Runge-Kutta Methods Adapted to Manifolds and Based on Rigid Frames

We consider numerical integration methods for differentiable manifolds as proposed by Crouch and Grossman. The differential system is phrased by means of a system of frame vector fields E₁, ... , E_n on the manifold. The numerical approximation is obtained by composing flows of certain vector fields in the linear span of E₁, ... , E_n that are tangent to the differential system at various points. The methods reduce to traditional Runge-Kutta methods if the frame vector fields are chosen as the standard basis of euclidean ⁿ. A complete theory for the order conditions involving ordered rooted trees is developed. Examples of explicit and diagonal implicit methods are presented, along with some numerical results.