We study the dynamical behavior of compressible fluids evolving on the outer domain of communication of a Schwarzschild background. For both the relativistic Burgers equation and the relativistic Euler system, assuming spherical symmetry we introduce numerical methods that take the Schwarzschild geometry and, specifically, the steady state solutions into account. The schemes we propose preserve the family of steady state solutions and enable us to study the nonlinear stability of fluid equilibria and the behavior of solutions near the black hole horizon. We state and numerically demonstrate several properties about the late-time behavior of perturbed steady states.
"A numerical study of the relativistic Burgers and Euler equations on a Schwarzschild black hole exterior." Commun. Appl. Math. Comput. Sci. 13 (2) 271 - 301, 2018. https://doi.org/10.2140/camcos.2018.13.271