Error of Rosenbrock methods for stiff problems studied via differential algebraic equations
type: Item
instance of: scholarly article
title: Error of Rosenbrock methods for stiff problems studied via differential algebraic equations
zbMATH Open document ID: 0674.65039
DOI: 10.1007/BF01932707
published in: BIT
publication date: 1989-01-01T00:00:00Z
review text: This paper studies Rosenbrock methods when they are applied to stiff problems in singular perturbation form \(y'=f(y,z),\quad \epsilon z'=g(y,z)\) \((0<\epsilon \ll 1)\) with initial values y(0), z(0) admitting a smooth solution. Let the Rosenbrock methods have classical order p and differential algebraic order r and satisfy a certain condition on the stability function R(\(\omega)\) (e.g. which is fulfilled if the method is A-stable). Then the following interesting convergence results hold: \(y_ n-y(x_ n)=O(h^ r)+O(\epsilon h^ 2),\quad z_ n-z(x_ n)=O(h^ r)+O(\epsilon h),\) and, in the case of \(p>r=2\), \(y_ n-y(x_ n)=O(h^ 3),\quad z_ n-z(x_ n)=O(h^ 2).\) The basic ideas and techniques of the proof are the same as those developed for Runge-Kutta methods in an earlier paper of the authors [ibid. 28, 678-700 (1988; Zbl 0657.65093)], but there is often considerable difference in the details. The results are illustrated by numerical examples.
Mathematics Subject Classification ID: 65L05
Mathematics Subject Classification ID: 65L20
Mathematics Subject Classification ID: 34A34
Mathematics Subject Classification ID: 34E15
zbMATH DE Number: 4104281
zbMATH Keywords: differential algebraic equations
zbMATH Keywords: error estimates
zbMATH Keywords: Rosenbrock methods
zbMATH Keywords: stiff problems
zbMATH Keywords: singular perturbation
zbMATH Keywords: stability
zbMATH Keywords: convergence
zbMATH Keywords: Runge- Kutta methods
zbMATH Keywords: numerical examples
author: Ernst Hairer
author: Christian Lubich
author: Michel Roche
reviewed by: Rudolf Scherer
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01932707
OpenAlex ID: W4233696007
description: scientific article; zbMATH DE number 4104281