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ABSTRACT The way supermassive black holes (SMBHs) in Galactic Centres (GCs) 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 multidimensional parameter space needed to frame the phenomenon. The aims of this paper are to extend our investigation of the importance of SNe for inducing accretion on to an 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 GC 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 an SN occurring at a galactocentric distance of 5 pc typically drives 1–3 M⊙ into the central 1 pc around the GC. 

Aims. We simulate shells created by supernovae expanding into the interstellar medium of the nuclear region of a galaxy, and analyze how the shell evolution is influenced by the supernova position relative to the galactic center, by the interstellar matter density, and by the combined gravitational pull of the nuclear star cluster and supermassive black hole (SMBH). Methods. We adopted simplified hydrodynamical simulations using the infinitesimally thin layer approximation in 3D (code RING) and determined whether and where the shell expansion may bring new gas into the inner parsec around the SMBH. Results. The simulations show that supernovae occurring within a conical region around the rotational axis of the galaxy can feed the central accretion disk surrounding the SMBH. For ambient densities between 10 3 and 10 5 cm −3 , the average mass deposited into the central parsec by individual supernovae varies between 10 and 1000 solar masses depending on the ambient density and the spatial distribution of supernova events. Supernovae occurring in the aftermath of a starburst event near a galactic center can supply two to three orders of magnitude more mass into the central parsec, depending on the magnitude of the starburst. The deposited mass typically encounters and joins an accretion disk. The fate of that mass is then divided between the growth of the SMBH and an energetically driven outflow from the disk.