Journal
FOUNDATIONS OF COMPUTATIONAL MATHEMATICS
Volume 16, Issue 1, Pages 151-181Publisher
SPRINGER
DOI: 10.1007/s10208-014-9241-9
Keywords
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
Funding
- Natural Science Foundation of China [11271186, 11401333]
- NSFC
- RS International Exchanges Project [113111162]
- Specialized Research Foundation for the Doctoral Program of Higher Education [20130091110041]
- 985 Project at Nanjing University [9112020301]
- Priority Academic Program Development of Jiangsu Higher Education Institutions
- Natural Science Foundation of Shandong Province [ZR2014AQ003]
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We rigorously study a novel type of trigonometric Fourier collocationmethods for solving multi-frequency oscillatory second-order ordinary differential equations (ODEs) q '' (t) + Mq(t) = f (q(t)) with a principal frequencymatrix M is an element of R-dxd. If M is symmetric and positive semi-definite and f (q) = -del 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 -> 0, each trigonometric Fourier collocation method creates a particular Runge-Kutta-Nystrom-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.
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