Summary
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].
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References
Frank, R.: Schätzungen des globalen Diskretisierungsfehlers bei Runge-Kutta-Verfahren. ISNM, Vol. 27, pp. 45–70. Basel-Stuttgart: Birkhäuser 1975
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Hairer, E. On the order of iterated defect correction. Numer. Math. 29, 409–424 (1978). https://doi.org/10.1007/BF01432878
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DOI: https://doi.org/10.1007/BF01432878