Geometric numerical integration illustrated by the Störmer–Verlet method

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 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. 400 E. Hairer, Ch. Lubich and G. Wanner CONTENTS

Geometric methods and formulations in computational multibody system dynamics

“…In the field of MBS geometric integration, special attention is devoted to structure preserving methods that exploit rich geometric structure of rigid body rotational dynamics (see [7,16,25,58,67,73,84,85,87,89,90,97,104,135,146,147,152] and references cited therein). To this end, rigid body rotational dynamics is studied most conveniently as Lie-Poisson system that is defined on so * (3) (the dual space of so (3)).…”

“…These two constraints imply that the equations (105) constitute a Hamiltonian system constrained on the manifold K H = {(P, R) ∈ R 3,3 × R 3,3 |R T R = I, R T PD −1 + D −1 P T R = 0} (106) which, however, is not the cotangent bundle T * SO (3) of the manifold SO(3). Discretization of (105) that is based on standard Störmer-Verlet integration algorithm [73,74,136] leads to symplectic RATTLE scheme for rotational rigid body dynamics that exhibits excellent structure preserving properties [88]. structure preserving scheme for rigid body dynamics that also treats rigid body as constrained mechanical system is proposed in [13].…”

A demonstration of 4D‐Var using a time‐distributed background term

“…The model was integrated forward using a Störmer-Verlet method (Hairer et al, 2003), with a time step of 0.01. The resulting trajectories are plotted in Figure 1(a) in coordinates relative to the centre of mass.…”

Geometric structure-preserving optimal control of a rigid body

“…The necessary optimality conditions for the problem of minimizing (6) subject to the dynamics (7) and the boundary conditions (8) are given by the single fourth order 2 differential equation…”

“…(6) subject to the dynamics (7) and the boundary conditions (8) are given by Proof. In Theorem 2.2, differentiate Λ 2 once and then use all three differential equations to replace Λ 1 and Λ 2 with expressions involving only τ , M and Ω.…”