BIT Numerical Mathematics

, Volume 47, Issue 1, pp 197–212 | Cite as

Adaptive methods for piecewise polynomial collocation for ordinary differential equations

Article
First Online:
Received:
Accepted:

Abstract

Various adaptive methods for the solution of ordinary differential boundary value problems using piecewise polynomial collocation are considered. Five different criteria are compared using both interval subdivision and mesh redistribution. The methods are all based on choosing sub-intervals so that the criterion values have (approximately) equal values in each sub-interval. In addition to the main comparison it is shown by example that at least when accuracy is low then equidistribution may not give a unique solution.

The main results that using interval size times maximum residual as criterion gives very much better results than using maximum residual itself. It is also shown that a criterion based on a global error estimate while giving very good results in some cases, is unsatisfactory in other cases. The other criteria considered are that given by De Boor and the last Chebyshev series coefficient.

Key words

ordinary differential boundary value problems piecewise polynomial collocation adaptive algorithms 
This is a preview of subscription content, log in to check access.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    A. H. Ahmed and K. Wright, Error estimation for collocation solution of linear ordinary differential equations, Comp. Math. Appl., 12B (1986), pp. 1053–1059.CrossRefMathSciNetGoogle Scholar
  2. 2.
    A. H. Ahmed and K. Wright, Further asymptotic properties of collocation matrix norms, IMA J. Numer. Anal., 5 (1985), pp. 235–246.MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    U. M. Ascher, J. Christiansen, and R. D. Russell, Collocation software for boundary value ordinary differential equations, ACM Trans. Math. Softw., 7 (1981), pp. 209–222.MATHCrossRefGoogle Scholar
  4. 4.
    G. E. Carey and D. L. Humphrey, Finite element mesh refinement algorithm using element residuals, in Codes for Boundary-Value Problems in Ordinary Differential Equations, pp. 243–256. B. Childs et al., eds., Springer, New York, 1979.Google Scholar
  5. 5.
    Y. Ceo and L. Petzold, A posteriori error estimation and global error control for ordinary differential equations by the adjoint method, SIAM J. Sci. Comput., 26 (2004), pp. 359–374.CrossRefMathSciNetGoogle Scholar
  6. 6.
    C. de Boor, Good approximation by splines with variable knots II, in Proceedings of Conference on the Numerical Solution of Differential Equations, Dundee 1973, G. A. Watson, eds., Lect. Notes Math., vol. 363, pp. 12–20, Springer, Berlin, Heidelberg, New York, 1974.Google Scholar
  7. 7.
    C. de Boor and B. Swartz, Collocation at gaussian points, SIAM J. Numer. Anal., 10 (1973), pp. 582–606.MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    K. Eriksson, D. Estep, P. Hansbo, and C. Johnson, Introduction to adaptive methods for differential equations, Acta Numer., 4 (1995), pp. 105–158.MathSciNetCrossRefGoogle Scholar
  9. 9.
    E. Hermansyah, An Investigation of Collocation Algorithms for Solving Boundary Value Problems for Systems of ODEs, Ph.D. thesis, University of Newcastle upon Tyne, 2001.Google Scholar
  10. 10.
    C. Lanczos, Trigonometric interpolation of empirical and analytical functions, J. Math. Phys., 17 (1938), pp. 123–199.MATHGoogle Scholar
  11. 11.
    A. Logg, Mult-adaptive methods for ODEs 1, SIAM J. Sci. Comput., 24 (2003), pp. 1879–1902.MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    K.-S. Moon, A. Szepessy, R. Tempone, and G. E. Zouraris, Convergence rates for adaptive approximation of ordinary differential equations, Numer. Math., 96 (2003), pp. 99–129.MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    W. C. Rheinboldt, On a theory of mesh-refinement processes, SIAM J. Numer. Anal., 17 (1980), pp. 766–778.MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    W. C. Rheinboldt, Adaptive mesh refinement process for finite element solution, Int. J. Numer. Methods Eng., 17 (1981), pp. 649–662.MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    R. D. Russell and J. Christiansen, Adaptive mesh selection strategies for solving boundary value problems, SIAM J. Numer. Anal., 15 (1978), pp. 59–80.MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    K. Wright, A review of some developments in collocation algorithms, in Proceedings of IMA conference on computational ordinary differential equations, pp. 215–223, London 1989, J. R. Cash and I. Gladwell, eds., Oxford University Press, Oxford, 1992.Google Scholar
  17. 17.
    K. Wright, A. H. Ahmed, and A. H. Seleman, Mesh selection in collocation for boundary value problems, IMA J. Numer. Anal., 11 (1991), pp. 7–20.MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science + Business Media B.V. 2006

Authors and Affiliations

  1. 1.School of Computing ScienceUniversity of Newcastle upon TyneNewcastle upon TyneUK

Personalised recommendations