Efficient energy-preserving integrators for oscillatory Hamiltonian systems

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• 作者：Xinyuan Wu ; ^{xywu@nju.edu.cn} ; Bin Wang ^{wangbinmaths@gmail.com} ; Wei Shi ^{shuier628@163.com}
• 关键词：Hamiltonian system ; Energy-preserving formula ; Structure-preserving algorithm ; Oscillatory differential equation ; Average Vector Field formula ; Fermi-Pasta-Ulam problem ; Sine-Gordon equation
• 刊名：Journal of Computational Physics
• 出版年：2013
• 出版时间：15 February, 2013
• 年：2013
• 卷：235
• 期：Complete
• 页码：587-605
• 全文大小：650 K

文摘

In this paper, we focus our attention on deriving and analyzing an efficient energy-preserving formula for the system of nonlinear oscillatory or highly oscillatory second-order differential equations , where M is a symmetric positive semi-definite matrix with and is the negative gradient of a real-valued function . This system is a Hamiltonian system with the Hamiltonian . The energy-preserving formula exactly preserves the Hamiltonian. We analyze in detail the properties of the energy-preserving formula and propose new efficient energy-preserving integrators in the sense of numerical implementation. The convergence analysis of the fixed-point iteration is presented for the implicit integrators proposed in this paper. It is shown that the convergence of implicit Average Vector Field methods is dependent on , whereas the convergence of the new energy-preserving integrators is independent of . The Fermi-Pasta-Ulam problem and the sine-Gordon equation are carried out numerically to show the competence and efficiency of the novel integrators in comparison with the well-known Average Vector Field methods in the scientific literature.