TY - JOUR T1 - High-Order Symplectic and Symmetric Composition Methods for Multi-Frequency and Multi-Dimensional Oscillatory Hamiltonian Systems AU - Liu , Kai AU - Wu , Xinyuan JO - Journal of Computational Mathematics VL - 4 SP - 356 EP - 378 PY - 2015 DA - 2015/08 SN - 33 DO - http://doi.org/10.4208/jcm.1502-m2014-0082 UR - https://global-sci.org/intro/article_detail/jcm/9848.html KW - Symplectic and symmetric composition methods, Multi-frequency and multi-dimensional ERKN integrators, ARKN integrators, Multi-frequency oscillatory Hamiltonian systems. AB -

The multi-frequency and multi-dimensional adapted Runge-Kutta-Nyström (ARKN) integrators, and multi-frequency and multi-dimensional extended Runge-Kutta-Nyström(ERKN) integrators have been developed to efficiently solve multi-frequency oscillatory Hamiltonian systems. The aim of this paper is to analyze and derive high-order symplectic and symmetric composition methods based on the ARKN integrators and ERKN integrators. We first consider the symplecticity conditions for the multi-frequency and multi-dimensional ARKN integrators. We then analyze the symplecticity of the adjoint integrators of the multi-frequency and multi-dimensional symplectic ARKN integrators and ERKN integrators, respectively. On the basis of the theoretical analysis and by using the idea of composition methods, we derive and propose four new high-order symplectic and symmetric methods for the multi-frequency oscillatory Hamiltonian systems. The numerical results accompanied in this paper quantitatively show the advantage and efficiency of the proposed high-order symplectic and symmetric methods.