Order conditions for commutator-free Lie group methods

We derive order conditions for commutator-free Lie group integrators. These schemes can for certain problems be good alternatives to the Runge-Kutta-Munthe-Kaas schemes, especially when applied to stiff problems or to homogeneous manifolds with large isotropy groups. The order conditions correspond to a certain subsets of the set of ordered rooted trees. We discuss ways to select these subsets and their combinatorial properties. We also suggest how the reuse of flow calculations can be included in order to reduce the computational cost. In the case that at most two flow calculations are admitted in each stage, the order conditions simplify substantially. We derive families of fourth order schemes which effectively use only 5 flow calculations per step.