Invariant curves for variable step size integrators
type: Item
instance of: scholarly article
title: Invariant curves for variable step size integrators
zbMATH Open document ID: 0724.65070
DOI: 10.1007/BF01952792
author: Daniel Stoffer
author: Kaspar Nipp
published in: BIT
publication date: 1991-01-01T00:00:00Z
review text: The behaviour of the numerical integration of the ordinary differential equation (1) \(\dot x=f(x)\), \(x\in \Omega \subset {\mathbb{R}}^ n\), (with a hyperbolic periodic solution) applying a one-step method with variable step size is investigated. It is shown that there exists an attractive (hyperbolic) invariant closed curve for the numerical method as in the case of one-step method with constant step size. If \(\phi\) is the integration method of order p and H is the step-size function it is shown that under general assumptions on H there is a function s(x,\(\epsilon\)) such that the set \(M_{\epsilon}:=\{(h,x)| h=\epsilon s(x,\epsilon)\), \(x\in \Omega \}\), is an attractive invariant manifold for the map \(\Psi:\;\left( \begin{matrix} h\\ x\end{matrix} \right)\to \left( \begin{matrix} \bar h\\ \bar x\end{matrix} \right)=\left( \begin{matrix} H(h,x,TOL)\\ \phi (h,x)\quad \quad \quad \end{matrix} \right),\) where \(\epsilon:=TOL^{1/p}\), \(TOL=| l(h,x)| /h\), \(\rho (h,x)=\phi (h,x)-\phi (h,x)\) with \(\rho\) (t,x) the solution of equation (1) with initial condition x and \(\phi\) (h,x) a map of an integration method (one- step method) of order p. So the step-size, asymptotically, does not depend on the previous step size but only depends on the position x and the parameter TOL (or \(\epsilon\)). Under time transformations which depend on the position x equation (1) takes the form \((2)\quad \frac{dx}{d\tau}=r(x)f(x),\) with r(x) a positive scalar function. With a proper choice of r(x) the variable step size method \(\phi\) of equation (1) is transformed into an integration method \(\phi\) of equation (2) with constant step size. So, general properties of constant step size methods carry over to variable step size methods. The results of one-step methods with constant step size may be applied to equation (2) and may be transferred to equation (1) where an integration method with variable step is applied.
reviewed by: Maria Gousidou-Koutita
Mathematics Subject Classification ID: 65L05
Mathematics Subject Classification ID: 65L50
Mathematics Subject Classification ID: 34C05
Mathematics Subject Classification ID: 34A34
zbMATH DE Number: 4195074
zbMATH Keywords: invariant curves
zbMATH Keywords: system
zbMATH Keywords: hyperbolic periodic solution
zbMATH Keywords: one-step method
zbMATH Keywords: variable step size
zbMATH Keywords: attractive invariant manifold
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01952792
OpenAlex ID: W2088953326
description: scientific article; zbMATH DE number 4195074