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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: Ernst.Hairer@math.unige.ch
Christian Lubich
Affiliation:
Mathematisches Institut, Universität Tübingen, Germany E-mail: Lubich@na.uni-tuebingen.de
Gerhard Wanner
Affiliation:
Section de Mathématiques, Université de Genève, Switzerland E-mail: Gerhard.Wanner@math.unige.ch

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

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