On the qualitative behaviour of symplectic integrators. I: Perturbed linear systems
type: Item
instance of: scholarly article
title: On the qualitative behaviour of symplectic integrators. I: Perturbed linear systems
zbMATH Open document ID: 0885.65088
author: Daniel Stoffer
published in: Numerische Mathematik
publication date: 1997-12-10T00:00:00Z
review text: This paper deals with the qualitative behaviour of symplectic Runge-Kutta methods in the numerical integration of perturbed linear Hamiltonian systems with dissipative perturbations. The differential systems considered here have the form \({\mathbf x}'= {\mathbf f}^0(x)+ \varepsilon{\mathbf f}^1({\mathbf x}, \varepsilon)\) where \({\mathbf f}^0(x)= {\mathbf {Cx}}\) is a linear Hamiltonian vector field and \(\varepsilon{\mathbf f}^1\) the dissipative perturbation. In the first section the author proves that, describing the system in terms of the action-angle variables of the unperturbed problem \((\varepsilon=0)\), under suitable assumptions the perturbed problem admits an attractive invariant torus \(T_\varepsilon\), for \(\varepsilon\) sufficiently small. Next the existence of an invariant torus \(T_{h,\varepsilon}\) (close to \(T_\varepsilon)\) in the flow defined by a Runge-Kutta method applied to this problem with a fixed stepsize is considered. The remarkable result proved here is that for linearly symplectic methods (or else linearly symmetric) for all \(\varepsilon\in (0,\varepsilon_0)\) the discretized problem has an invariant torus \(T_{h,\varepsilon}\) (close to \(T_\varepsilon\)) for all \(h\leq h_0\) with some positive \(h_0\) independent of the perturbation parameter \(\varepsilon\), while for non symplectic methods \(h_0= h_0 (\varepsilon)\) depends on \(\varepsilon\), in a way that the local discretization error of the method must be \(\leq\varepsilon\). Thus, in the case of non symplectic methods to ensure the existence of these tori for \(\varepsilon\to 0\) a simultaneous restriction on the maximum stepsize must be imposed. This fact shows the favourable behaviour of symplectic Runge-Kutta methods in order to preserve the qualitative properties of the flow in the above systems.
Mathematics Subject Classification ID: 65L06
Mathematics Subject Classification ID: 65L05
Mathematics Subject Classification ID: 37C10
Mathematics Subject Classification ID: 34C30
Mathematics Subject Classification ID: 37J99
zbMATH DE Number: 1096028
zbMATH Keywords: invariant manifolds
zbMATH Keywords: symplectic Runge-Kutta methods
zbMATH Keywords: linear Hamiltonian systems
zbMATH Keywords: dissipative perturbations
zbMATH Keywords: attractive invariant torus
zbMATH Keywords: flow
reviewed by: Manuel Calvo
MaRDI profile type: Publication
full work available at URL: s002110050299
OpenAlex ID: W2040475467
DOI: 10.1007/S002110050299
description: scientific article; zbMATH DE number 1096028