A fast multipole method for stellar dynamics

The approximate computation of all gravitational forces between N interacting particles via the fast multipole method (FMM) can be made as accurate as direct summation, but requires less than O(N) operations. FMM groups particles into spatially bounded cells and uses cell-cell interactions to approximate the force at any position within the sink cell by a Taylor expansion obtained from the multipole expansion of the source cell. By employing a novel estimate for the errors incurred in this process, I minimise the computational effort required for a given accuracy and obtain a well-behaved distribution of force errors. For relative force errors of ~ 10⁻⁷, the computational costs exhibit an empirical scaling of ∝ N^{0.87}. My implementation (running on a 16 core node) out-performs a GPU-based direct summation with comparable force errors for N ≳ 10⁵