Arbitrary-Order Trigonometric Fourier Collocation Methods for Multi-Frequency Oscillatory Systems

We rigorously study a novel type of trigonometric Fourier collocation methods for solving multi-frequency oscillatory second-order ordinary differential equations (ODEs) q (t) + Mq(t) = f(q(t)) with a principal frequency matrix M ∈ Rd×d . If M is symmetric and positive semi-definite and f(q) = − U(q) for a smooth function U(q), then this is a multi-frequency oscillatory Hamiltonian system with the Hamiltonian H(q, p) = pT p/2 + qT Mq/2 + U(q), where p = q . The solution of this system is a nonlinear multi-frequency oscillator. The new trigonometric Fourier collocation method takes advantage of the special structure brought by the linear term Mq and its construction incorporates the idea of collocation methods, the variation-of-constants formula and the local Fourier expansion of the system. The properties of the new methods are analysed. The analysis in the paper demonstrates an important feature, namely that the trigonometric Fourier collocation methods can be of an arbitrary o...

• Type: Journal
• Year: 2016
• Where: FOCM
• Authors: Bin Wang, Arieh Iserles, Xinyuan Wu
• Keywords: Automated Reasoning, FOCM 2016