Energy conserving, Liouville, and symplectic integrators
type: Item
instance of: scholarly article
title: Energy conserving, Liouville, and symplectic integrators
zbMATH Open document ID: 0836.65088
author: Daniel Okunbor
published in: Journal of Computational Physics
publication date: 1996-04-29T00:00:00Z
review text: In the last few years most research in the numerical solution of ordinary differential equations has been addressed to the development of methods adapted to special problems. In particular, a complete theory of symplectic methods for Hamiltonian systems has been constructed [see e.g. \textit{J. M. Sanz-Serna} and \textit{M. P. Calvo}, Numerical Hamiltonian problems (1994; Zbl 0816.65042)]. The present author explores, from a numerical point of view, the possibilities of some numerical methods satisfying a weaker requirement than symplecticity. Thus, in the first part he introduces a particular technique to conserve the energy in a Hamiltonian system by rescaling the velocity at each step so that the energy remains constant along the numerical integration. Some numerical results are presented to show that for the Henon-Heiles problem this technique destroys some important dynamical properties of the phase space. Secondly, the author explores two-stage Runge-Kutta-Nyström methods conserving the volume in the phase-space. Since for problems with dimension \(\geq 2\) this property is a weaker requirement than symplecticity, the author compares a second-order preserving volume method with the Stormer-Verlet symplectic method. Numerical experiments for the two-body problem and a pseudospectral discretization of the sine- Gordon equation show that both methods have a similar behavior for these problems.
Mathematics Subject Classification ID: 65L05
Mathematics Subject Classification ID: 65L06
Mathematics Subject Classification ID: 37J99
Mathematics Subject Classification ID: 70F05
zbMATH DE Number: 818565
zbMATH Keywords: numerical experiments
zbMATH Keywords: symplectic methods
zbMATH Keywords: Hamiltonian systems
zbMATH Keywords: numerical results
zbMATH Keywords: Henon-Heiles problem
zbMATH Keywords: Runge-Kutta-Nyström methods
zbMATH Keywords: Stormer- Verlet symplectic method
zbMATH Keywords: two-body problem
zbMATH Keywords: sine-Gordon equation
reviewed by: Manuel Calvo
MaRDI profile type: Publication
full work available at URL: jcph.1995.1172
OpenAlex ID: W2041383953
DOI: 10.1006/JCPH.1995.1172
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description: scientific article; zbMATH DE number 818565