Spectral methods for time-dependent studies of accretion flows.: II.: Two-dimensional hydrodynamic disks with self-gravity

Spectral methods are well suited for solving hydrodynamic problems in which the self-gravity of the flow needs to be considered. Because Poisson's equation is linear, the numerical solution for the gravitational potential for each individual mode of the density can be precomputed, thus reducing substantially the computational cost of the method. In this second paper, we describe two different approaches to computing the gravitational field of a two-dimensional flow with pseudospectral methods. For situations in which the density profile is independent of the third coordinate (i.e., an infinite cylinder), we use a standard Poisson solver in spectral space. On the other hand, for situations in which the density profile is a delta-function along the third coordinate (i.e., an infinitesimally thin disk), or any other function known a priori, we perform a direct integration of Poisson's equation using a Green's functions approach. We devise a number of test problems to verify the implementations of these two methods. Finally, we use our method to study the stability of polytropic, self-gravitating disks. We find that when the polytropic index Gamma is <= 4/3, Toomre's criterion correctly describes the stability of the disk. However, when Gamma > 4/3 and for large values of the polytropic constant K, the numerical solutions are always stable, even when the linear criterion predicts the contrary. We show that in the latter case, the minimum wavelength of the unstable modes is larger than the extent of the unstable region, and hence the local linear analysis is inapplicable.