Invariant manifolds and global error estimates of numerical integration schemes applied to stiff systems of singular perturbation type. I: RK- methods
type: Item
instance of: scholarly article
title: Invariant manifolds and global error estimates of numerical integration schemes applied to stiff systems of singular perturbation type. I: RK- methods
zbMATH Open document ID: 0824.65058
published in: Numerische Mathematik
publication date: 1995-11-13T00:00:00Z
review text: For implicit Runge-Kutta (RK) methods applied to singularly perturbed systems of ordinary differential equations it is shown that the resulting discrete systems preserve the geometric properties of the underlying differential equation. In particular, if the perturbation parameter \(\varepsilon\) is small, under the usual assumptions, the singularly perturbed system has a highly attractive invariant manifold \(M_ \varepsilon\). For stepsize \(h\) fixed and \(\varepsilon\) small enough, the discrete system given by an implicit RK-method admits an attractive invariant manifold \(M_{h,\varepsilon}\) close to \(M_ \varepsilon\). This invariant manifold result is used to derive sharp bounds on the global error of RK-solutions.
Mathematics Subject Classification ID: 65L06
Mathematics Subject Classification ID: 65L50
Mathematics Subject Classification ID: 34A34
Mathematics Subject Classification ID: 34E13
Mathematics Subject Classification ID: 34C30
zbMATH DE Number: 770158
zbMATH Keywords: global error estimates
zbMATH Keywords: stiff systems
zbMATH Keywords: singular perturbation
zbMATH Keywords: implicit Runge-Kutta methods
zbMATH Keywords: highly attractive invariant manifold
author: Kaspar Nipp
author: Daniel Stoffer
reviewed by: Stephen L. Campbell
MaRDI profile type: Publication
full work available at URL: s002110050118
OpenAlex ID: W2079168690
DOI: 10.1007/S002110050118
description: scientific article; zbMATH DE number 770158