  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.
