Since the beginning of the nineties, there has been a growing interest in numerical solution techniques for ODEs on manifolds based on actions by Lie groups. In this talk, we will summarize the major contributions to the field, and in particular focus on how the computational cost has decreased through the continuing introduction of new schemes and method formats.

Initially there were the methods of Crouch and Grossman which need computation of a large number of flows (e.g.\ by computation of the matrix exponential) and the methods of Munthe-Kaas which still require the same type of flow computations, but fewer in number. Recently, many improvements have been made, some of them reduce substantially the computational effort. In particular, we will focus in this talk on the use of canonical coordinates of the second kind, and we will show that these new methods can be competitive to existing ones, also when applied to problems phrased on $\mathbb{R}^n$.

MINISIMPOSIUM SESSION: 12. Lie Group methods