  The way supermassive black holes (SMBH) in galactic centers accumulate their
mass is not completely determined. At large scales, it is governed by galactic
encounters, mass inflows connected to spirals arms and bars, or due to
expanding shells from supernova (SN) explosions in the central parts of
galaxies. The investigation of the latter process requires an extensive set of
gas dynamical simulations to explore the muti-dimensional parameter space
needed to frame the phenomenon. The aims of this paper are to extend our
investigation of the importance of supernovae for inducing accretion onto a
SMBH and carry out a comparison between the fully hydrodynamic code Flash and
the much less computationally intensive code Ring, which uses the thin shell
approximation. We simulate 3D expanding shells in a gravitational potential
similar to that of the Galactic Center with a variety of homogeneous and
turbulent environments. In homogeneous media, we find convincing agreement
between Flash and Ring in the shapes of shells and their equivalent radii
throughout their whole evolution until they become subsonic. In highly
inhomogeneous, turbulent media, there is also a good agreement of shapes and
sizes of shells, and of the times of their first contact with the central 1 pc
sphere, where we assume that they join the accretion flow. The comparison
supports the proposition that a SN occurring at a galactocentric distance of 5
pc typically drives 1 - 3 $M_\odot$ into the central 1 pc around the galactic
center.
