Self-Gravitational Force Calculation of Second Order Accuracy for Infinitesimally Thin Gaseous Disks in Polar Coordinates

TL;DR

This paper introduces a second order accurate, efficient algorithm for calculating self-gravitational forces in infinitesimally thin gaseous disks using polar coordinates, improving upon previous methods without artificial boundary conditions.

Contribution

The authors develop a direct, second order accurate algorithm for self-gravitational force calculation in polar coordinates, avoiding artificial boundary conditions and softening length.

Findings

The algorithm achieves second order accuracy verified by analytic solutions.

The convolution form reduces computational complexity to nearly linear with FFT.

The modified particle method improves accuracy without using softening length.

Abstract

Investigating the evolution of disk galaxies and the dynamics of proto-stellar disks can involve the use of both a hydrodynamical and a Poisson solver. These systems are usually approximated as infinitesimally thin disks using two- dimensional Cartesian or polar coordinates. In Cartesian coordinates, the calcu- lations of the hydrodynamics and self-gravitational forces are relatively straight- forward for attaining second order accuracy. However, in polar coordinates, a second order calculation of self-gravitational forces is required for matching the second order accuracy of hydrodynamical schemes. We present a direct algorithm for calculating self-gravitational forces with second order accuracy without artifi- cial boundary conditions. The Poisson integral in polar coordinates is expressed in a convolution form and the corresponding numerical complexity is nearly lin- ear using a fast Fourier transform. Examples with analytic solutions are used to verify that the truncated error of this algorithm is of second order. The kernel integral around the singularity is applied to modify the particle method. The use of a softening length is avoided and the accuracy of the particle method is significantly improved.