Foundations of Computational Mathematics

, Volume 16, Issue 1, pp 151–181 | Cite as

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

  • Bin Wang
  • Arieh Iserles
  • Xinyuan Wu


We rigorously study a novel type of trigonometric Fourier collocation methods for solving multi-frequency oscillatory second-order ordinary differential equations (ODEs) \(q^{\prime \prime }(t)+Mq(t)=f(q(t))\) with a principal frequency matrix \(M\in \mathbb {R}^{d\times d}\). If \(M\) is symmetric and positive semi-definite and \(f(q) = -\nabla U(q)\) for a smooth function \(U(q)\), then this is a multi-frequency oscillatory Hamiltonian system with the Hamiltonian \(H(q,p)=p^{T}p/2+q^{T}Mq/2+U(q),\) 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 order and when \(M\rightarrow 0\), each trigonometric Fourier collocation method creates a particular Runge–Kutta–Nyström-type Fourier collocation method, which is symplectic under some conditions. This allows us to obtain arbitrary high-order symplectic methods to deal with a special and important class of systems of second-order ODEs in an efficient way. The results of numerical experiments are quite promising and show that the trigonometric Fourier collocation methods are significantly more efficient in comparison with alternative approaches that have previously appeared in the literature.


Second-order ordinary differential equations Multi-frequency oscillatory systems Trigonometric Fourier collocation methods Multi-frequency oscillatory Hamiltonian systems Quadratic invariant Variation-of-constants formula  Symplectic methods 

Mathematics Subject Classification

65L05 65L20 65M20 65P10 



Bin Wang and Xinyuan Wu were supported in part by the Natural Science Foundation of China under Grants 11271186 and 11401333, by NSFC and RS International Exchanges Project under Grant 113111162, by the Specialized Research Foundation for the Doctoral Program of Higher Education under Grant 20130091110041, by the 985 Project at Nanjing University under Grant 9112020301, by A Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions, by the Natural Science Foundation of Shandong Province under Grant ZR2014AQ003. Bin Wang is sincerely thankful to Numerical Analysis Group at University of Cambridge since the work was partly done when he was studying in this group as a visiting student. The revised version of the manuscript was completed during Bin Wang and Xinyuan Wu were visiting to Numerical Analysis Group at University of Cambridge in July and August, 2014. The authors are sincerely thankful to two anonymous reviewers for their valuable suggestions, which help improve the presentation of the manuscript significantly.


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Copyright information

© SFoCM 2014

Authors and Affiliations

  1. 1.School of Mathematical SciencesQufu Normal UniversityQufuPeople’s Republic of China
  2. 2.Department of Mathematics, Nanjing UniversityState Key Laboratory for Novel Software Technology at Nanjing UniversityNanjingPeople’s Republic of China
  3. 3.Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical SciencesUniversity of CambridgeCambridgeUK

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