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 talk illustrates concepts and results of geometric numerical integration on the important example of the Störmer-Verlet method, the integrator of choice in molecular dynamics.

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 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.

The talk is based on joint work with Ernst Hairer and Gerhard Wanner (see our monograph Geometric Numerical Integration, Springer Berlin 2002).

Christian Lubich

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