  We study the dynamical behavior of compressible fluids evolving on the outer
domain of communication of a Schwarzschild background. To this end, we design
several numerical methods which take the Schwarzschild geometry into account
and we treat, both, the relativistic Burgers equation and the relativistic
Euler system under the assumption that the flow is spherically symmetric. All
the schemes we construct are proven to be well-balanced and therefore to
preserve the family of steady state solutions for both models. They enable us
to study the nonlinear stability of fluid equilibria, and in particular to
investigate the behavior of the fluid near the blackhole horizon. We state and
numerically demonstrate several conjectures about the late-time behavior of
perturbations of steady solutions.
