The existence of symplectic general linear methods
type: Item
instance of: scholarly article
title: The existence of symplectic general linear methods
zbMATH Open document ID: 1176.65080
author: L. L. Hewitt
author: J. C. Butcher
published in: Numerical Algorithms
publication date: 2009-06-30T00:00:00Z
review text: The paper obtains a criterion that any general linear integration method must satisfy if it is symplectic. After deriving a criterion for symplectic behavior based on a standard conservative problem of the form, \[ \begin{cases} z_1'(t)= -\alpha(z_3,z_4,\dots,z_N)z_2(t), \\ z_2'(t)= \alpha(z_3,z_4,\dots,z_N)z_1(t), \\ z_i'(t)=f_i(z_3,z_4,\dots,z_N)z_1(t), \quad i=3,4\dots,N, \end{cases} \] where the linear factor \(\alpha\) can take on different values at the stages of the numerical method, the authors give a linear algebraic condition satisfied by all symplectic general methods. As a result of this discussion, the authors address the reducibility problem of a general linear method with \(s\) stages and \(r\) inputs. Although the question of full reducibility is not considered in general, the main contribution of the paper is to prove that if a method with \(r>1\) exhibits symplectic behavior, then it is reducible to a method with \(r=1\). In other words, the main theorem states that a stable, irreducible, general linear method with \(r>1\) cannot be symplectic, and so linking the criterion of the paper to that for Runge-Kutta methods.
reviewed by: Josep J. Masdemont
Mathematics Subject Classification ID: 65L06
Mathematics Subject Classification ID: 65L05
Mathematics Subject Classification ID: 34A34
Mathematics Subject Classification ID: 65P10
Mathematics Subject Classification ID: 37M15
zbMATH DE Number: 5571631
zbMATH Keywords: general linear methods
zbMATH Keywords: symplectic
zbMATH Keywords: reducibility
MaRDI profile type: Publication
full work available at URL: s11075-008-9250-3
OpenAlex ID: W2047542526
cites work: Q4531870
DOI: 10.1007/S11075-008-9250-3
description: scientific article; zbMATH DE number 5571631