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A highly accurate explicit symplectic ERKN method for multi-frequency and multidimensional oscillatory Hamiltonian systems

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Abstract

The numerical integration of Hamiltonian systems with multi-frequency and multidimensional oscillatory solutions is encountered frequently in many fields of the applied sciences. In this paper, we firstly summarize the extended Runge–Kutta–Nyström (ERKN) methods proposed by Wu et al. (Comput. Phys. Comm. 181:1873–1887, (2010)) for multi-frequency and multidimensional oscillatory systems and restate the order conditions and symplecticity conditions for the explicit ERKN methods. Secondly, we devote to exploring the explicit symplectic multi-frequency and multidimensional ERKN methods of order five based on the symplecticity conditions and order conditions. A five-stage explicit symplectic multi-frequency and multidimensional ERKN method of order five with some small residuals is proposed and its stability and phase properties are analyzed. It is shown that the new method is dispersive of order six. Numerical experiments are carried out and the numerical results demonstrate that the new method is much more efficient than the methods appeared in the scientific literature.

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Authors and Affiliations

  1. School of Mathematics and Physics, Qingdao University of Science and Technology, Qingdao, 266061, People’s Republic of China

    Bin Wang

  2. Department of Mathematics, State Key Laboratory for Novel Software Technology at Nanjing University, Nanjing University, Nanjing, 210093, People’s Republic of China

    Bin Wang & Xinyuan Wu

Authors
  1. Bin Wang
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  2. Xinyuan Wu
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Correspondence to Xinyuan Wu.

Additional information

Dedicated to John’s 80th Birthday

The research of the first author was supported by the Doctoral Found of Qingdao University of Science & Technology. The research of the second author was supported in part by the Natural Science Foundation of China under Grant 11271186, by the Specialized Research Foundation for the Doctoral Program of Higher Education under Grant 20100091110033, 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.

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Wang, B., Wu, X. A highly accurate explicit symplectic ERKN method for multi-frequency and multidimensional oscillatory Hamiltonian systems. Numer Algor 65, 705–721 (2014). https://doi.org/10.1007/s11075-013-9811-y

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