Family of symplectic implicit Runge-Kutta formulae
type: Item
instance of: scholarly article
title: Family of symplectic implicit Runge-Kutta formulae
zbMATH Open document ID: 0761.65057
DOI: 10.1007/BF02074888
author: Satoshi Saito
author: Hiroshi Sugiura
author: Taketomo Mitsui
published in: BIT
publication date: 1993-01-16T00:00:00Z
review text: This paper is concerned with the construction of canonical Runge-Kutta (RK) methods, i.e. implicit RK methods which integrate Hamiltonian systems of ordinary differential equations preserving the symplectic structure of the phase space. Firstly, the authors show that the canonicity conditions [see e.g. \textit{J. M. Sanz-Serna}, BIT 28, No. 4, 877-883 (1988; Zbl 0655.70013)] of an \(s\)-stages method defined by its Butcher tableau \(A=(a_{ij})\), \(b=(b_ i)\), \((1\leq i,j\leq s)\) can be written as a linear system in the \(s^ 2\) coefficients \(a_{ij}\), proving that the dimension of its kernel is \((s- 1)(s-2)/2\). This fact implies that the set of \(s\)-stages canonical RK methods depends on \((s-1)(s-2)/2\) parameters. Finally, taking into account the usual order conditions the construction of canonical methods with \(s\leq 3\) stages is studied.
Mathematics Subject Classification ID: 65L06
Mathematics Subject Classification ID: 65L05
Mathematics Subject Classification ID: 65L20
Mathematics Subject Classification ID: 34A34
Mathematics Subject Classification ID: 37J99
zbMATH DE Number: 93335
zbMATH Keywords: Runge-Kutta methods
zbMATH Keywords: Hamiltonian systems
zbMATH Keywords: \(s\)-stages method
zbMATH Keywords: Butcher tableau
zbMATH Keywords: order conditions
zbMATH Keywords: canonical methods
reviewed by: Manuel Calvo
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02074888
OpenAlex ID: W2086616053
description: scientific article; zbMATH DE number 93335