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Geometric numerical integration illustrated by the Stoermer-Verlet method

Lubich, Christian
Published in Acta Numerica. 2003, vol. 12, p. 399-450
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 method: long-time energy conservation, linear error growth and preservation of invariant tori in near-integrable systems, a discrete virial theorem, preservation of adiabatic invariants.
Keywords Geometric numerical integrationStörmer/Verlet methodSymplecticitySymmetry and reversibilityConservation of first integrals and adiabatic invariants backward error analysisShakeNumerical experiments
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HAIRER, Ernst, LUBICH, Christian, WANNER, Gerhard. Geometric numerical integration illustrated by the Stoermer-Verlet method. In: Acta Numerica, 2003, vol. 12, p. 399-450. doi: 10.1017/S0962492902000144 https://archive-ouverte.unige.ch/unige:12277

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Deposited on : 2010-11-01

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