge support by grant SNF-147597. Matthew Emmett and Michael Minion were supported by the Applied Mathematics Program of the DOE Office of Advanced Scientific Computing Research under the U.S. Department of Energy under contract DE-AC02-05CH11231. Michael Minion was also supported by the U.S. National Science Foundation grant DMS-1217080.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Speck, R., Ruprecht, D., Emmett, M. et al. A multi-level spectral deferred correction method. Bit Numer Math 55, 843–867 (2015). https://doi.org/10.1007/s10543-014-0517-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10543-014-0517-x

Keywords

Mathematics Subject Classification

Search

Navigation