On the relation between stability and contractivity
type: Item
instance of: scholarly article
title: On the relation between stability and contractivity
zbMATH Open document ID: 0557.65044
DOI: 10.1007/BF01934922
author: Marc N. Spijker
published in: BIT
publication date: 1984-01-01T00:00:00Z
review text: The author treats linear initial value problems \[ \frac{d}{dt}U(t)=A(t)U(t),\quad U(0)=u_ 0, \] where, for an arbitrary fixed norm on \({\mathbb{R}}^ s\), \(\| U(t)\|\) is non-increasing. A general type of method is considered which generates a sequence \(\{u_ n\}\) of approximations to \(\{U(t_ n)\}\) where, with arbitrary steplengths, \(t_ n=h_ 1+h_ 2+...+h_ n\). A method is weakly stable if there is a bound \(\| u_ n\| \leq \gamma n^ q\| u_ 0\|,\) \(n=1,2,3,..\).. It is shown that for single step methods where \(u_ n=\phi (A_ n)u_{n-1},\) \(n=1,2,3,...\), with \(\phi\) a rational function, weak stability implies the stronger contractivity bound \(\| u_ n\| \leq \| u_{n-1}\|\), \(n=1,2,3,..\).. It is known that this implies that the order is at most one. It is proved that this is also the case for more general methods under the assumption of weak stability. The order barrier does not hold if attention is restricted to norms generated by inner products.
Mathematics Subject Classification ID: 65L05
Mathematics Subject Classification ID: 65L20
Mathematics Subject Classification ID: 34A30
zbMATH DE Number: 3888824
zbMATH Keywords: single step methods
zbMATH Keywords: weak stability
zbMATH Keywords: contractivity bound
zbMATH Keywords: order barrier
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01934922
OpenAlex ID: W2011995202
description: scientific article; zbMATH DE number 3888824