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

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• 作者：Bin Wang ; Arieh Iserles ; Xinyuan Wu
• 关键词：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 ; 65L05 ; 65L20 ; 65M20 ; 65P10
• 刊名：Foundations of Computational Mathematics
• 出版年：2016
• 出版时间：February 2016
• 年：2016
• 卷：16
• 期：1
• 页码：151-181
• 全文大小：1,012 KB
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• 作者单位：Bin Wang (1) (2)
  Arieh Iserles (3)
  Xinyuan Wu (2)
  
  1. School of Mathematical Sciences, Qufu Normal University, Qufu, 273165, Shandong, People’s Republic of China
  2. Department of Mathematics, Nanjing University, State Key Laboratory for Novel Software Technology at Nanjing University, Nanjing, 210093, People’s Republic of China
  3. Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Rd, Cambridge, CB3 0WA, UK
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Numerical Analysis
  Computer Science, general
  Math Applications in Computer Science
  Linear and Multilinear Algebras and Matrix Theory
  Applications of Mathematics
  
• 出版者：Springer New York
• ISSN：1615-3383

文摘

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. 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