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We design well-balanced numerical algorithms with first-, second-, and third-order of accuracy for the spherically symmetry evolution of a compressible fluid on a Schwarzschild black hole. We treat both the relativistic Burgers-Schwarzschild model and the relativistic Euler-Schwarzschild model. Our schemes follow the finite volume methodology and preserve the stationary solutions and allow us to investigate the global asymptotic behavior of such flows and reach definite conclusions about the behavior of the mass density and velocity field.

For the evolution of a compressible fluid in spherical symmetry on a Schwarzschild curved background, we design a class of well-balanced numerical algorithms up to third-order 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.