Stability of numerical methods for ordinary differential equations
type: Item
instance of: scholarly article
title: Stability of numerical methods for ordinary differential equations
zbMATH Open document ID: 1016.65059
DOI: 10.1023/A:1021108006254
author: A. D. Heard
author: J. C. Butcher
published in: Numerical Algorithms
publication date: 2003-03-10T00:00:00Z
review text: This paper is concerned with the stability properties of three multivalue methods on non uniform grids. In the first case the authors consider the \(A(\alpha)\)-stability of the two step backward differentiation formula (BDF) method with variable coefficients. Here the \( 2 \times 2 \) stability matrix \( A = A(r,z)\) depends on two parameters: the step size ratio \( r= h_n/h_{n-1}\) between two consecutive steps and the complex variable \(z\). By choosing a suitable real matrix \(T\) the authors prove that the norm \( \|A \|= \|T^{-1} A T \|_1 \leq 0\) for all \( r \in [0, ( 1 + \sqrt 3)/ 2 ]\) and \( |\arg z - \pi |\leq 70^0\) which implies \(A(\alpha)\)-stability for arbitrary step size ratios in the range \([0, ( 1 + \sqrt 3)/ 2 ]\). The second multivalue method is a first order two step general linear method whose stability matrix \( M(r,z)\) is a \(( 2 \times 2)\) matrix of rank one. Here by using the factorization \( M(r,z)= v u^T \) the authors show that any product of stability matrices is \(A(0)\)-stable for all \( r \in [0, 2.94].\) Finally the zero stability of the three step BDF method with variable coefficients is considered. Now the propagation matrix can be written as a \((2 \times 2)\) matrix depending on two consecutive step size ratios \( C (r_n, r_{n-1})\) and using a discrete Lyapunov type function the authors show that zero stability holds for arbitrary \( r \leq ( 1 + \sqrt 5)/2 \) which is an optimal step size bound for arbitrary step sizes. It must be noticed that similar results for BDF methods (not mentioned in the present paper) have been obtained, among others, by Brayton and Conley, Dahlquist-Liniger-Nevanlinna, Grigorieff, Calvo-Grande-Grigorieff, and Sand.
Mathematics Subject Classification ID: 65L20
Mathematics Subject Classification ID: 65L06
Mathematics Subject Classification ID: 65L05
Mathematics Subject Classification ID: 34A34
Mathematics Subject Classification ID: 65L50
zbMATH DE Number: 1879359
zbMATH Keywords: initial value problems
zbMATH Keywords: multivalue methods
zbMATH Keywords: variable stepsize
zbMATH Keywords: backward differentiation formula method
zbMATH Keywords: BDF methods
reviewed by: Manuel Calvo
MaRDI profile type: Publication
full work available at URL: a:1021108006254
OpenAlex ID: W1522407833
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description: scientific article; zbMATH DE number 1879359