Geometric numerical integration illustrated by the Störmer-Verlet method

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.