Multi-step hybrid methods adapted to the numerical integration of oscillatory second-order systems

Multi-step hybrid methods adapted to the numerical integration of oscillatory second-order systems y′′(t)+My(t)=g(t,y(t))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$y''(t)+My(t)=g(t,y(t))$$\end{document} are proposed and developed. The new methods inherit the basic framework of multi-step hybrid methods proposed by Li et al. (Numer Algorithms 73:711–733, 2016) and take account into the special oscillatory feature of the true flows. These methods contain the information from the previous steps and are designed specifically for oscillatory problem. The key property is that these methods are able to integrate exactly unperturbed oscillators y′′(t)+My(t)=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$y''(t)+My(t)=\mathbf {0}$$\end{document}. The order conditions of the new methods are deduced by using the theory of extended Nyström-series defined on the set of extended Nyström-trees. The linear stability properties are examined. Based on the order conditions, two explicit adapted four-step hybrid methods with order six and seven, respectively, are constructed. Numerical results show the superiority of the new methods over other methods from the scientific literature for oscillatory second-order systems.