An FFT-based Solution Method for the Poisson Equation on 3D Spherical Polar Grids

TL;DR

This paper introduces a non-iterative, FFT-based method for solving the 3D Poisson equation on spherical polar grids, providing exact solutions for complex density distributions and improving computational efficiency.

Contribution

The paper presents a novel eigenfunction-based solver that overcomes limitations of truncated spherical harmonics, enabling accurate solutions for off-center point masses in 3D spherical grids.

Findings

Handles off-center point masses effectively

Computationally competitive with traditional methods

Parallel implementation available

Abstract

The solution of the Poisson equation is a ubiquitous problem in computational astrophysics. Most notably, the treatment of self-gravitating flows involves the Poisson equation for the gravitational field. In hydrodynamics codes using spherical polar grids, one often resorts to a truncated spherical harmonics expansion for an approximate solution. Here we present a non-iterative method that is similar in spirit, but uses the full set of eigenfunctions of the discretized Laplacian to obtain an exact solution of the discretized Poisson equation. This allows the solver to handle density distributions for which the truncated multipole expansion fails, such as off-center point masses. In three dimensions, the operation count of the new method is competitive with a naive implementation of the truncated spherical harmonics expansion with $N_\ell \approx 15$ multipoles. We also discuss the parallel implementation of the algorithm. The serial code and a template for the parallel solver are made publicly available.