Authors: A. Iserles, H. Z. Munthe-Kaas, S. P. Nørsett, A. Zanna

Journal: Acta Numerica (2000), pp. 215–365, CUP.

Abstract: 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 geo- metry 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.

Hindsight notes: This paper surveys the state of knowledge of Lie group methods up to year 2000. Later came other developments such as generalized polar decompositions and commutator free methods, which are not discussed in this paper.