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

  • Xiongbiao TuEmail author
  • Ander Murua
  • Yifa Tang


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.


High order Separable Hamiltonian Quadratic 

Mathematics Subject Classification




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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© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.LSEC, ICMSEC, Academy of Mathematics and Systems ScienceChinese Academy of SciencesBeijingChina
  2. 2.School of Mathematical SciencesUniversity of Chinese Academy of SciencesBeijingChina
  3. 3.Konputazio Zientziak eta A. A. Saila, Informatika FakultateaUPV/EHUDonostia–San SebastiánSpain

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