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Arbeitsgruppe Angewandte Mathematik / Numerische Analysis
Bergische Universität Wuppertal
Fakultät 04
Gaußstraße 20
D-42119 Wuppertal
Deutschland

Telefon: +49 202 439 5296
Fax: +49 202 439 5201
E-Mail: sek-amna{at}math.uni-wuppertal.de

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Geometric numerical integration on Lie groups

In many physical applications, differential equations on Lie groups occur.

Due to the fact that a Lie group is nonlinear, the differential equation can not be directly solved. A way out is a mapping of its elements into its appropriate Lie algebra which is a linear space. In doing so, the original differential equation in the Lie group has to be replaced by another one in the Lie algebra such that it can immediately be solved by a numerical integration method. Thereby, it has to be taken into account that a Lie algebra generally is non-abelian.

Usually, some important properties of the physical system have to be preserved. Therefore, geometric numerical integration methods are used. First of all, these methods are symmetric (i.e. time-reversible). Moreover, geometric integrators are symplectic (i.e area- or volume-preserving).

Geometric integrators on Lie groups are a combination of the aforementioned integration methods. Additionally, one aims at a high convergence order with low computational effort. As an example, the St\"ormer-Verlet (i. e. Leapfrog-) method is one of the most commonly known methods and has convergence order 2. In our working group, especially higher-order Runge-Kutta methods are investigated.

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