High order Runge-Kutta methods on manifolds

We present a family of Runge-Kutta type integration schemes of arbitrarily high order for differential equations evolving on manifolds. We prove that any classical Runge-Kutta method can be turned into an invariant method of the same order on a general homogeneous manifold, and present a family of algorithms that are relatively simple to implement. These are defined in a general abstract framework, based on a Lie algebra acting on the manifold. The general framework gives rise to a wide range of different concrete applications; we present some examples. (C) 1999 Elsevier Science B.V. and IMACS. All rights reserved.