Pseudo-high-order symplectic integrators

Symplectic N-body integrators are widely used to study problems in celestial mechanics. The most popular algorithms are of second and fourth order, requiring two and six substeps per time step, respectively. The number of substeps increases rapidly with order in time step, rendering higher order methods impractical. However, symplectic integrators are often applied to systems in which perturbations between bodies are a small factor epsilon of the force due to a dominant central mass. In this case, it is possible to create optimized symplectic algorithms that require fewer substeps per time step. This is achieved by only considering error terms of order epsilon and neglecting those of order epsilon(2), epsilon(3), etc. Here we devise symplectic algorithms with four and six substeps per step which effectively behave as fourth- and sixth-order integrators when epsilon is small. These algorithms are more efficient than the usual second- and fourth-order methods when applied to planetary systems.