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Symplectic exponentially fitted RK and RKN methods have been extensively studied by many researchers. Due to their good property, they have been applied to many problems such as pendulum, Morse oscillator, harmonic oscillator, Lennard–Jones oscillator, Kepler’s orbit problem, and so on. In this paper, we construct an implicit symmetric and symplectic exponentially fitted modified Runge–Kutta–Nyström (ISSEFMRKN) method. The new integrator integrates exactly differential systems whose solutions can be expressed as linear combinations of functions from the set    \documentclass[12pt]{minimal} 				\usepackage{amsmath} 				\usepackage{wasysym}  				\usepackage{amsfonts}  				\usepackage{amssymb}  				\usepackage{amsbsy} 				\usepackage{mathrsfs} 				\usepackage{upgreek} 				\setlength{\oddsidemargin}{-69pt} 				\begin{document}$\{\exp(\lambda t),\exp(-\lambda t)\}$\end{document}   {  exp  (  λ  t  )  ,  exp  (  −  λ  t  )  },    \documentclass[12pt]{minimal} 				\usepackage{amsmath} 				\usepackage{wasysym}  				\usepackage{amsfonts}  				\usepackage{amssymb}  				\usepackage{amsbsy} 				\usepackage{mathrsfs} 				\usepackage{upgreek} 				\setlength{\oddsidemargin}{-69pt} 				\begin{document}$\lambda\in\mathbb{C}$\end{document}   λ  ∈  C, or equivalently    \documentclass[12pt]{minimal} 				\usepackage{amsmath} 				\usepackage{wasysym}  				\usepackage{amsfonts}  				\usepackage{amssymb}  				\usepackage{amsbsy} 				\usepackage{mathrsfs} 				\usepackage{upgreek} 				\setlength{\oddsidemargin}{-69pt} 				\begin{document}$\{\sin(\omega t),\cos(\omega t)\}$\end{document}   {  sin  (  ω  t  )  ,  cos  (  ω  t  )  }when    \documentclass[12pt]{minimal} 				\usepackage{amsmath} 				\usepackage{wasysym}  				\usepackage{amsfonts}  				\usepackage{amssymb}  				\usepackage{amsbsy} 				\usepackage{mathrsfs} 				\usepackage{upgreek} 				\setlength{\oddsidemargin}{-69pt} 				\begin{document}$\lambda=i\omega$\end{document}   λ  =  i  ω,    \documentclass[12pt]{minimal} 				\usepackage{amsmath} 				\usepackage{wasysym}  				\usepackage{amsfonts}  				\usepackage{amssymb}  				\usepackage{amsbsy} 				\usepackage{mathrsfs} 				\usepackage{upgreek} 				\setlength{\oddsidemargin}{-69pt} 				\begin{document}$\omega \in\mathbb{R}$\end{document}   ω  ∈  R. When    \documentclass[12pt]{minimal} 				\usepackage{amsmath} 				\usepackage{wasysym}  				\usepackage{amsfonts}  				\usepackage{amssymb}  				\usepackage{amsbsy} 				\usepackage{mathrsfs} 				\usepackage{upgreek} 				\setlength{\oddsidemargin}{-69pt} 				\begin{document}$z=\lambda h$\end{document}   z  =  λ  happroaches zero, the ISSEFMRKN method reduces to the classical symplectic, symmetric RKN integrator. Numerical experiments are accompanied to show the efficiency and competence of the new method compared with some efficient codes in the literature.

Implicit symmetric and symplectic exponentially fitted modified Runge–Kutta–Nyström methods for solving oscillatory problems