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Legendre spectral collocation in space and time for PDEs

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Abstract

Spectral methods solve partial differential equations numerically with errors bounded by an exponentially decaying function of the number of modes when the solution is analytic. For time dependent problems, almost all focus has been on low-order finite difference schemes for the time derivative and spectral schemes for spatial derivatives. Spectral methods which converge spectrally in both space and time have appeared recently. This paper shows that a Legendre spectral collocation method of Tang and Xu for the heat equation converges exponentially quickly when the solution is analytic. We also derive a condition number estimate of the method. Another space-time spectral scheme which is easier to implement is proposed. Numerical experiments verify the theoretical results.

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Acknowledgments

I am indebted to Professor Lijun Yi for discussions of this work and for his pointers to the literature. I also thank Tim Hoffman for commenting on an earlier draft of this paper. Finally, I am grateful to the referees for their careful reading of the manuscript and for their numerous suggestions which have improved this paper.

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Authors and Affiliations

  1. Department of Mathematics, University of Manitoba, Winnipeg, MB, R3T 2N2, Canada

    S. H. Lui

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  1. S. H. Lui
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Correspondence to S. H. Lui.

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This work was in part supported by a Grant from NSERC.

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Lui, S.H. Legendre spectral collocation in space and time for PDEs. Numer. Math. 136, 75–99 (2017). https://doi.org/10.1007/s00211-016-0834-x

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  • DOI: https://doi.org/10.1007/s00211-016-0834-x

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