On the order of iterated defect correction
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SummaryIn 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].
[1] Reinhard Frank. Schätzungen des Globalen Diskretisierungsfehlers bei Runge-Kutta-Methoden , 1975 .
[2] Hans J. Stetter. Economical Global Error Estimation , 1974 .
[3] Error Estimation and Iterative Improvement for the Numerical Solution of Operator Equations. , 1976 .
[4] P. Zadunaisky. On the estimation of errors propagated in the numerical integration of ordinary differential equations , 1976 .