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A multi-level spectral deferred correction method

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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.

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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.

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Acknowledgments

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

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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
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  2. Daniel Ruprecht
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  4. Michael Minion
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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.

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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

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