Inexact Spectral Deferred Corrections

  • Robert SpeckEmail author
  • Daniel Ruprecht
  • Michael Minion
  • Matthew Emmett
  • Rolf Krause
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 104)


Implicit integration methods based on collocation are attractive for a number of reasons, e.g. their ideal (for Gauss-Legendre nodes) or near ideal (Gauss-Radau or Gauss-Lobatto nodes) order and stability properties. However, straightforward application of a collocation formula with M nodes to an initial value problem with dimension d requires the solution of one large Md × Md system of nonlinear equations.


Stiff Problem Multigrid Solver Euler Step Viscous Burger Spectral Deferred 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 work was supported by the Swiss National Science Foundation (SNSF) under the lead agency agreement through the project “ExaSolvers” within the Priority Programme 1648 “Software for Exascale Computing” (SPPEXA) of the Deutsche Forschungsgemeinschaft (DFG). 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. The authors acknowledge support from Matthias Bolten, who provided the employed multigrid solver.


  1. 1.
    R. Alexander, Diagonally implicit Runge-Kutta methods for stiff O.D.E.’s. SIAM J. Numer. Anal. 14(6), 1006–1021 (1977)Google Scholar
  2. 2.
    A. Bourlioux, A. Layton, M. Minion, High-order multi-implicit spectral deferred correction methods for problems of reactive flow. J. Comput. Phys. 189(2), 651–675 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    E. Bouzarth, M. Minion, A multirate time integrator for regularized stokeslets. J. Comput. Phys. 229(11), 4208–4224 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    A. Dutt, L. Greengard, V. Rokhlin, Spectral deferred correction methods for ordinary differential equations. BIT Numer. Math. 40(2), 241–266 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    M. Emmett, M. Minion, Toward an efficient parallel in time method for partial differential equations. Commun. Appl. Math. Comput. Sci. 7, 105–132 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    M. Emmett, M.L. Minion, Efficient Implementation of a Multi-Level Parallel in Time Algorithm. Domain Decomposition Methods in Science and Engineering XXI, Lecture Notes in Computational Science and Engineering, vol. 98 (Springer, Switzerland, 2014), pp. 359–366Google Scholar
  7. 7.
    J. Huang, J. Jia, M. Minion, Accelerating the convergence of spectral deferred correction methods. J. Comput. Phys. 214(2), 633–656 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    A. Layton, M. Minion, Conservative multi-implicit spectral deferred correction methods for reacting gas dynamics. J. Comput. Phys. 194(2), 697–715 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    J.-L. Lions, Y. Maday, G. Turinici, A “parareal” in time discretization of PDE’s. C. R. l’Académie Sci. Math. 332, 661–668 (2001)MathSciNetzbMATHGoogle Scholar
  10. 10.
    M. Minion, Semi-implicit spectral deferred correction methods for ordinary differential equations. Commun. Math. Sci. 1(3), 471–500 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    M. Minion, A hybrid parareal spectral deferred corrections method. Commun. Appl. Math. Comput. Sci. 5(2), 265–301 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    C. Oosterlee, T. Washio, On the use of multigrid as a preconditioner, in Proceedings of 9th International Conference on Domain Decomposition Methods, pp. 441–448 (1996)Google Scholar
  13. 13.
    D. Ruprecht, R. Speck, M. Emmett, M. Bolten, R. Krause, Poster: Extreme-scale space-time parallelism, in Proceedings of the 2013 Supercomputing Companion, SC ‘13 Companion, 2013 Google Scholar
  14. 14.
    R. Speck, D. Ruprecht, M. Emmett, M. Bolten, R. Krause. A space-time parallel solver for the three-dimensional heat equation. in Parallel Computing: Accelerating Computational Science and Engineering (CSE), Advances in Parallel Computing, vol. 25 (IOS Press, 2014), pp. 263–272. doi:10.3233/978-1-61499-381-0-263Google Scholar
  15. 15.
    R. Speck, D. Ruprecht, M. Emmett, M. Minion, M. Bolten, R. Krause, A multi-level spectral deferred correction method. BIT Numer. Math. (2014)zbMATHGoogle Scholar
  16. 16.
    W. Spotz, G. Carey, A high-order compact formulation for the 3D Poisson equation. Numer. Meth. PDEs 12(2), 235–243 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Y. Xia, Y. Xu, C.-W. Shu, Efficient time discretization for local discontinuous Galerkin methods. Disc. Cont. Dyn. Syst. 8(3), 677–693 (2007)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Robert Speck
    • 1
    • 2
    Email author
  • Daniel Ruprecht
    • 2
  • Michael Minion
    • 3
  • Matthew Emmett
    • 4
  • Rolf Krause
    • 2
  1. 1.Jülich Supercomputing Centre, Forschungszentrum Jülich GmbHJülichGermany
  2. 2.Institute of Computational ScienceUniversità della Svizzera italianaLuganoSwitzerland
  3. 3.Institute for Computational and Mathematical EngineeringStanford UniversityStanfordUSA
  4. 4.Center for Computational Sciences and EngineeringLawrence Berkeley National LaboratoryBerkeleyUSA

Personalised recommendations