  During galaxy mergers the gas falls to the center, triggers star formation,
and feeds the rapid growth of supermassive black holes (SMBHs). SMBHs respond
to this fueling by supplying energy back to the ambient gas. Numerical studies
suggest that this feedback is necessary to explain why the properties of SMBHs
and the formation of bulges are closely related. This intimate link between the
SMBH's mass and the large scale dynamics and luminosity of the host has proven
to be a difficult issue to tackle with simulations due to the inability to
resolve all the relevant length scales simultaneously. In this paper we
simulate SMBH growth at high-resolution with {\it FLASH}, accounting for the
gravitational focusing effects of nuclear star clusters (NSCs), which appear to
be ubiquitous in galactic nuclei. In the simulations, the NSC core is resolved
by a minimum cell size of about 0.001 pc or approximately $10^{-3}$ of the
cluster's radius. We discuss the conditions required for effective gas
funneling to occur, which are mainly dominated by a relationship between NSC
velocity dispersion and the local sound speed, and provide a sub-grid
prescription for the augmentation of central SMBH accretion rates in the
presence of NSCs. For the conditions expected to persist in the centers of
merging galaxies, the resultant large central gas densities in NSCs should
produce drastically enhanced embedded SMBH accretion rates - up to an order of
magnitude increase can be achieved for gas properties resembling those in
large-scale galaxy merger simulations. This will naturally result in faster
black hole growth rates and higher luminosities than predicted by the commonly
used Bondi-Hoyle-Lyttleton accretion formalism.
