Efficient sixth order methods for nonlinear oscillation problems
type: Item
instance of: scholarly article
title: Efficient sixth order methods for nonlinear oscillation problems
zbMATH Open document ID: 0667.65064
DOI: 10.1007/BF01954909
author: S. H. Smith
published in: BIT
publication date: 1988-01-01T00:00:00Z
review text: By applying two-step (hybrid) methods to solve initial value problems of the kind (1) \(\ddot x=f(t,x)\), \(x(0)\), \(\dot x(0)\) given with nonlinear f, one has to solve a nonlinear algebraic system at each step. This may be done by a modified Newton iteration scheme. The author derives a class of fourth order, P-stable methods which require only two function evaluation per iteration and one matrix factorization at each step. For a definition of P-stability see \textit{J. D. Lambert} and \textit{I. A. Watson} [J. Inst. Math. Appl. 18, 189-202 (1976; Zbl 0359.65060)]. Furthermore there is propsed an extension to sixth order P-stable methods which require only three function evaluation per iteration. These methods have been implemented on a new variable step, local error controlling code.
Mathematics Subject Classification ID: 65L05
Mathematics Subject Classification ID: 65L20
Mathematics Subject Classification ID: 34A34
zbMATH DE Number: 4092688
zbMATH Keywords: two-step (hybrid) methods
zbMATH Keywords: Newton iteration
zbMATH Keywords: matrix factorization
zbMATH Keywords: P- stability
zbMATH Keywords: sixth order P-stable methods
zbMATH Keywords: variable step, local error controlling code
reviewed by: Hans Ade
MaRDI profile type: Publication
cites work: High order P-stable formulae for the numerical integration of periodic initial value problems
cites work: High-accuracy P-stable Methods for y″ = f(t, y)
cites work: Efficiency of Methods for Second-Order Problems
cites work: Symmetric Multistip Methods for Periodic Initial Value Problems
cites work: Implementation of Implicit Formulas for the Solution of ODE<scp>s</scp>
cites work: Efficient fourth order P-stable formulae
full work available at URL: https://doi.org/10.1007/bf01954909
OpenAlex ID: W2031223160
description: scientific article; zbMATH DE number 4092688