Characterizations and construction of Poisson/symplectic and symmetric multi-revolution implicit Runge-Kutta methods of high order
type: Item
instance of: scholarly article
title: Characterizations and construction of Poisson/symplectic and symmetric multi-revolution implicit Runge-Kutta methods of high order
zbMATH Open document ID: 1149.65096
author: Min Li
author: Aiguo Xiao
published in: Applied Numerical Mathematics
publication date: 2008-06-11T00:00:00Z
review text: This paper is concerned with the explicit construction of symplectic and/or symmetric multirevolution Runge-Kutta (RK) type algorithms. For a near identity map \( \varphi\), \((\varphi (y) = y + \varepsilon f(y) , 0 < \varepsilon <<1)\) typically associated to the one period Poincaré map of a dynamical system, multirevolution RK algorithms attempt to approximate \( \varphi^N \) (with \(N\) large) by using the \( \varphi\)-map at a few selected points. A study of the preservation of symplecticness of \( \varphi \) by these methods has been developed by \textit{M. Calvo, L. O. Jay, J. I. Montijano} and \textit{L. Rández} [Numer. Math. 97, No. 4, 635--666 (2004; Zbl 1067.65067)] together with some families of symplectic multirevolution RK methods. In the paper under consideration the authors give a step further by using the W-transformation to construct several families Poisson/symplectic and symmetric methods. They generalize the standard W-transformation to construct families of high order RK methods by using the \( B(p)\), \(C(q)\) and \(D(r)\) simplifying assumptions and select the available parameters so that satisfy the symplecticness and symmetry requirements. Several Gauss, Radau and Lobatto type methods are derived and the results of some numerical experiments are presented to confirm the order of some methods.
reviewed by: Manuel Calvo
Mathematics Subject Classification ID: 65P10
Mathematics Subject Classification ID: 65L05
Mathematics Subject Classification ID: 65L06
Mathematics Subject Classification ID: 34A34
Mathematics Subject Classification ID: 37M15
zbMATH DE Number: 5287842
zbMATH Keywords: multi-revolution Runge-Kutta methods
zbMATH Keywords: symplectic methods
zbMATH Keywords: modified W-transformation
zbMATH Keywords: near identity map
zbMATH Keywords: Gauss type method
zbMATH Keywords: Radau type method
zbMATH Keywords: Lobatto type method
zbMATH Keywords: Poisson type method
describes a project that uses: RODAS
MaRDI profile type: Publication
full work available at URL: j.apnum.2007.04.001
OpenAlex ID: W2071789734
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DOI: 10.1016/J.APNUM.2007.04.001
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description: scientific article; zbMATH DE number 5287842