Stability and B-convergence of linearly implicit Runge-Kutta methods
type: Item
instance of: scholarly article
title: Stability and B-convergence of linearly implicit Runge-Kutta methods
zbMATH Open document ID: 0611.65045
DOI: 10.1007/BF01389669
published in: Numerische Mathematik
publication date: 1986-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/133140
review text: The investigation of B-convergence has been initiated by \textit{R. Frank}, \textit{J. Schneid} and \textit{C. W. Ueberhuber} [SIAM J. Numer. Anal. 18, 753-780 (1981; Zbl 0467.65032)] for implicit Runge-Kutta methods. They considered general stiff differential equations satisfying a one-sided Lipschitz condition. Since similar results cannot be expected for linearly implicit methods (such as W-methods, Rosenbrock methods,...), the author restricts his analysis to semi-linear problems, where the stiffness is present only in the linear part of the differential equation. This article presents a stability analysis which includes stability with respect to internal perturbations. This allows to prove that (under suitable stability assumptions) linearly implicit methods are B-convergent of order at least 1, i.e. the global error is O(h), where the constant in the O(.)-symbol is independent of the stiffness. It is left as an open question whether B-convergence of an order \(p>1\) can be achieved. For linear differential equations with constant coefficients a complete characterization of the order of B-convergence is given.
reviewed by: Ernst Hairer
Mathematics Subject Classification ID: 65L05
Mathematics Subject Classification ID: 65L20
Mathematics Subject Classification ID: 34A34
zbMATH DE Number: 3988633
zbMATH Keywords: B-convergence
zbMATH Keywords: implicit Runge-Kutta methods
zbMATH Keywords: stiff differential equations
zbMATH Keywords: linearly implicit methods
zbMATH Keywords: W-methods
zbMATH Keywords: Rosenbrock methods
zbMATH Keywords: global error
author: Willem Hundsdorfer
MaRDI profile type: Publication
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cites work: An Attempt to Avoid Exact Jacobian and Nonlinear Equations in the Numerical Solution of Stiff Differential Equations
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description: scientific article; zbMATH DE number 3988633