Advertisement

BIT Numerical Mathematics

, Volume 40, Issue 2, pp 241–266 | Cite as

Spectral Deferred Correction Methods for Ordinary Differential Equations

  • Alok Dutt
  • Leslie Greengard
  • Vladimir Rokhlin
Article

Abstract

We introduce a new class of methods for the Cauchy problem for ordinary differential equations (ODEs). We begin by converting the original ODE into the corresponding Picard equation and apply a deferred correction procedure in the integral formulation, driven by either the explicit or the implicit Euler marching scheme. The approach results in algorithms of essentially arbitrary order accuracy for both non-stiff and stiff problems; their performance is illustrated with several numerical examples. For non-stiff problems, the stability behavior of the obtained explicit schemes is very satisfactory and algorithms with orders between 8 and 20 should be competitive with the best existing ones. In our preliminary experiments with stiff problems, a simple adaptive implementation of the method demonstrates performance comparable to that of a state-of-the-art extrapolation code (at least, at moderate to high precision).

Deferred correction methods based on the Picard equation appear to be promising candidates for further investigation.

Spectral methods initial value problems deferred correction stiffness 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

REFERENCES

  1. 1.
    K. E. Brenan, S. L. Campbell, and L. R. Petzold, Numerical Solution of Initial-Value Problems in Differential-Algebraic Equations, SIAM, Philadelphia, PA, 1995.Google Scholar
  2. 2.
    J. Butcher, The Numerical Analysis of Ordinary Differential Equations: Runge– Kutta and General Linear Methods, Wiley, New York, 1987.Google Scholar
  3. 3.
    K. Böhmer and H. J. Stetter, eds., Defect Correction Methods, Theory and Applications, Springer-Verlag, New York, 1984.Google Scholar
  4. 4.
    P. Deuflhard, Recent progress in extrapolation methods for ordinary differential equations, SIAM Rev., 27 (1985), pp. 505–535.Google Scholar
  5. 5.
    A. Dutt, M. Gu, and V. Rokhlin, Fast algorithms for polynomial interpolation, integration, and differentiation, SIAM J. Numer. Anal., 33 (1996), pp. 1689–1711.Google Scholar
  6. 6.
    R. Frank and C. W. Ueberhuber, Iterated deferred correction for the efficient solution of stiff systems of ordinary differential equations, BIT, 17 (1977), pp. 146–159.Google Scholar
  7. 7.
    C. W. Gear, Numerical Initial Value Problems in Ordinary Differential Equations, Prentice-Hall, Englewood Cliffs, NJ, 1971.Google Scholar
  8. 8.
    D. Gottlieb and S. A. Orszag, Numerical Analysis of Spectral Methods, SIAM, Philadelphia, 1977.Google Scholar
  9. 9.
    L. Greengard, Spectral integration and two-point boundary value problems, SIAM J. Numer. Anal., 28 (1991), pp. 1071–1080.Google Scholar
  10. 10.
    E. Hairer, S. P. Nørsett, and G. Wanner, Solving Ordinary Differential Equations I, Non-Stiff Problems, Springer-Verlag, Berlin, 1993.Google Scholar
  11. 11.
    E. Hairer and G. Wanner, Solving Ordinary Differential Equations II, Springer-Verlag, Berlin, 1996.Google Scholar
  12. 12.
    D. J. Higham and L. N. Trefethen, Stiffness of ODEs, BIT, 33 (1993), pp. 285–303.Google Scholar
  13. 13.
    A. Iserles, A First Course in the Numerical Analysis of Differential Equations, Cambridge University Press, 1996.Google Scholar
  14. 14.
    R. Jeltsch and O. Nevanlinna, Stability and accuracy of time discretizations for initial value problems, Numer. Math., 40 (1982), pp. 245–296.Google Scholar
  15. 15.
    J. D. Lambert, Numerical Methods for Ordinary Differential Equations, Wiley, New York, 1991.Google Scholar
  16. 16.
    B. Lindberg, Error estimation and iterative improvement for discretization algorithms, BIT, 20 (1980), pp. 486–500.Google Scholar
  17. 17.
    V. Pereyra, Iterated deferred correction for nonlinear boundary value problems, Numer. Math., 11 (1968), pp. 111–125.Google Scholar
  18. 18.
    R. D. Skeel, A theoretical framework for proving accuracy results for deferred correction, SIAM J. Numer. Anal., 19 (1976), pp. 171–196.Google Scholar
  19. 19.
    J. Stoer and R. Bulirsch, Introduction to Numerical Analysis, Springer-Verlag, Berlin, 1992.Google Scholar
  20. 20.
    L. N. Trefethen and M. R. Trummer, An instability phenomenon in spectral methods, SIAM J. Numer. Anal., 24 (1987), pp. 1008–1023.Google Scholar
  21. 21.
    P. E. Zadunaisky, On the estimation of errors propagated in the numerical integration of ordinary differential equations, Numer. Math., 27 (1976), pp. 21–40.Google Scholar

Copyright information

© Swets & Zeitlinger 2000

Authors and Affiliations

  • Alok Dutt
    • 1
  • Leslie Greengard
    • 2
  • Vladimir Rokhlin
    • 3
  1. 1.Bank of AmericaLondonEngland
  2. 2.Courant Institute of Mathematical SciencesNew York UniversityNew YorkUSA.
  3. 3.Departments of Mathematics and Computer ScienceYale UniversityNew HavenUSA

Personalised recommendations