46 research outputs found
Numerical Integrators for Highly Oscillatory Hamiltonian Systems: A Review
Numerical methods for oscillatory, multi-scale Hamiltonian systems are reviewed. The construction principles are described, and the algorithmic and analytical distinction between problems with nearly constant high frequencies and with time- or state-dependent frequencies is emphasized. Trigonometric integrators for the first case and adiabatic integrators for the second case are discussed in more detail
Important aspects of geometric numerical integration
At the example of Hamiltonian differential equations, geometric properties of the flow are discussed that are only preserved by special numerical integrators (such as symplectic and/or symmetric methods). In the ‘non-stiff' situation the long-time behaviour of these methods is well-understood and can be explained with the help of a backward error analysis. In the highly oscillatory (‘stiff') case this theory breaks down. Using a modulated Fourier expansion, much insight can be gained for methods applied to problems where the high oscillations stem from a linear part of the vector field and where only one (or a few) high frequencies are present. This paper terminates with numerical experiments at space discretizations of the sine-Gordon equation, where a whole spectrum of frequencies is presen
Phase-fitted Discrete Lagrangian Integrators
Phase fitting has been extensively used during the last years to improve the
behaviour of numerical integrators on oscillatory problems. In this work, the
benefits of the phase fitting technique are embedded in discrete Lagrangian
integrators. The results show improved accuracy and total energy behaviour in
Hamiltonian systems. Numerical tests on the long term integration (100000
periods) of the 2-body problem with eccentricity even up to 0.95 show the
efficiency of the proposed approach. Finally, based on a geometrical evaluation
of the frequency of the problem, a new technique for adaptive error control is
presented