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Investigating the evolution of disk galaxies and the dynamics ofproto-stellar disks can involve the use of both a hydrodynamical and a Poissonsolver. These systems are usually approximated as infinitesimally thin disksusing two- dimensional Cartesian or polar coordinates. In Cartesiancoordinates, the calcu- lations of the hydrodynamics and self-gravitationalforces are relatively straight- forward for attaining second order accuracy.However, in polar coordinates, a second order calculation of self-gravitationalforces is required for matching the second order accuracy of hydrodynamicalschemes. We present a direct algorithm for calculating self-gravitationalforces with second order accuracy without artifi- cial boundary conditions. ThePoisson integral in polar coordinates is expressed in a convolution form andthe corresponding numerical complexity is nearly lin- ear using a fast Fouriertransform. Examples with analytic solutions are used to verify that thetruncated error of this algorithm is of second order. The kernel integralaround the singularity is applied to modify the particle method. The use of asoftening length is avoided and the accuracy of the particle method issignificantly improved.

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