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New high order symplectic integrators via generating functions with its application in many-body problem

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Abstract

A new family of high order one-step symplectic integration schemes for separable Hamiltonian systems with Hamiltonians of the form \(T(p) + U(q)\) is presented. The new integration methods are defined in terms of an explicitly defined generating function (of the third kind). They are implicit in q (but explicit in p and the internal states), and require the evaluation of the gradients of T(p) and U(q) and the actions of their Hessians on vectors (the later being relatively cheap in the case of many-body problems). A time-symmetric symplectic method is constructed that has order 10 when applied to Hamiltonian systems with quadratic kinetic energy T(p). It is shown by numerical experiments that the new methods have the expected order of convergence.

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Acknowledgements

This work of Yifa Tang and Xiongbiao Tu is supported by the National Natural Science Foundation of China (Grant No. 11771438). Ander Murua has received funding from the Ministerio de Economía y Competitividad (Spain) and through project MTM2016-77660-P (AEI/the Department of Education of the Basque Government through the Consolidated Research Group MATHMODE (IT1294-19).

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Authors and Affiliations

  1. LSEC, ICMSEC, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, China

    Xiongbiao Tu & Yifa Tang

  2. School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing, 100049, China

    Xiongbiao Tu & Yifa Tang

  3. Konputazio Zientziak eta A. A. Saila, Informatika Fakultatea, UPV/EHU, E-20018, Donostia–San Sebastián, Spain

    Ander Murua

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  1. Xiongbiao Tu
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  3. Yifa Tang
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Correspondence to Xiongbiao Tu.

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Appendix A

Appendix A

Table 5 The initial values of the sun and the eight planets in the Solar System in the rectangular coordinates (x, y, x)
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Tu, X., Murua, A. & Tang, Y. New high order symplectic integrators via generating functions with its application in many-body problem. Bit Numer Math 60, 509–535 (2020). https://doi.org/10.1007/s10543-019-00785-0

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