Explicit near-symplectic integrators for post-Newtonian Hamiltonian systems

Explicit symplectic integrators are powerful and widely used for Hamiltonian systems. However, once the post-Newtonian (PN) effect is considered to provide more precise modeling for the N-body problem, explicit symplectic methods cannot be constructed due to the nonseparability of the Hamiltonian. Thus, the available symplectic method is either fully implicit or semi-implicit, which decreases the efficiency because of the implicit iteration used during the evolution. In this paper, we aim to explore efficient explicit methods whose performance is mostly like symplectic methods for PN Hamiltonian systems. Taking the small parameter epsilon\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon $$\end{document} appearing in PN terms into consideration, we replace the implicit symplectic solver with explicit solvers in the mixed symplectic method to solve the PN term and then derive three explicit methods. It is theoretically shown that the proposed methods are respectively second-order, fourth-order, and pseudo-fourth-order, and that their closeness to the corresponding symplectic methods are O(epsilon 3h3),\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathscr {O}}(\varepsilon <^>{3}h<^>{3}),$$\end{document}O(epsilon 5h5),\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathscr {O}}(\varepsilon <^>{5}h<^>{5}),$$\end{document} and O(epsilon 3h3).\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathscr {O}}(\varepsilon <^>{3}h<^>{3}).$$\end{document} That is, they are explicit near-symplectic methods with the presence of the small parameter epsilon.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon .$$\end{document} Numerical experiments with the Hamiltonian problem of spinning compact binaries show that the energy errors and orbital errors of the proposed explicit near-symplectic methods are indistinguishable from the corresponding mixed semi-implicit symplectic methods. The very small magnitude of the difference between the proposed explicit near-symplectic methods and the mixed symplectic methods confirms our theoretical analysis of their closeness to symplecticity. Finally, the much less CPU time consumed by the proposed methods highlights their most important advantage of high efficiency over the mixed symplectic methods.