Symplectic integrators: T plus V revisited and round-off reduced

Symplectic integrators separate a problem into parts that can be solved in isolation, alternately advancing these sub-problems to approximate the evolution of the complete system. Problems with a single, dominant mass can use mixed-variable symplectic (MVS) integrators that separate the problem into Keplerian motion of satellites about the primary, and satellite-satellite interactions. Here, we examine T + V algorithms, where the problem is separated into kinetic T and potential energy V terms. T + V integrators are typically less efficient than MVS algorithms. This difference is reduced by using different step sizes for primary-satellite and satellite-satellite interactions. The T + V method is improved further using fourth and sixth-order algorithms that include force gradients and symplectic correctors. We describe three sixth-order algorithms, containing two or three force evaluations per step, that are competitive with MVS in some cases. Round-off errors for T + V integrators can be reduced by several orders of magnitude, at almost no computational cost, using a simple modification that keeps track of accumulated changes in the coordinates and momenta. This makes T + V algorithms desirable for long term, high-accuracy calculations.