High-Order Symplectic and Symmetric Composition Methods for Multi-Frequency and Multi-Dimensional Oscillatory Hamiltonian Systems (Journal of Computational Mathematics, 2018)
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 oscillatoryHamiltonian systems. The aim of this paper is to analyze and derive high-order symplectic and symmetric composition methods based on the ARKN integrators and ERKNintegrators. We first consider the symplecticity conditions for the multi-frequency andmulti-dimensional ARKN integrators. We then analyze the symplecticity of the adjoint integrators of the multi-frequency and multi-dimensional symplectic ARKN integrators andERKN integrators, respectively. On the basis of the theoretical analysis and by using theidea of composition methods, we derive and propose four new high-order symplectic andsymmetric methods for the multi-frequency oscillatory Hamiltonian systems. The numerical results accompanied in this paper quantitatively show the advantage and efficiency ofthe proposed high-order symplectic and symmetric methods.