Geometric numerical integration illustrated by the Störmer–Verlet method

• Published in: Acta numerica Vol. 12; pp. 399 - 450
• Main Authors: Hairer, Ernst, Lubich, Christian, Wanner, Gerhard
• Format: Journal Article
• Language: English
• Published: Cambridge, UK Cambridge University Press 01.05.2003; Cambridge University Press (CUP)
• Subjects: Mathematics
• ISSN: 0962-4929; 1474-0508
• DOI: 10.1017/S0962492902000144

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.