Commutator-free Lie group methods

RKMK methods and Crouch-Grossman methods are two classes of Lie group methods. The former is using flows and commutators of a Lie algebra of vector fields as a part of the method definition. The latter uses only compositions of flows of such vector fields, but the number of flows which needs to be computed is much higher than in the RKMK methods. We present a new type of methods which avoids the use of commutators, but which has a much lower number of flow computations than the Crouch-Grossman methods. We argue that the new methods may be particularly useful when applied to problems on homogeneous manifolds with large isotropy groups, or when used for stiff problems. Numerical experiments verify these claims when applied to a problem on the orthogonal Stiefel manifold, and to an example arising from the semidiscretisation of a linear inhomogeneous heat conduction problem.