Vol. 13, No. 2, 2018

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A numerical study of the relativistic Burgers and Euler equations on a Schwarzschild black hole exterior

Philippe G. LeFloch and Shuyang Xiang

Vol. 13 (2018), No. 2, 271–301
Abstract

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.

Keywords
relativistic fluid, Schwarzschild black hole, steady state solution, generalized Riemann problem, random choice method, finite volume scheme
Mathematical Subject Classification 2010
Primary: 35L60, 35L65, 65M08, 76L05, 76M12
Milestones
Received: 11 August 2017
Revised: 17 April 2018
Accepted: 25 June 2018
Published: 25 September 2018
Authors
Philippe G. LeFloch
Laboratoire Jacques-Louis Lions
Centre National de la Recherche Scientifique
Sorbonne Université
Paris
France
Shuyang Xiang
Mathematics Division
Center for Computational Engineering Science
Rheinisch-Westfälische Technische Hochschule Aachen
Aachen
Germany