  We consider the relativistic Euler equations governing spherically symmetric,
perfect fluid flows on the outer domain of communication of Schwarzschild
spacetime, and we introduce a version of the finite volume method which is
formulated geometrically (without choosing coordinates a priori) and is
well--balanced, in the sense that it preserves steady solutions to the Euler
equations on the curved geometry under consideration. In order to formulate our
method, we first derive a closed formula describing all steady and spherically
symmetric solutions to the Euler equations posed on Schwarzschild spacetime.
Second, we describe a finite volume method which is formulated geometrically
from the family of steady solutions to the Euler system. Our scheme is
second--order accurate and, as required, preserves the family of steady
solutions at the discrete level. Numerical experiments are presented which
demonstrate the efficiency and robustness of the proposed method even for
solutions containing shock waves and nonlinear interacting wave patterns. As an
application, we investigate the late--time asymptotics of perturbed steady
solutions and demonstrate its convergence for late time toward another steady
solution, taking the overall effect of the perturbation into account.
