Superstable single-step methods for second-order initial-value problems
type: Item
instance of: scholarly article
title: Superstable single-step methods for second-order initial-value problems
zbMATH Open document ID: 0653.65053
DOI: 10.1016/0377-0427(88)90337-8
author: Rakesh Goel
author: Rakesh K. Jain
published in: Journal of Computational and Applied Mathematics
publication date: 1988-01-01T00:00:00Z
review text: This paper is concerned with the numerical solution of initial value problems for general second-order differential equations. Two implicit one step methods of orders 4 and 6 which are superstable (except for isolated values of the parameters) in the sense of Chawla are proposed. Here the term superstability means that when the method is applied to the test equation \(y''+2\alpha y'+\beta \quad 2y=0\) with \(\alpha\),\(\beta\geq 0\), \(\alpha +\beta >0\), we get a linear difference equation whose characteristic polynomial \(\rho (x)=A_ 1x\quad 2+A_ 2x+A_ 3\) \((A_ i=A_ i(h\alpha,h\beta))\) satisfies: i) If \(\rho (x)=0\), then \(| x| <1\) for all \(h\alpha >0\), \(h\beta >0\). ii) If \(\alpha =0\), the roots of \(\rho\) are complex conjugate and with modulus one. iii) If \(\beta =0\) the two roots of \(\rho\) (x) are such that \(x_ 1=1\) and \(| x_ 2| <1\).
Mathematics Subject Classification ID: 65L05
Mathematics Subject Classification ID: 65L20
Mathematics Subject Classification ID: 34A34
zbMATH DE Number: 4066249
zbMATH Keywords: second-order differential equations
zbMATH Keywords: implicit one step methods
zbMATH Keywords: superstability
MaRDI profile type: Publication
full work available at URL: 0377-0427(88)90337-8
OpenAlex ID: W2068967193
cites work: Superstable two-step methods for the numerical integration of general second order initial value problems
cites work: Unconditionally stable Noumerov-type methods for second order differential equations
cites work: Q3915947
cites work: A fourth order method for y''=f(x,y,y')
cites work: Symmetric Multistip Methods for Periodic Initial Value Problems
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description: scientific article; zbMATH DE number 4066249