This paper gives a review of integration algorithms for finite dimensional mechanical systems that are based on discrete variational principles. The variational technique gives a unified treatment of many symplectic schemes, including those of higher order, as well as a natural treatment of the discrete Noether theorem. The approach also allows us to include forces, dissipation and constraints in a natural way. Amongst the many specific schemes treated as examples, the Verlet, SHAKE, RATTLE, Newmark, and the symplectic

Many differential equations of practical interest evolve on Lie groups or on manifolds acted upon by Lie groups. The retention of Lie-group structure under discretization is often vital in the recovery of qualitatively correct geometry and dynamics and in the minimization of numerical error. Having introduced requisite elements of differential geometry, this paper surveys the novel theory of numerical integrators that respect Lie-group structure, highlighting theory, algorithmic issues and a number of applications.

We consider backward error analysis of numerical approximations to ordinary differential equations, i.e., the numerical solution is formally interpreted as the exact solution of a modified differential equation. A simple recursive definition of the modified equation is stated. This recursion is used to give a new proof of the exponentially closeness of the numerical solutions and the solutions to an appropriate truncation of the modified equation. We also discuss qualitative properties of the modified equation and apply these results to the symplectic variable step-size integration of Hamiltonian systems, the conservation of adiabatic invariants, and numerical chaos associated to homoclinic orbits. 3 S. Reich, Backward error analysis 4

We discuss the integration of autonomous Hamiltonian systems via dynamical rescaling of the vector field (reparameterization of time). Appropriate rescalings (e.g., based on normalization of the vector field or on minimum particle separation in an N-body problem) do not alter the time-reversal symmetry of the flow, and it is desirable to maintain this symmetry under discretization. For standard form mechanical systems without rescaling, this can be achieved by using the explicit leapfrog-Verlet method; we show that explicit time-reversible integration of the reparameterized equations is also possible if the parameterization depends on positions or velocities only. For general rescalings, a scalar nonlinear equation must be solved at each step, but only one force evaluation is needed. The new method also conserves the angular momentum for an N-body problem. The use of reversible schemes, together with a step control based on normalization of the vector field (arclength reparameterization), is demonstrated in several numerical experiments, including a double pendulum, the Kepler problem, and a three-body problem.

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.

Commencing from a global-error formula, originally due to Henrici, we investigate the accumulation of global error in the numerical solution of linear highly-oscillating systems of the form y 00 + g(t)y = 0, where g(t) t!1 \Gamma! 1. Using WKB analysis we derive an explicit form of the global-error envelope for Runge-Kutta and Magnus methods. Our results are closely matched by numerical experiments. Motivated by the superior performance of Lie-group methods, we present a modification of the Magnus expansion which displays even better long-term behaviour in the presence of oscillations.

Symplectic methods for Hamiltonian systems are known to have favourable properties concerning long-time integrations (no secular terms in the error of the energy integral, linear error growth in the angle variables instead of quadratic growth, correct qualitative behaviour) if they are applied with constant step sizes, while all of these properties are lost in a standard variable step size implementation. In this article we present a "meta-algorithm" which allows us to combine the use of variable steps with symplectic integrators, without destroying the above mentioned favourable properties. We theoretically justify the algorithm by a backward error analysis, and illustrate its performance by numerical experiments. Keywords: Hamiltonian systems, symplectic integration, variable step sizes, backward error analysis, Kepler's problem, Verlet scheme 1 Introduction Hamiltonian systems of ordinary differential equations arise in many applications (e.g., mechanics, astrophysics, molecular dy...

. Besides preserving the energy, the flow of a conservative multibody system possesses important geometric (symplectic) invariants. Symplectic discretization schemes that mimic the corresponding feature of the true flow have been shown to be effective alternatives to standard methods for many conservative problems. For systems of rigid bodies, the development of such schemes can be complicated or costly to implement, depending on the choice of problem formulation. In this article, we demonstrate that a special formulation of the multibody system (based on a particle representation) together with a symplectic discretization for constrained problems borrowed from molecular dynamics offers an efficient alternative to standard approaches. Numerical experiments illustrating this approach are described. 1. Introduction In this paper, we consider efficient numerical integrators for systems of rigid bodies interconnected by various types of mechanical joints and subject to the forces of natur...

One of the most fruitful ways to analyze the effects of discretization error in the numerical solution of a system of differential equations is to examine the "modified equations," which are the equations that are exactly satisfied by the (approximate) discrete solution. These do not actually exist in general, but rather are defined by an asymptotic expansion in powers of the discretization parameter. Nonetheless, if the expansion is suitably truncated, the resulting modified equations have a solution which is remarkably close to the discrete solution. In the case of a Hamiltonian system of ordinary differential equations, the modified equations are also Hamiltonian if and only if the integrator is symplectic. The existence of a modified Hamiltonian is an indicator of the validity of statistical estimates calculated from long time integration of chaotic Hamiltonian systems. Evidence for the existence of a Hamiltonian for a particular calculation is obtained by calculating modified Hami...

. For numerical integrators of ordinary differential equations we compare the theory of asymptotic expansions of the global error with backward error analysis. On a formal level both approaches are equivalent. If, however, the arising divergent series are truncated, important features such as the semigroup property, structure perservation and exponentially small estimates over long times are valid only for the backward error analysis. We consider one-step methods as well as multistep methods, and we illustrate the theoretical results on several examples. In particular, we study the preservation of weakly stable limit cycles by symmetric methods. Key words. Asymptotic expansions, backward error analysis, one-step methods, multistep methods, long-time behavior. 1. Introduction. Together with an autonomous system of ordinary differential equations y 0 = f(y); y(0) = y 0 (1.1) we consider a numerical solution sequence y 0 ; y 1 ; y 2 ; y 3 ; : : : ; obtained either by a one-step meth...