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The defect correction principle and discretization methods

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Summary

Recently, a number of closely related techniques for error estimation and iterative improvement in discretization algorithms have been proposed. In this article, we expose the common structural principle of all these techniques and exhibit the principal modes of its implementation in a discretization context.

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References

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

    Google Scholar 

  2. Zadunaisky, P.E.: On the accuracy in the numerical computation of orbits. In: Periodic orbits, stability and resonances (G.E.O. Giacaglia, ed.), pp. 216–227. Dordrecht: Reidel 1972

    Google Scholar 

  3. Zadunaisky, P.E.: On the estimation of errors propagated in the numerical integration of ordinary differential equations. Numer. Math.27, 21–39 (1976)

    Google Scholar 

  4. Stetter, H.J.: Economical global error estimation. In: Stiff differential systems (R.A. Willoughby, ed.), pp. 245–258. New York-London: Plenum Press 1974

    Google Scholar 

  5. Stetter, H.J.: Analysis of discretization methods for ordinary differential equations. Berlin-Heidelberg-New York: Springer 1973

    Google Scholar 

  6. Frank, R.: Schätzungen des globalen Diskretisierungsfehlers bei Runge-Kutta-Methoden, ISNM Vol. 27, pp. 45–70. Basel-Stuttgart: Birkhäuser 1975

    Google Scholar 

  7. Frank, R.: The method of iterated defect-correction and its application to two-point boundary value problems. Part I: Numer. Math.25, 409–419 (1976); Part II: Numer. Math.27, 407–420 (1977)

    Google Scholar 

  8. Frank, R., Ueberhuber, C.W.: Iterated defect correction for Runge-Kutta methods. Report No. 14/75, Inst. f. Numer. Math., Technical University of Vienna, 22 pp., 1975

  9. Frank, R., Ueberhuber, C.W.: Iterated defect correction for the efficient solution of stiff systems of ordinary differential equations. Nordisk Tidskr. Informationsbehandling (BIT)17, 146–159 (1977)

    Google Scholar 

  10. Frank, R., Hertling, J., Ueberhuber, C.W.: Iterated defect correction based on estimates of the local discretization error. Report No. 18/76, Inst. f. Numer. Math., Technical University of Vienna, 21 pp., 1976

  11. Frank, R., Ueberhuber, C.W.: Collocation and iterated defect correction. In: Numerical treatment of differential equations (R. Bulirsch, R.D. Grigorieff, J. Schröder, eds.), pp. 19–34. Lecture notes in Mathematics Vol. 631. Berlin-Heidelberg-New York: Springer 1978

    Google Scholar 

  12. Frank, R., Hertling, J., Ueberhuber, C.W.: An extension of the applicability of iterated deferred corrections. Report No. 23/76, Inst. f. Numer. Math., Technical University of Vienna, 20 pp., 1976 (to appear in Math. Comput.).

  13. Fox, L.: The numerical solution of two-point boundary value problems in ordinary differential equations. Oxford: University Press 1957

    Google Scholar 

  14. Pereyra, V.L.: On improving an approximate solution of a functional equation by deferred corrections. Numer. Math.8, 376–391 (1966)

    Google Scholar 

  15. Pereyra, V.L.: Iterated deferred corrections for nonlinear operator equations. Numer. Math.10, 316–323 (1967)

    Google Scholar 

  16. Lindberg, B.: Error estimation and iterative improvement for the numerical solution of operator equations. Report UIUCDCS-R-76-820, Dept. of Computer Science, Univ. of Ill., Urbana, 1976

    Google Scholar 

  17. Daniel, J.W., Martin, A.J.: Implementing deferred corrections for Numeroy's difference method for second-order two-point boundary-value problems. Report CNA-107, Center for Numerical Analysis, Univ. of Texas, Austin, 1975

    Google Scholar 

  18. Lentini, M., Pereyra, V.L.: Boundary problem solvers for first order systems based on deferred corrections. In: Numerical solutions of boundary value problems for ordinary differential equations, (A.K. Aziz, ed.), New York: Academic Press 1975

    Google Scholar 

  19. Pereyra, V.L.: Iterated deferred corrections for nonlinear boundary value problems. Numer. Math.11, 111–125 (1968)

    Google Scholar 

  20. Stetter, H.J.: Global error estimation in Adams PC-codes. TOMS (to appear)

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Authors and Affiliations

  1. Institut für Numerische Mathematik der Technischen Universität, Gusshausstr. 27, A-1040, Wien, Austria

    Hans J. Stetter

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  1. Hans J. Stetter
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Written during a sabbatical stay at Oxford University partially supported by the British Science and Research Council

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Stetter, H.J. The defect correction principle and discretization methods. Numer. Math. 29, 425–443 (1978). https://doi.org/10.1007/BF01432879

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  • DOI: https://doi.org/10.1007/BF01432879

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