We consider the relativistic Euler equations governing spherically symmetric, perfect fluid flows on the outer domain of communication of Schwarzschild space- time, and we introduce a version of the finite volume method which is formulated from the geometric formulation (and thus takes the geometry into account at the discretization level) 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 sym- metric solutions to the Euler equations posed on Schwarzschild spacetime. Second, we describe a geometry-preserving, finite volume method which is based 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 pro- posed 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
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