# A Fast Poisson Solver of Second-Order Accuracy for Isolated Systems in Three-Dimensional Cartesian and Cylindrical Coordinates

## TL;DR

This paper introduces a fast, second-order accurate Poisson solver for calculating gravitational potentials in 3D Cartesian and cylindrical coordinates, suitable for isolated systems with vacuum boundary conditions.

## Contribution

It develops a novel eigenfunction expansion and boundary screening method, including a discrete Green's function in cylindrical coordinates, integrated into Athena++ for improved accuracy and efficiency.

## Key findings

- The solver achieves second-order accuracy in tests.
- It demonstrates good parallel performance.
- It effectively computes gravitational potential for isolated systems.

## Abstract

We present an accurate and efficient method to calculate the gravitational potential of an isolated system in three-dimensional Cartesian and cylindrical coordinates subject to vacuum (open) boundary conditions. Our method consists of two parts: an interior solver and a boundary solver. The interior solver adopts an eigenfunction expansion method together with a tridiagonal matrix solver to solve the Poisson equation subject to the zero boundary condition. The boundary solver employs James's method to calculate the boundary potential due to the screening charges required to keep the zero boundary condition for the interior solver. A full computation of gravitational potential requires running the interior solver twice and the boundary solver once. We develop a method to compute the discrete Green's function in cylindrical coordinates, which is an integral part of the James algorithm to maintain second-order accuracy. We implement our method in the {\tt Athena++} magnetohydrodynamics code, and perform various tests to check that our solver is second-order accurate and exhibits good parallel performance.