Families of two-step fourth order \(P\)-stable methods for second order differential equations
type: Item
instance of: scholarly article
title: Families of two-step fourth order \(P\)-stable methods for second order differential equations
zbMATH Open document ID: 0599.65044
DOI: 10.1016/0377-0427(86)90028-2
author: Man M. Chawla
author: Beny Neta
published in: Journal of Computational and Applied Mathematics
publication date: 1986-01-01T00:00:00Z
review text: In the numerical integration of initial value problems for second order ordinary differential equations, \textit{J. D. Lambert} and \textit{I. A. Watson} [J. Inst. Math. Appl. 18, 189-202 (1976; Zbl 0359.65060)] introduced the concept of \(P\)-stability as a desirable property for numerical methods to possess. P-stable linear multistep methods suffer from an order barrier of two. This barrier can however be overcome within the class of hybrid methods, as shown, independently, by \textit{M. M. Chawla} [BIT 21, 190-193 (1981; Zbl 0457.65053)] and \textit{J. R. Cash} [Numer. Math. 37, 355-370 (1981; Zbl 0488.65029)], who constructed fourth order, two-step methods which normally require three function evaluations per step. The present contribution shows how an appropriate choice of the free parameters in the Chawla-Cash family can result in computational savings.
Mathematics Subject Classification ID: 65L05
Mathematics Subject Classification ID: 65L20
Mathematics Subject Classification ID: 34A34
zbMATH DE Number: 3967819
zbMATH Keywords: second order equations
zbMATH Keywords: two-step fourth order methods
zbMATH Keywords: economical methods
zbMATH Keywords: efficient methods
zbMATH Keywords: P-stability
zbMATH Keywords: linear multistep methods
zbMATH Keywords: order barrier
zbMATH Keywords: hybrid methods
reviewed by: Jesús María Sanz-Serna
MaRDI profile type: Publication
full work available at URL: 0377-0427(86)90028-2
OpenAlex ID: W2075074779
cites work: High order P-stable formulae for the numerical integration of periodic initial value problems
cites work: Efficient P-stable methods for periodic initial value problems
cites work: Two-step fourth order P-stable methods for second order differential equations
cites work: Unconditionally stable Noumerov-type methods for second order differential equations
cites work: Numerical integration of ordinary differential equations based on trigonometric polynomials
cites work: Unconditionally stable methods for second order differential equations
cites work: Symmetric Multistip Methods for Periodic Initial Value Problems
cites work: Families of methods for ordinary differential equations based on trigonometric polynomials
cites work: Stabilization of Cowell's classical finite difference method for numerical integration
Recommended article: A two-stage fourth-order ``almost'' P-stable method for oscillatory problems
Recommended article: A family of second order methods for variable coefficient fourth order parabolic partial differential equations
Recommended article: Q3719739
Recommended article: Attainable order of theP-stable family of certain two-step methods for periodic second order initial value problems
Recommended article: Two-step fourth order methods for linear ODEs of the second order
Recommended article: Q3738491
Recommended article: Characterization of a class of P-stable methods for differential equations of second order
Recommended article: Stability of a Family of Iterative Methods of Fourth-Order
Recommended article: \(P\)-stable Obrechkoff methods of arbitrary order for second-order differential equations
Recommended article: A New Explicit Singularly P-Stable Four-Step Method for the Numerical Solution of Second Order IVPs
description: scientific article; zbMATH DE number 3967819