Lie group methods and their expansions - a unified approach

Brynjulf Owren

Department of Mathematical Sciences, NTNU, N-7491 Trondheim, NTNU

Brynjulf.Owren@math.ntnu.no

http://www.math.ntnu.no/ bryn

Invited talk

The integration methods on manifolds introduced by Crouch and Grossman [1] 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, the most general formulation is found in the paper [3] 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 in [3] is such that the order problem can be settled by classical RK theory. What is perhaps less well-known is that before Munthe-Kaas discovered this formulation, he developed an expansion theory in [2].

In this talk, present a unified approach to Lie group methods starting at an elementary level. In the last part, we will extend results from [4], 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 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. We address also the connection between the expansions of [2] and those derived in [4].