G-Symplectic Integration of Many Body Problems

The purpose of this paper is to derive a symmetric parasitic-free G-symplectic general linear method of order 4 and apply it to problems in celestial mechanics. The numerical method is constructed by satisfying the G-symplectic conditions for general linear methods, together with relevant order conditions while making sure that there is zero parasitism. The internal stages are designed to be diagonally implicit to make the method more efficient. Being multivalue in nature, a starting method is required for the implementation of general linear method and this is also calculated using rooted trees. The general linear method is applied to many body problems and acceptable error in energy and global error are observed.