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BIT Numerical Mathematics

, Volume 55, Issue 3, pp 843–867 | Cite as

A multi-level spectral deferred correction method

  • Robert Speck
  • Daniel Ruprecht
  • Matthew Emmett
  • Michael Minion
  • Matthias Bolten
  • Rolf Krause
Article

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.

Keywords

Spectral deferred corrections Multi-level spectral deferred corrections FAS correction PFASST 

Mathematics Subject Classification

65M55 65M70 65Y05 

Notes

Acknowledgments

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

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Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  • Robert Speck
    • 1
    • 2
  • Daniel Ruprecht
    • 2
  • Matthew Emmett
    • 3
  • Michael Minion
    • 4
  • Matthias Bolten
    • 5
  • Rolf Krause
    • 2
  1. 1.Jülich Supercomputing CentreForschungszentrum JülichJülichGermany
  2. 2.Institute of Computational ScienceUniversità della Svizzera italianaLuganoSwitzerland
  3. 3.Center for Computational Sciences and EngineeringLawrence Berkeley National LaboratoryBerkeleyUSA
  4. 4.Institute for Computational and Mathematical EngineeringStanford UniversityStanfordUSA
  5. 5.Department of Mathematics and ScienceUniversity of WuppertalWuppertalGermany

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