  For the evolution of a compressible fluid in spherical symmetry on a
Schwarzschild curved background, we design a class of well-balanced numerical
algorithms with first-order or second-order of accuracy. We treat both the
relativistic Burgers-Schwarzschild model and the relativistic
Euler-Schwarzschild model and take advantage of the explicit or implicit forms
available for the stationary solutions of these models. Our schemes follow the
finite volume methodology and preserve the stationary solutions. Importantly,
they allow us to investigate the global asymptotic behavior of such flows and
determine the asymptotic behavior of the mass density and velocity field of the
fluid.
