Abstract:
Lie group methods are formulated by means of a Lie group action on a
manifold. If the dimension of the Lie group is greater than the
dimension of the manifold, there is a certain freedom in the
formulation of the method. We focus on Runge-Kutta-Munthe-Kaas Lie
group methods.
Background from differential topology, Lie and matrix groups is
provided and from there a presentation of Runge-Kutta-Munthe-Kaas
which methods are given.
Lewis and Olver (2001) has recently shown how to improve the accuracy for algorithms on the sphere by means of an SO(3)-action. We elaborate and extend their approach by using Lie series, and find an equation that can be used to determine a choice of the isotropy freedom leading to better numerical behavior.
The same methodology is then applied to an SL(2)-based Lie group method on R². We use the Lotka-Volterra system and the Duffing oscillator as examples and obtain excellent long time behaviour comparable to Symplectic Euler by a careful choice of isotropy.
Thesis downloads:
August 2002 I gave a talk summarizing the results of my diploma thesis at the conference "Foundations of Computational Mathematics" held in Minneapolis (link).
Slides downloads: