SciCADE 2013
International Conference on Scientific Computation and Differential Equations
September 16-20, 2013, Valladolid (Spain)

Invited Talk

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Integrating Highly-Oscillatory Mechanical Systems with Solution-Dependent Frequencies

D. Weiss

Abstract
The talk is about the application of several integrators to highly-oscillatory mechanical systems with solution-dependent frequencies. As an example we use the stiff spring double pendulum: two mass points are attached serially by stiff springs to one another. The numerical behaviour of several integrators such as FLAVORS, the impulse method, and the mollified impulse method is studied. It is explained that a correct approximation of the actual motion relies on an almost-invariance property of the actions in the system. This almost-invariance property also guarantees the existence of an underlying effective system, which is derived. The analysis is done using canonical transformations proposed by K. Lorenz and Ch. Lubich (see [1, 2]).

Bibliography
[1] E. Hairer, Ch. Lubich and G. Wanner, Geometric Numerical Integration. Structure-Preserving Algorithms for Ordinary Differential Equations, Springer, 2006.
[2] Lorenz, K. Adiabatische Integratoren für hochoszillatorische Hamilton-Systeme, Dissertation, Universität Tübingen, 2006.

Organized by         Universidad de Valladolid     IMUVA