Diagonally implicit general linear methods for ordinary differential equations
type: Item
instance of: scholarly article
title: Diagonally implicit general linear methods for ordinary differential equations
zbMATH Open document ID: 0795.65043
DOI: 10.1007/BF01990528
published in: BIT
publication date: 1994-03-07T00:00:00Z
review text: The authors construct linear approximation methods of the type: \((*)\) \(Y_ i= h\sum^ s_{j=1} a_{ij} f(Y_ j)+ \sum u_{ij} y_ j^{[n-1]}\), \(j=1,2,\dots,s\), \(y_ i^{[n]}= h\sum^ s_{j=1} b_{ij} f(Y_ j)+ \sum v_{ij} y_ j^{[n-1]}\), \(i=1,2,\dots,r\), for \(y'(x)= f(y(x))\), \(x\in [x_ 0,X]\), \(y(x_ 0)= y_ 0\). A method of the type \((*)\) is said to have \(s\) internal stages and \(r\) external approximations. This class \((*)\) includes many known numerical methods such as Runge-Kutta, multistep, etc.. Suppose \(q\) is the stage approximation order and \(p\) that of the external approximations. The authors construct new interesting methods corresponding to the cases: \(s+ 1= r=q\), \(p=q\) or \(p= q+1\); \(s= r+1= q\), \(p= q\) or \(p= q+1\) and \(s= r= q\), \(p= q+1\). Effective methods are given and their stability properties investigated.
Mathematics Subject Classification ID: 65L06
Mathematics Subject Classification ID: 65L05
Mathematics Subject Classification ID: 65L20
Mathematics Subject Classification ID: 34A34
zbMATH DE Number: 503584
zbMATH Keywords: diagonally implicit general linear methods
zbMATH Keywords: Runge-Kutta
zbMATH Keywords: multistep
zbMATH Keywords: stability
author: J. C. Butcher
author: Zdzislaw Jackiewicz
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01990528
OpenAlex ID: W2091615270
description: scientific article; zbMATH DE number 503584