Geometric numerical integration illustrated by the Stoermer-Verlet method

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Reference

Geometric numerical integration illustrated by the Stoermer-Verlet method

HAIRER, Ernst, LUBICH, Christian, WANNER, Gerhard

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 an 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 [...]

HAIRER, Ernst, LUBICH, Christian, WANNER, Gerhard. Geometric numerical integration illustrated by the Stoermer-Verlet method. Acta Numerica , 2003, vol. 12, p. 399-450

DOI : 10.1017/S0962492902000144

Available at:

http://archive-ouverte.unige.ch/unige:12277

Disclaimer: layout of this document may differ from the published version.

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