# Self-gravitational Force Calculation of High-Order Accuracy for Infinitesimally Thin Gaseous Disks

## TL;DR

This paper introduces a fast, high-order numerical algorithm for calculating self-gravitational forces in infinitesimally thin disks, crucial for galactic and protoplanetary disk studies, using convolution and FFT techniques.

## Contribution

The authors develop a linear-complexity, high-order accurate method leveraging B-splines and FFT, avoiding artificial boundary conditions for force calculations in thin disks.

## Key findings

- The method achieves significant accuracy improvements with higher order.
- Numerical results agree well with analytic solutions for generalized disks.
- The approach is efficient and stable for practical astrophysical simulations.

## Abstract

Self-gravitational force calculation for infinitesimally thin disks is important for studies on the evolution of galactic and protoplanetary disks. Although high-order methods have been developed for hydrodynamic and magneto-hydrodynamic equations, high-order improvement is desirable for solving self-gravitational forces for thin disks. In this work, we present a new numerical algorithm that is of linear complexity and of high-order accuracy. This approach is fast since the force calculation is associated with a convolution form, and the fast calculation can be achieved using Fast Fourier Transform. The nice properties, such as the finite supports and smoothness, of B-splines are exploited to stably interpolate a surface density and achieve a high-order accuracy in forces. Moreover, if the mass distribution of interest is exclusively confined within a calculation domain, the method does not require artificial boundary values to be specified before the force calculation. To validate the proposed algorithm, a series of numerical tests, ranging from 1st- to 3rd-order implementations, are performed and the results are compared with analytic expressions derived for 3rd- and 4th-order generalized Maclaurin disks. We conclude that the improvement on the numerical accuracy is significant with the order of the method, with only little increase of the complexity of the method.