Diploma thesis

My diploma thesis was delivered 14th of June 2002, finalizing my education to the level of "sivilingeniør" (Masters degree).

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: