  We present 1D numerical simulations, which consider the effects of radiative
cooling and gravity on the hydrodynamics of the matter reinserted by stellar
winds and supernovae within young nuclear starbursts with a central
supermassive black hole (SMBH). The simulations confirm our previous
semi-analytic results for low energetic starbursts, evolving in a
quasi-adiabatic regime, and extend them to more powerful starbursts evolving in
the catastrophic cooling regime. The simulations show a bimodal hydrodynamic
solution in all cases. They present a quasi-stationary accretion flow onto the
black hole, defined by the matter reinserted by massive stars within the
stagnation volume and a stationary starburst wind, driven by the high thermal
pressure acquired in the region between the stagnation and the starburst radii.
In the catastrophic cooling regime, the stagnation radius rapidly approaches
the surface of the starburst region, as one considers more massive starbursts.
This leads to larger accretion rates onto the SMBH and concurrently to powerful
winds able to inhibit interstellar matter from approaching the nuclear
starburst.
  Our self-consistent model thus establishes a direct physical link between the
SMBH accretion rate and the nuclear star formation activity of the host galaxy
and provides a good upper limit to the accretion rate onto the central black
hole.
