An improved starting step of the G-B-S-method for the solution of ordinary differential equations
type: Item
instance of: scholarly article
title: An improved starting step of the G-B-S-method for the solution of ordinary differential equations
zbMATH Open document ID: 0662.65065
DOI: 10.1007/BF02238735
published in: Computing
publication date: 1989-01-01T00:00:00Z
review text: This paper proposes a modification of the starting step when the midpoint rule is applied to extrapolation methods in the numerical solution of ordinary differential equations. In traditional extrapolation procedures [cf. \textit{R. Bulirsch} and \textit{J. Stoer}, Numer. Math. 8, 1-13 (1966; Zbl 0135.379)], \textit{P. Deuflhard} [Preprint Univ. Heidelberg (1980)] the additional approximation required to apply the midpoint rule is computed by Euler's method and a final smoothing procedure is included to improve the stability of the method. Here the final smoothing has been skipped and the Euler step is replaced by \(y_ 1=y_ 0+hf(x_ 0,y_ 0)+(h^ 2/2)c(x_ 0,y_ 0),\) where \(c(x_ 0,y_ 0)\) is an approximation to \(y''(x_ 0)\) given by \(H^{-1}[f(x_ 0+H,y_ 0+Hf(x_ 0,y_ 0)-f(x_ 0,y_ 0)].\) The aim of this new starting step is to reduce the oscillating terms in the \(h^ 2\)-expansion of the error. The paper ends comparing the extrapolation code DIFEX1 (due to Deuflhard) with two modified versions of it for the test problems of DETEST. It is shown that, with the modified code, an average saving of about 8 \% in the total number of function calls is obtained.
Mathematics Subject Classification ID: 65L05
Mathematics Subject Classification ID: 65B05
Mathematics Subject Classification ID: 65G50
Mathematics Subject Classification ID: 34A34
zbMATH DE Number: 4082870
zbMATH Keywords: Gregg-Bulirsch-Stoer method
zbMATH Keywords: starting step
zbMATH Keywords: midpoint rule
zbMATH Keywords: extrapolation methods
zbMATH Keywords: Euler's method
zbMATH Keywords: final smoothing
zbMATH Keywords: stability
author: Martin Kiehl
author: Christoph Zenger
MaRDI profile type: Publication
cites work: Numerical treatment of ordinary differential equations by extrapolation methods
cites work: Order and stepsize control in extrapolation methods
cites work: On Extrapolation Algorithms for Ordinary Initial Value Problems
cites work: Comparing Numerical Methods for Ordinary Differential Equations
cites work: Smoothing the extrapolated midpoint rule
cites work: Q4403232
full work available at URL: https://doi.org/10.1007/bf02238735
OpenAlex ID: W207043387
description: scientific article; zbMATH DE number 4082870