Explicit symplectic multidimensional exponential fitting modified Runge-Kutta-Nyström methods

This paper is concerned with multidimensional exponential fitting modified Runge-Kutta-Nyström (MEFMRKN) methods for the system of oscillatory second-order differential equations q″(t)+Mq(t)=f(q(t)), where M is a d×d symmetric and positive semi-definite matrix and f(q) is the negative gradient of a potential scalar U(q). We formulate MEFMRKN methods and show clearly the relationship between MEFMRKN methods and multidimensional extended Runge-Kutta-Nyström (ERKN) methods proposed by Wu et al. (Comput. Phys. Comm. 181:1955–1962, 2010). Taking into account the fact that the oscillatory system is a separable Hamiltonian system with Hamiltonian \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$H(p,q)=\frac{1}{2}p^{T}p+ \frac{1}{2}q^{T}Mq+U(q)$\end{document}, we derive the symplecticity conditions for the MEFMRKN methods. Two explicit symplectic MEFMRKN methods are proposed. Numerical experiments accompanied demonstrate that our explicit symplectic MEFMRKN methods are more efficient than some well-known numerical methods appeared in the scientific literature.