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

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.