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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 pre-computed, 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 pseudo-spectral 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.Comment: 13 pages, 9 figures. To appear in the ApJ. High resolution plots and
animations of the simulations are available at
http://www.physics.arizona.edu/~chan/research/astro-ph/0512448/index.htm
Spectral methods for CFD
One of the objectives of these notes is to provide a basic introduction to spectral methods with a particular emphasis on applications to computational fluid dynamics. Another objective is to summarize some of the most important developments in spectral methods in the last two years. The fundamentals of spectral methods for simple problems will be covered in depth, and the essential elements of several fluid dynamical applications will be sketched
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