Lie group integrators

Most conventional time integrators (ODE solvers) in linear spaces are constructed by using a clever combination of essentially two primitive operations (i) Evaluating the ODE vector field (ii) Taking linear combinations of such derivative evaluations. Of course we must allow for the computation of the vector field at "unknown points" thus leading to implicit schemes. But in this way we can recover Runge-Kutta-, multistep-, collocation-, projection- and general linear methods as well as many other types of schemes. Suppose that we replace the linear space by a manifold. Then it is no longer possible to take linear combinations of the vector field at different points, since they are now tangent vectors which belong to different spaces. One way of seeing Lie group integrators is a way to generalize the two primitive building blocks in such a way that the "same methods" work also on manifolds. In this talk, we shall give a first introduction to Lie group integrators. We begin by presenting a few examples where the methods have shown a good potential. Secondly, we explain the right setting for these integrators, using the language of group actions on a manifold. Trying to generalize conventional methods, one immediately runs into many challenges, we will explain some of them. Many problems have been solved in the last decade, but there are still open problems. We will discuss both.