Weakly regular fluid flows with bounded variation on the domain of outer communication of a Schwarzschild black hole spacetime. A numerical study

TL;DR

This paper develops numerical methods to study the behavior of spherically symmetric compressible fluids near a Schwarzschild black hole, focusing on stability and late-time dynamics.

Contribution

It introduces well-balanced numerical schemes tailored for relativistic fluid equations on Schwarzschild backgrounds, enabling stability analysis near black hole horizons.

Findings

Numerical schemes preserve steady states accurately.

Fluid perturbations tend to stabilize or exhibit specific late-time behaviors.

The study provides insights into fluid dynamics in curved spacetime near black holes.

Abstract

We study the dynamical behavior of compressible fluids evolving on the outer domain of communication of a Schwarzschild background. To this end, we design several numerical methods which take the Schwarzschild geometry into account and we treat, both, the relativistic Burgers equation and the relativistic Euler system under the assumption that the flow is spherically symmetric. All the schemes we construct are proven to be well-balanced and therefore to preserve the family of steady state solutions for both models. They enable us to study the nonlinear stability of fluid equilibria, and in particular to investigate the behavior of the fluid near the blackhole horizon. We state and numerically demonstrate several conjectures about the late-time behavior of perturbations of steady solutions.