Lie group methods and their expansions

The integration methods on manifolds introduced by Crouch and Grossman were expressed in terms of a frame, that is, a set of $d$ vector fields which span the tangent space at each point on the manifold. Munthe-Kaas introduced a different type of methods, where he uses the language of Lie group actions to express his methods for ODEs on homogeneous manifolds. Crouch and Grossman used Lie series to analyse the order conditions of their methods, whereas the method format of Munthe-Kaas is such that the order problem can be settled by classical RK theory. In this talk, we present a unified approach to Lie group methods starting at an elementary level. In the last part, we will extend results by Owren and Marthinsen, and focus on expansions of the methods and the exact solution, which are series of the form \sum_{t\in T_O} h^{\rho(t)}\mathbf{a}(t)\mathbb{F}(t) similar to to the Butcher theory, except now $T_O$ is the set of \emph{ordered} rooted trees, and $\mathbb{F}(t)$ is a higher order derivation operator replacing the elementary differentials. There is an interesting and useful algebraic structure on $T_O$ that we will discuss. It can be used to characterise dependencies between the coefficients $\mathbf{a}(t)$ and as a tool in backward error analysis.