Geometric Integration for ODEs

Geometric integration is the numerical integration of a differential equation, while preserving one or more of its geometric properties exactly, i.e. to within round-off error. Many of these geometric properties are of crucial importance in physical applications: preservation of energy, momentum, angular momentum, phase space volume, symmetries, time-reversal symmetry, symplectic structure and dissipation are examples. In this paper we present a survey of geometric numerical integration methods for ordinary differential equations. Our aim has been to make the review of use for both the novice and the more experienced practitioner interested in the new developments and directions of the past decade. To this end, the reader who is interested in reading up on detailed technicalities will be provided with numerous signposts to the relevant literature.