  We have developed a numerical code to study the evolution of self-gravitating
matter in dynamic black hole axisymmetric spacetimes in general relativity. The
matter fields are evolved with a high-resolution shock-capturing scheme that
uses the characteristic information of the general relativistic hydrodynamic
equations to build up a linearized Riemann solver. The spacetime is evolved
with an axisymmetric ADM code designed to evolve a wormhole in full general
relativity. We discuss the numerical and algorithmic issues related to the
effective coupling of the hydrodynamical and spacetime pieces of the code, as
well as the numerical methods and gauge conditions we use to evolve such
spacetimes. The code has been put through a series of tests that verify that it
functions correctly. Particularly, we develop and describe a new set of testbed
calculations and techniques designed to handle dynamically sliced,
self-gravitating matter flows on black holes, and subject the code to these
tests. We make some studies of the spherical and axisymmetric accretion onto a
dynamic black hole, the fully dynamical evolution of imploding shells of dust
with a black hole, the evolution of matter in rotating spacetimes, the
gravitational radiation induced by the presence of the matter fields and the
behavior of apparent horizons through the evolution.

General Relativity and Quantum Cosmology;Astrophysics