A scheme for the implementation of implicit Runge-Kutta methods
type: Item
instance of: scholarly article
title: A scheme for the implementation of implicit Runge-Kutta methods
zbMATH Open document ID: 0726.65083
DOI: 10.1007/BF02238800
author: G. J. Cooper
author: R. Vignesvaran
published in: Computing
publication date: 1990-01-01T00:00:00Z
review text: The application of an s-stage implicit Runge-Kutta method to a stiff ordinary differential equation (dimension n) leads to a nonlinear system in \({\mathbb{R}}^{s\cdot n}\). For its solution, an iterative scheme is proposed which requires only one LU-decomposition of a real \(n\times n\) matrix (also in the case where the Runge-Kutta matrix has different and/or complex eigenvalues). Compared to the iteration proposed by \textit{G. J. Cooper} and \textit{J. C. Butcher} [IMA J. Numer. Anal. 3, 127-140 (1983; Zbl 0525.65052)] less matrix vector multiplications (with a cost of \({\mathcal O}(s^ 2n)\) operations) are needed. For the Gauss methods of order 2s optimal values for the free parameters in the iteration are presented. The convergence rates are confirmed by numerical examples.
Mathematics Subject Classification ID: 65L06
Mathematics Subject Classification ID: 65H10
Mathematics Subject Classification ID: 65L05
Mathematics Subject Classification ID: 34A34
Mathematics Subject Classification ID: 34E13
zbMATH DE Number: 4199515
zbMATH Keywords: implementation
zbMATH Keywords: stiff equations
zbMATH Keywords: implicit Runge-Kutta method
zbMATH Keywords: nonlinear system
zbMATH Keywords: iterative scheme
zbMATH Keywords: LU-decomposition
zbMATH Keywords: Gauss methods
zbMATH Keywords: convergence rates
zbMATH Keywords: numerical examples
reviewed by: Ernst Hairer
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02238800
OpenAlex ID: W1607052731
description: scientific article; zbMATH DE number 4199515