Hostname: page-component-586b7cd67f-t7czq Total loading time: 0 Render date: 2024-11-27T11:26:41.396Z Has data issue: false hasContentIssue false

Geometric numerical integration illustrated by the Störmer–Verlet method

Published online by Cambridge University Press:  29 July 2003

Ernst Hairer
Affiliation:
Section de Mathématiques, Université de Genève, Switzerland E-mail: [email protected]
Christian Lubich
Affiliation:
Mathematisches Institut, Universität Tübingen, Germany E-mail: [email protected]
Gerhard Wanner
Affiliation:
Section de Mathématiques, Université de Genève, Switzerland E-mail: [email protected]

Abstract

The subject of geometric numerical integration deals with numerical integrators that preserve geometric properties of the flow of a differential equation, and it explains how structure preservation leads to improved long-time behaviour. This article illustrates concepts and results of geometric numerical integration on the important example of the Störmer–Verlet method. It thus presents a cross-section of the recent monograph by the authors, enriched by some additional material.

After an introduction to the Newton–Störmer–Verlet–leapfrog method and its various interpretations, there follows a discussion of geometric properties: reversibility, symplecticity, volume preservation, and conservation of first integrals. The extension to Hamiltonian systems on manifolds is also described. The theoretical foundation relies on a backward error analysis, which translates the geometric properties of the method into the structure of a modified differential equation, whose flow is nearly identical to the numerical method. Combined with results from perturbation theory, this explains the excellent long-time behaviour of the method: long-time energy conservation, linear error growth and preservation of invariant tori in near-integrable systems, a discrete virial theorem, and preservation of adiabatic invariants.

Type
Research Article
Copyright
© Cambridge University Press 2003

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)