Iterated deferred correction for linear two-point boundary value problems
type: Item
instance of: scholarly article
title: Iterated deferred correction for linear two-point boundary value problems
zbMATH Open document ID: 0848.65059
author: H. H. M. Silva
author: Jeff R. Cash
published in: Computational and Applied Mathematics
publication date: 1996-10-27T00:00:00Z
review text: This paper deals with the numerical solution of two-point boundary value problems by the so-called iterated deferred correction. [For a precise statement of this procedure and a study of its convergence see: \textit{R. D. Skeel}, SIAM J. Numer. Anal. 19, 171-196 (1982; Zbl 0489.65051)]. Here the authors propose deferred corrections based on successive application of symmetric implicit Runge-Kutta methods which increase the order by two units in each iteration. In the first part of the paper a study of the linear stability of Gaussian and Lobatto IIIC formulas is presented. More precisely they require that the numerical solutions provided by the method for \(y'(x) = \lambda y (x)\), \(x \in [x_0, x_N]\) be bounded for \(y_0\) given and \(\text{Re }\lambda \leq 0\) and \(y_\nu\) given and \(\text{Re } \lambda \geq 0\). Then, they show that Lobatto IIC methods can be written as mono-implicit Runge-Kutta methods and by using the previous experience of one of the authors with this type of methods in the numerical solution of two-point boundary value problems, they have constructed a code that uses Lobatto formulas up to order eight. Finally, some numerical experiments comparing the behaviour of two standard codes: COLSYS and TWPBVP (available from NETLIB) and two new codes based on successive application of Gauss and Lobatto formulas, are presented. The main conclusions derived from these numerical experiments are that the new codes have a good stability. Further, the code based on Lobatto formulas has the advantage of a lower computational cost than the Gaussian code.
Mathematics Subject Classification ID: 65L10
Mathematics Subject Classification ID: 65L06
Mathematics Subject Classification ID: 34B15
zbMATH DE Number: 890678
zbMATH Keywords: two-point boundary value problems
zbMATH Keywords: iterated deferred correction
zbMATH Keywords: symmetric implicit Runge-Kutta methods
zbMATH Keywords: linear stability
zbMATH Keywords: Gaussian and Lobatto IIIC formulas
zbMATH Keywords: numerical experiments
reviewed by: Manuel Calvo
describes a project that uses: twpbvp
describes a project that uses: Netlib
describes a project that uses: COLSYS
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 890678