Runge-Kutta methods on Lie groups

We construct generalized Runge-Kutta methods for integration of differential equations evolving on a Lie group. The methods are using intrinsic operations on the group, and we are hence guaranteed that the numerical solution will evolve on the correct manifold. Our methods must satisfy two different criteria to achieve a given order: Coefficients A(i,j) and b(j) must satisfy the classical order conditions. This is done by picking the coefficients of any classical RK scheme of the given order. We must construct functions to correct for certain non-commutative effects to the given order. These tasks are completely independent, so once correction functions are found to the given order, we can turn any classical RK scheme into an RK method of the same order on any Lie group. The theory in this paper shows the tight connections between the algebraic structure of the order conditions of RK methods and the algebraic structure of the so called 'universal enveloping algebra' of Lie algebras. This may give important insight also into the classical RK theory.