Geometric numerical integration illustrated by
the Störmer/Verlet method
Ernst Hairer, Christian Lubich and Gerhard Wanner
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.
Key Words.
Geometric numerical integration, Störmer/Verlet method,
symplecticity, symmetry and reversibility, conservation of
first integrals and adiabatic invariants
backward error analysis, Shake, numerical experiments.