  We investigate the dynamics of self-gravitating, spherically-symmetric
distributions of fluid through numerical means. In particular, systems
involving neutron star models driven far from equilibrium in the strong-field
regime of general relativity are studied. Hydrostatic solutions of Einstein's
equations using a stiff, polytropic equation of state are used for the stellar
models. Many of the scenarios we examine involve highly-relativistic flows that
require improvements upon previously published numerical methods to simulate.
Here our particular focus is on the physical behavior of the coupled
fluid-gravitational system at the threshold of black hole formation--so-called
black hole critical phenomena. To investigate such phenomena starting from
conditions representing stable stars, we must drive the star far from its
initial stable configuration. We use one of two different mechanisms to do
this: setting the initial velocity profile of the star to be in-going, or
collapsing a shell of massless scalar field onto the star. Both of these
approaches give rise to a large range of dynamical scenarios that the star may
follow. These scenarios have been extensively surveyed by using different
initial star solutions, and by varying either the magnitude of the velocity
profile or the amplitude of the scalar field pulse. In addition to illuminating
the critical phenomena associated with the fluid collapse, the resulting phase
diagram of possible outcomes provides an approximate picture of the stability
of neutron stars to large, external perturbations that may occur in nature.
