Commun. Comput. Phys., 15 (2014), pp. 827-852.

A Geometry-Preserving Finite Volume Method for Compressible Fluids on Schwarzschild Spacetime

Philippe G. LeFloch 1*, Hasan Makhlof 1

1 Laboratoire Jacques-Louis Lions & Centre National de la Recherche Scientifique, Universite Pierre et Marie Curie (Paris 6), 4 Place Jussieu, 75252 Paris, France.

Received 29 December 2012; Accepted (in revised version) 16 September 2013
Available online 4 December 2013


We consider the relativistic Euler equations governing spherically symmetric, perfect fluid flows on the outer domain of communication of Schwarzschild spacetime, and we introduce a version of the finite volume method which is formulated from the geometric formulation (and thus takes the geometry into account at the discretization level) and is well-balanced, in the sense that it preserves steady solutions to the Euler equations on the curved geometry under consideration. In order to formulate our method, we first derive a closed formula describing all steady and spherically symmetric solutions to the Euler equations posed on Schwarzschild spacetime. Second, we describe a geometry-preserving, finite volume method which is based from the family of steady solutions to the Euler system. Our scheme is second-order accurate and, as required, preserves the family of steady solutions at the discrete level. Numerical experiments are presented which demonstrate the efficiency and robustness of the proposed method even for solutions containing shock waves and nonlinear interacting wave patterns. As an application, we investigate the late-time asymptotics of perturbed steady solutions and demonstrate its convergence for late time toward another steady solution, taking the overall effect of the perturbation into account.

AMS subject classifications: 76L05, 35L65

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Key words: Compressible fluid, Euler system, Schwarzschild spacetime, geometry-preserving, finite volume method, steady state.

*Corresponding author.
Email: (P. G. LeFloch), (H. Makhlof)

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