Numerische Mathematik

, Volume 136, Issue 1, pp 75–99 | Cite as

Legendre spectral collocation in space and time for PDEs

  • S. H. LuiEmail author


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.

Mathematics Subject Classification

65M70 65L05 35K20 35L20 41A10 



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.


  1. 1.
    Bartels, R.H., Stewart, G.W.: A solution of the equation \(AX+XB=C\). Commun. ACM 15, 820–826 (1972)CrossRefGoogle Scholar
  2. 2.
    Bernardi, C., Maday, Y.: Spectral methods. In: Ciarlet, P.G., Lions, J.L. (eds.) Handbook of Numerical Analysis. pp, vol. 5, pp. 209–485. North-Holland, Amsterdam (1997)Google Scholar
  3. 3.
    Boyd, J.P.: Chebyshev and Fourier Spectral Methods, 2nd Rev. Ed. Dover, Mineola (2001)Google Scholar
  4. 4.
    Brugnano, L., Iavernaro, F., Trigiante, D.: Analysis of Hamiltonian boundary value methods (HBVMs): a class of energy-preserving Runge-Kutta methods for the numerical solution of polynoial Hamiltonian systems. Commum. Nonlinear Sci. Numer. Simul. 20, 650–667 (2015)CrossRefzbMATHGoogle Scholar
  5. 5.
    Canuto, C., Hussaini, M.Y., Quarteroni, A., Zang, T.A.: Spectral Methods-Fundamentals in Single Domains. Springer, New York (2006)zbMATHGoogle Scholar
  6. 6.
    Carpenter, M.H., Gottlieb, D.: Spectral methods on arbitrary grids. J. Comput. Phys. 129, 74–86 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Christlieb, A.J., Haynes, R.D., Ong, B.W.: A parallel space-time algorithm. SIAM J. Sci. Stat. Comput. 34, C233–C248 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Dai, X., Maday, Y.: Stable parareal in time method for first- and second-order hyperbolic systems. SIAM J. Sci. Comput. 35, A52–A78 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Dutt, A., Greengard, L., Rokhlin, V.: Spectral deferred correction methods for ordinary differential equations. BIT 40, 241–266 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Falgout, R.D., Friedhoff, S., Kolev, Tz.V., Maclachlan, S.P., Schroder, J.B.: Parallel time integration with multigrid. SIAM J. Sci. Comput. 36, C625–C661 (2014)Google Scholar
  11. 11.
    Fornberg, B.: A Practical Guide to Pseudospectral Methods. Cambridge University Press, Cambridge (1996)CrossRefzbMATHGoogle Scholar
  12. 12.
    Funaro, D.: Spectral Elements for Transport-Dominated Equations. Springer, Berlin (1997)CrossRefzbMATHGoogle Scholar
  13. 13.
    Gander, M.J., Vandewalle, S.: Analysis of the parareal time-parallel time-integration method. SIAM J. Sci. Comput. 29, 556–578 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Golub, G.H., Nash, S., Van Loan, C.F.: Hessenberg–Schur method for the problem \(ax+xb=c\). IEEE Trans. Autom. Control, AC-24, pp. 909–913 (1979)Google Scholar
  15. 15.
    Gottlieb, D., Orszag, S.A.: Numerical Analysis of Spectral Methods. SIAM, Philadelphia (1977)CrossRefzbMATHGoogle Scholar
  16. 16.
    Guo, B.Y.: Spectral Methods and Their Applications. World Scientific, Singapore (1998)CrossRefzbMATHGoogle Scholar
  17. 17.
    Guo, B.-Y., Wang, Z.-Q.: Legendre-Gauss collocation methods for ordinary differential equations. Adv. Comput. Math. 30, 249–280 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Hesthaven, J., Gottlieb, S., Gottlieb, D.: Spectral Methods for Time-Dependent Problems. Cambridge University Press, Philadelphia (2007)CrossRefzbMATHGoogle Scholar
  19. 19.
    Horton, G., Vandewalle, S.: A space-time multigrid method for parabolic partial differential equations. SIAM J. Sci. Comput. 16, 848–864 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Liu, W., Sun, J., Wu, B.: Galerkin–chebyshev spectral method and block boundary value methods for two-dimensional semilinear parabolic equations. Numer. Algorithms 71, 437–455 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Liu, W., Wu, B., Sun, J.: Space-time spectral collocation method for the one-dimensional Sine–Gordon equation. Numer. Methods PDEs 31, 670–690 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Lui, S.H.: Numerical Analysis of Partial Differential Equations. Wiley, Hoboken (2011)CrossRefzbMATHGoogle Scholar
  23. 23.
    McDonald, E.G., Wathen, A.J.: A simple proposal for parallel computation over time of an evolutionary process with implicit time stepping. Technical report, The Mathematical Institute, University of Oxford Technical Report, vol. 1860 (2014)Google Scholar
  24. 24.
    Shen, J., Tang, T., Wang, L.-L.: Spectral Methods. Springer, Berlin (2011)CrossRefzbMATHGoogle Scholar
  25. 25.
    Shen, J., Wang, L.-L.: Fourierization of the Legendre–Galerkin method and a new space-time spectral method. Appl. Numer. Math. 57, 710–720 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Tal-Ezer, H.: Spectral methods in time for hyperbolic equations. SIAM J. Numer. Anal. 23, 11–26 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Tal-Ezer, H.: Spectral methods in time for parabolic problems. SIAM J. Numer. Anal. 26, 1–11 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Tang, J.-G., Ma, H.-P.: Single and multi-interval Legendre \(\tau \)-methods in time for parabolic equations. Adv. Comput. Math. 17, 349–367 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Tang, J.-G., Ma, H.-P.: A Legendre spectral method in time for first-order hyperbolic equations. Appl. Numer. Math. 57, 1–11 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Tang, T., Xu, X.: Accuracy enhancement using spectral postprocessing for differential equations and integral equations. Commun. Comput. Phys. 5, 779–792 (2009)MathSciNetGoogle Scholar
  31. 31.
    Trefethen, L.N.: Spectral Methods in Matlab. SIAM, Philadelphia (2000)CrossRefzbMATHGoogle Scholar
  32. 32.
    Wang, L.-L., Samson, M.D., Zhao, X.: A well-conditioned collocation method using a pseudospectral integration matrix. SIAM J. Sci. Comput. 36, 907–929 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Wu, S., Liu, X.: Convergence of spectral method in time for Burgers’ equation. Acta Math. Appl. Sin. 13, 314–320 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Xie, Z., Wang, L.-L., Zhao, X.: On exponential convergence of Gegenbauer interpolation and spectral differentiation. Math. Comput. 82, 1017–1036 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Yi, L., Wang, Z.: Legendre–Gauss-type spectral collocation algorithms for nonlinear ordinary/partial differential equations. Int. J. Comput. Math. 91, 1434–1460 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Yi, L., Wang, Z.: Legendre spectral collocation method for second-order nonlinear ordinary/partial differential equations. Discrete Contin. Dyn. Syst. B 19, 299–322 (2014)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of ManitobaWinnipegCanada

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