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 whichthe self-gravity of the flow needs to be considered. Because Poisson's equationis linear, the numerical solution for the gravitational potential for eachindividual mode of the density can be pre-computed, thus reducing substantiallythe computational cost of the method. In this second paper, we describe twodifferent approaches to computing the gravitational field of a two-dimensionalflow with pseudo-spectral methods. For situations in which the density profileis independent of the third coordinate (i.e., an infinite cylinder), we use astandard Poisson solver in spectral space. On the other hand, for situations inwhich 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 performa 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 twomethods. 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, thenumerical solutions are always stable, even when the linear criterion predictsthe contrary. We show that, in the latter case, the minimum wavelength of theunstable modes is larger than the extent of the unstable region and hence thelocal linear analysis is inapplicable.