Author: Hans Munthe-Kaas

Journal: BIT 38:1 (1998), 92-111.

Abstract: 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.

Remarks in hindsight: We did not at the time know how to go beyond order 4. Arbitrary order was found by the (much simpler) approach of THIS paper.