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

The geometric finite volume method for compressible fluid flows on Schwarzschild spacetime