Efficient Direct Method for Self-gravity in 3D, Accelerated by a Fast Fourier Transform

Self-gravity calculations for 3D are expensive in terms of computational time. Several methods exist for this computation, for example multigrid and spectral methods. Unfortunately, these approaches require the imposition of boundary conditions, which can be either numerically expensive (direct Newtonian sums), artificial (periodicity assumptions), or potentially imprecise (multipolar expansions). In this work we present a novel direct numerical method to calculate the gravitational potential and forces by solving the Poisson equation without the need to prescribe artificial boundary conditions; this method, despite being direct, turns out to be efficient due to the possibility of using a fast Fourier transform for its implementation. For a grid having N zones in each dimension, the computational complexity of the method presented here is $O({N}^{3}\mathrm{log}{N}^{3})$ , which is comparable with multigrid methods under no consideration of boundary settings. Finally, a numerical study shows this proposed method can achieve second order for calculations of both potential and forces.