ERKN methods for long-term integration of multidimensional orbital problems

This paper is devoted to introducing ERKN methods for long-term integration of multidimensional orbital problems. For the general multidimensional perturbed oscillators y^″+My=f(t,y)$y^{″}+\mathit{My}=f(t\text{,}y)$ with M∈R^m×m$M\in {\mathbb{R}}^{m\times m}$, the extended Runge–Kutta–Nyström (ERKN) methods are proposed by Wu et al. X. Wu, X. You, W. Shi, B. Wang, ERKN integrators for systems of oscillatory second-order differential equations, Comput. Phys. Commun. 181 (2010) 1873–1887]. These methods exactly integrate the multidimensional unperturbed oscillators and are highly efficient when the perturbing forces are small. In this paper, we pay attention to the applications of ERKN methods to multidimensional orbital problems. Numerical experiments accompanied demonstrate that for long-term integration of multidimensional orbital problems the multidimensional ERKN methods are more efficient compared with high-quality codes proposed in the scientific literature. In particular, when an orbital problem under consideration is a Hamiltonian system, the symplectic ERKN methods preserve the Hamiltonian very well, and has better accuracy than the high-quality codes with the same computational cost.