Skip to main content
SpringerLink
Log in

Error estimation and iterative improvement for discretization algorithms

  • Part II Numerical Mathematics
  • Published:
BIT Numerical Mathematics Aims and scope Submit manuscript

Abstract

A technique for estimating and iteratively correct for the smooth errors of discretization algorithms is presented. The theoretical foundation is given as a number of theorems. Some problems for ordinary differential equations are used as illustrative examples.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. C. Ballester and V. Pereyra,On the construction of discrete approximations to linear differential expressions, Math. Comp. 21 (1967), 297–302.

    Google Scholar 

  2. Å. Björck and G. Dahlquist,Numerical Methods, Prentice-Hall, Englewood Cliffs, New Jersey (1973).

    Google Scholar 

  3. Å. Björck and V. Pereyra,Solution of Vandermonde systems of equations, Math. Comp. 24 (1970), 893–903.

    Google Scholar 

  4. L. Fox,Some improvements in the use of relaxation methods for the solution of ordinary and partial differential equations, Proc. Roy. Soc. London Ser. A 190 (1947), 31–59.

    Google Scholar 

  5. R. Frank,The method of iterated defect-correction and its application for two-point boundary value problems I, Report 8/75, Institut für Numerische Mathematik, Technische Hochschule, Vienna (1975).

    Google Scholar 

  6. R. Frank, J. Hertling and C. Ueberhuber,An extension of the applicability of Iterated Deferred Corrections, Math. Comp. 31 (1977), 907–915.

    Google Scholar 

  7. G. Galimberti and V. Pereyra,Numerical differentiation and the solution of multidimensional Vandermonde systems, Math. Comp. 24 (1970), 357–364.

    Google Scholar 

  8. G. Galimberti and V. Pereyra,Solving confluent Vandermonde systems of Hermite type, Num. Math. 18 (1971), 44–60.

    Google Scholar 

  9. M. Lentini and V. Pereyra,A variable order finite difference method for nonlinear multipoint boundary value problems, Math. Comp. 28 (1974), 981–1003.

    Google Scholar 

  10. M. Lentini and V. Pereyra,Boundary problem solvers for first order systems based on deferred corrections, inNumerical Solution of Boundary Value Problems for Ordinary Differential Equations, Academic Press, New York (1975).

    Google Scholar 

  11. M. Lentini and V. Pereyra,An adaptive finite difference solver for nonlinear two-point boundary problems with mild boundary layers, Report STAN-CS-75-530, Computer Science Dept., Stanford University (1975).

  12. B. Lindberg,Error estimation and iterative improvement for the numerical solution of operator equations, Report UIUCDCS-R-76-820. Dept. of Computer Science, University of Illinois, Urbana (1976).

    Google Scholar 

  13. B. Lindberg,Compact deferred correction formulas, Report TRITA-NA-807, Dept. of Numerical Analysis and Computing Science, The Royal Inst. of Technology, Stockholm, Sweden (1980).

    Google Scholar 

  14. V. Pereyra,Iterated deferred corrections for nonlinear operator equations, Num. Math. 10 (1967), 316–323.

    Google Scholar 

  15. V. Pereyra,Accelerating the convergence of discretization algorithms, SIAM J. Numer. Anal. 4 (1967), 508–533.

    Google Scholar 

  16. V. Pereyra,Iterated deferred corrections for nonlinear boundary value problems, Num. Math. 11 (1968), 111–125.

    Google Scholar 

  17. V. Pereyra,Highly accurate numerical solution of quasilinear elliptic boundary value problems in n dimensions, Math. Comp. 24 (1970), 771–783.

    Google Scholar 

  18. V. Pereyra,High order finite difference solution of differential equations, Report STAN-CS-73-348, Computer Science Dept., Stanford University (1973).

  19. H. J. Stetter,Asymptotic expansions for the error of discretization algorithms for nonlinear functional equations, Num. Math. 7 (1965), 18–31.

    Google Scholar 

  20. H. J. Stetter,Analysis of Discretization Methods for Ordinary Differential Equations, Springer Verlag, New York (1973).

    Google Scholar 

  21. H. J. Stetter,Economical global error estimation, inStiff Differential Systems, R. A. Willoughby (ed.), Plenum Press, New York (1974).

    Google Scholar 

  22. H. J. Stetter,The defect correction principle and discretization methods, Num. Math. 29 (1978), 425–443.

    Google Scholar 

  23. P. E. Zadunaisky,A method for the estimation of errors propagated in the numerical solution of a system of ordinary differential equations, Proc. Astron. Union, Symposium No. 25, Academic Press, New York (1966).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dept. of Numerical Analysis and Computing Science, The Royal Inst. of Technology, Stockholm, Sweden

    Bengt Lindberg

Authors
  1. Bengt Lindberg
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Lindberg, B. Error estimation and iterative improvement for discretization algorithms. BIT 20, 486–500 (1980). https://doi.org/10.1007/BF01933642

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01933642

Keywords

Search

Navigation