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

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.

24 pages, 13 figures; accepted for publication in ApJS

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• An Extension of the Athena++ Framework for Fully Conservative Self-Gravitating Hydrodynamics
• A Magnetized, Moon-Forming Giant Impact
• Self-gravitational Force Calculation of Second Order Accuracy Using Multigrid Method on Nested Grids
• Future Simulations of Tidal Disruption Events
• Direct Calculation of Self-Gravitational Force for Infinitesimally Thin Gaseous Disks Using Adaptive Mesh Refinement
• An efficient spectral Poisson solver for the nirvana-III code: the shearing-box case with vertical vacuum boundary conditions
• Self-gravity in thin protoplanetary discs: 1. The smoothing-length approximation versus the exact self-gravity kernel