Numerische Mathematik

, Volume 29, Issue 4, pp 409–424 | Cite as

On the order of iterated defect correction

An algebraic proof
  • E. Hairer


In a recent article [2] Frank and Überhuber define and motivate the method of iterated defect correction for Runge-Kutta methods. They prove a theorem on the order of that method using the theory of asymptotic expansions.

In this paper we give similar results using the theory of Butcher series (see [4]). Our proofs are purely algebraic. We don't restrict our considerations to Runge-Kutta methods, but we admit arbitrary linear one-step methods. At the same time we consider more general defect functions as in [2].

Subject Classifications

AMS (MOS): 65LO5 CR: 5.17 


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  1. 1.
    Frank, R.: Schätzungen des globalen Diskretisierungsfehlers bei Runge-Kutta-Verfahren. ISNM, Vol. 27, pp. 45–70. Basel-Stuttgart: Birkhäuser 1975Google Scholar
  2. 2.
    Frank, R., Überhuber, C.W.: Iterated defect correction for Runge-Kutta methods. Report No. 14/75, Inst. f. Numer. Math., Technical University Vienna, 1975Google Scholar
  3. 3.
    Hairer, E., Wanner, G.: Multistep-multistage-multiderivative methods for ordinary differential equations. Computing11, 287–303 (1973)Google Scholar
  4. 4.
    Hairer, E., Wanner, G.: On the Butcher group and general multi-value methods. Computing13, 1–15 (1974)Google Scholar
  5. 5.
    Lindberg, B.: Error estimation and iterative improvement for the numerical solution of operator equations. Report R-76-820, Dept. of Computer Science, University of Illinois at Urbana-Champaign, July 1976Google Scholar
  6. 6.
    Stetter, H.J.: Economical global error estimation. In: Stiff differential systems. (R.A. Willoughby, ed.), pp. 245–258. New York-London: Plenum Press 1974Google Scholar
  7. 7.
    Wanner, G.: Integration gewöhnlicher Differentialgleichungen (B.I. 831/831 a). Mannheim: Bibl. Institut 1969Google Scholar
  8. 8.
    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

Copyright information

© Springer-Verlag 1978

Authors and Affiliations

  • E. Hairer
    • 1
  1. 1.Institut für Mathematik der Universität InnsbruckInnsbruckAustria

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