Backward Error Analysis for Lie--Group Methods

Backward error analysis has proven to be very useful in stability anal-ysis of numerical methods for ordinary dierential equations. However the analysis has so far been undertaken in the Euclidean space or open subsets thereof. In this paper we study dierential equations on manifolds. We prove a backward error analysis result for intrinsic numerical methods. Especially we are interested in Lie-group methods. The author proves elsewhere that if the Lie algebra is nilpotent a global stability analysis can be done in the Lie algebra. In the general case we must work on the nonlinear Lie group. In order to show that there is perturbed dierential equation on the Lie group with a solution that is exponentially close to the numerical integrator after several steps, we prove a generalised version of Alekseev{Grobner's theorem. A major motivation for this result is that it implies many stability properties of Lie-group methods