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.

Group members working on that field

• Michael Günther
• Michele Wandelt
• Dmitry Shcherbakov

Cooperation

• Francesco Knechtli, Theoretische Teilchenphysik, Bergische Universität Wuppertal

Ongoing projects

• SFB/Transregio 55 "Hadronenphysik mit Gitter-QCD", Project area B: Algorithms for Lattice QCD simulations.
• Marie Curie ITN STRONGnet "Strong Interaction Supercomputing Training Network " (01/2010 - 12/2014)

Literature

• Hairer, Lubich, Wanner: Geometric Numerical Integration - Structure-Preserving Algorithms for Ordinary Differential Equations, Springer Berlin, ISBN : 978-3-540-30663-4
• M. Wandelt, MasterThesis:
  Implicit partitioned Runge-Kutta integrators for simulations of gauge theories, BUW