Deferred corrections using uncentered end formulas

Abstract

Summary.The trapezoidal rule with deferred corrections using uncentered end formulas is shown to converge. While the proof technique is more specialized than the standard asymptotic expansion approach, it has some advantages. In addition to providing a more complete theoretical justification for current implementations of deferred corrections with the trapezoidal rule, the approach given here will hopefully apply for several other discretization methods.

Numerical Methods for Initial Value Problems.

“…They suggest how this might be proved but do not follow through because "Such a proof would be quite tedious." In this paper we sketch a proof of this fact, which we believe is less tedious than that of Christiansen and Russell (1979) due to the way in which we break down the proof into smaller sinpZy stated results.…”

“…(Lindberg; Van Rosendale, Skeel) 3. The Order of Accuracy for a Deferred Corrections Algorithm (Skeel) This work will constitute the future paper mentioned in Skeel (1980, section 4, second paragraph) in which we give an error analysis for a sequence of iterations for the algorithm considered by Christiansen and Russell (1979). This is intended to be a realistic example of the application of the theoretical framework for proving accuracy results, which because of its length was not included in .…”

“…satisfies the assumptions of Christiansen and Russell (1979) so that in particular the operator equation F(y) 0 has an isolated solution y.…”

“…In Christiansen and Russell (1979) a careful analysis of deferred corrections that does not involve asymptotic expansions is done for a realistic algorithm similar to the implementation of Lentini and Pereyra (1977) of iterated deferred corrections for two-point boundary value…”

Defect correction from a Galerkin viewpoint

“…Of course the key theoretical problem is to show that can be used h y to approximate T 1 y to 0(h 2 ), and similarly for the higher-order T k , and On the other hand defect correction [3,9,12,13] …”