An introduction to ``Almost Runge-Kutta'' methods
type: Item
instance of: scholarly article
title: An introduction to ``Almost Runge-Kutta'' methods
zbMATH Open document ID: 0905.65085
DOI: 10.1016/S0168-9274(97)00030-5
published in: Applied Numerical Mathematics
publication date: 1999-01-03T00:00:00Z
review text: The author introduces a new class of 4-stage methods of order \(p=4\) which are close to corresponding explicit Runge-Kutta methods but multivalue in character. Their basic idea is similar to that of using off-step points, e.g.\ one point from the previous step. The new methods are set up in such a way that the stage order is greater than achievable with Runge-Kutta methods, that the stability function is as simple as for Runge-Kutta methods and that very little information is passed between steps. The latter feature is important for economic stepsize changes while the first one is favourable for dense output extensions. In the paper the coefficients of four methods are given. An error control algorithm and an interpolatory dense output scheme is developed. Numerical experiments with the DETEST set of test problems are reported and a comparison with the classical Runge-Kutta method is made.
Mathematics Subject Classification ID: 65L06
Mathematics Subject Classification ID: 65L05
Mathematics Subject Classification ID: 65L20
Mathematics Subject Classification ID: 65L50
Mathematics Subject Classification ID: 65L70
Mathematics Subject Classification ID: 34A34
zbMATH DE Number: 1036663
zbMATH Keywords: Runge-Kutta methods
zbMATH Keywords: multivalue methods
zbMATH Keywords: stage order
zbMATH Keywords: continuous output
zbMATH Keywords: error estimation
zbMATH Keywords: stepsize control
zbMATH Keywords: numerical examples
zbMATH Keywords: stability
author: J. C. Butcher
reviewed by: Rolf Dieter Grigorieff
MaRDI profile type: Publication
cites work: On the Convergence of Numerical Solutions to Ordinary Differential Equations
cites work: A Multistep Generalization of Runge-Kutta Methods With Four or Five Stages
cites work: Pseudo-Runge-Kutta Methods Involving Two Points
cites work: A General Class of Two-Step Runge–Kutta Methods for Ordinary Differential Equations
full work available at URL: s0168-9274(97)00030-5
OpenAlex ID: W1989572481
description: scientific article; zbMATH DE number 1036663