Abstract:
The integration of highly oscillatory differential equations has been a numerical challenge for a long time.
In this thesis, integrators for highly oscillatory differential equations in classical mechanics are developed with the general case of Hamiltonians, where a constraining potential penalizes some directions of motion.
In chapter one to three, the case of a time-dependent Hamiltonian is considered and some useful transformations are presented to reach an almost-separation of the problem into slow and fast movements.
After again transforming the fast subsystem to smoother, so-called adiabatic variables, adiabatic integrators showing order two in the positions and one in some momenta are developed, and another two numerical integrators of global second order, called the adiabatic midpoint rule and the adiabatic Magnus method.
Second-order error bounds with step sizes significantly larger than the almost-period of the fastest oscillations are proved and illustrated by applying the new methods to a Fermi-Pasta-Ulam-Problem and a testproblem, where an almost-crossing of frequencies takes place.
An adaptive step size control is presented and successfully used in the case of almost-crossings and nonadiabatic transitions.
In order to benefit again of the good long-time-step behaviour of the adiabatic integrators, the frequency-dependent case is also transformed to adiabatic variables in chapter four. The integrators show similar accuracy, but in the case of an almost-crossing of frequencies, the problem shows a chaotic behaviour (the so-called Takens chaos) and raises new aspects.