Article: Lie-Butcher Therory for Runge-Kutta Methods
Author: Hans Munthe-Kaas
Journal: BIT 35 (1995), 572-587.
Abstract: Runge-Kutta methods are formulated via coordinate independent operations on manifolds. It is shown that there is an intimate connection between Lie series and
Lie groups on one hand and Butcher's celebrated theory of order conditions on the
other. In Butcher's theory the elementary differentials are represented as trees. In the
present formulation they appear as commutators between vector fields. This leads to
a theory for the order conditions, which can be developed in a completely coordinate
free manner. Although this theory is developed in a language that is not widely used
in applied mathematics, it is structurally simple. The recursion for the order conditions rests mainly on three lemmas, each with very short proofs. The techniques used in the analysis are prepared for studying RK-like methods on general Lie groups and homogeneous manifolds, but these themes are not studied in detail within the present paper.
Remarks in hindsight: This is the fastest paper I have ever written. Around January 1995, Brynjulf Owren invited me to give a talk in the Geiranger ‘Ode to Node’ meeting that summer. At this time I had done no work on ODE integration. My idea for the Geiranger talk was to explore if it would make sense to write RK methods using legal operations on a general manifolds instead of vector operations on R^n. It took less than three months to do all the work, write up and submit. The paper was published the same year! It was awarded the Carl-Erik Froberg prize for 1996. Although the methods of the paper are not very good (max order 2), the paper contains germs of ideas that were important in subsequent papers.
This paper initiated for me a line of work that has occupied me for more than a decade!
1 Mar 1995