Conservation properties of a time FE method. Part I: time-stepping schemes for N-body problems

In the present paper one-step implicit integration algorithms for the N-body problem are developed. The time-stepping schemes are based on a Petrov{Galerkin nite element method applied to the Hamiltonian formulation of the N-body problem. The approach furnishes algorithmic energy conservation in a natural way. The proposed time nite element method facilitates a systematic implementation of a family of time- stepping schemes. A particular algorithm is speci ed by the associated quadrature rule employed for the evaluation of time integrals. The in uence of various standard as well as non-standard quadrature formulas on algorithmic energy conservation and conservation of angular momentum is examined in detail for linear and quadratic time elements