  (Abridged) We study stability of gas accretion in Active Galactic Nuclei. Our
grid based simulations cover a radial range from 0.1 to 200 pc. Here, as in
previous studies by our group, we include gas radiative cooling as well as
heating by a sub-Eddington X-ray source near the central supermassive black
hole of 10^8 M_{\odot}. Our theoretical estimates and simulations show that for
the X-ray luminosity L_X \sim 0.008 L_{Edd}, the gas is thermally and
convectivelly unstable within the computational domain. In the simulations, we
observe that very tiny fluctuations in an initially smooth, spherically
symmetric, accretion flow, grow first linearly and then non-linearly.
Consequently, an initially one-phase flow relatively quickly transitions into a
two-phase/cold-hot accretion flow. For L_X = 0.015 L_{Edd} or higher, the cold
clouds continue to accrete but in some regions of the hot phase, the gas starts
to move outward. For L_X < 0.015 L_{Edd}, the cold phase contribution to the
total mass accretion rate only moderately dominates over the hot phase
contribution. This result might have some consequences for cosmological
simulations of the so-called AGN feedback problem. Our simulations confirm the
previous results of Barai et al. (2012) who used smoothed particle hydrodynamic
simulations to tackle the same problem. However here, because we use a grid
based code to solve equations in 1-D and 2-D, we are able to follow the gas
dynamics at much higher spacial resolution and for longer time in comparison to
the 3-D SPH simulations. One of new features revealed by our simulations is
that the cold condensations in the accretion flow initially form long
filaments, but at the later times, those filaments may break into smaller
clouds advected outwards within the hot outflow. These simulations may serve as
an attractive model for the so-called Narrow Line Region in AGN.
