A fast multipole method for stellar dynamics

TL;DR

This paper presents an efficient implementation of the fast multipole method (FMM) for stellar dynamics, achieving near-linear scaling and outperforming GPU-based direct summation for large N, with high accuracy in gravitational force calculations.

Contribution

The paper introduces a novel error estimation and minimization technique for FMM, improving computational efficiency and accuracy in stellar dynamics simulations.

Findings

FMM achieves ~10^{-7} force accuracy with sub-linear scaling (~N^{0.87})

Implementation outperforms GPU-based direct summation for N > 10^5

Error minimization leads to well-behaved force error distribution

Abstract

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 $\mathcal{O}(N)$ 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 $\sim10^{-7}$, the computational costs exhibit an empirical scaling of $\propto N^{0.87}$. My implementation (running on a 16 core node) out-performs a GPU-based direct summation with comparable force errors for $N\gtrsim10^5$.