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

Published online by Cambridge University Press:  03 June 2015

Philippe G. Le Floch  and
Hasan Makhlof
Philippe G. Le Floch*
Affiliation:
Laboratoire Jacques-Louis Lions & Centre National de la Recherche Scientifique, Université Pierre et Marie Curie (Paris 6), 4 Place Jussieu, 75252 Paris, France
Hasan Makhlof*
Affiliation:
Laboratoire Jacques-Louis Lions & Centre National de la Recherche Scientifique, Université Pierre et Marie Curie (Paris 6), 4 Place Jussieu, 75252 Paris, France
*
Corresponding author.Email:contact@philippelefloch.org
Email:makhlof@ann.jussieu.fr
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Abstract

We consider the relativistic Euler equations governing spherically symmetric, perfect fluid flows on the outer domain of communication of Schwarzschild space-time, 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.

Keywords

76L0535L65Compressible fluidEuler systemSchwarzschild spacetimegeometry-preservingfinite volume methodsteady state
Type
Research Article
Information
Communications in Computational Physics , Volume 15 , Issue 3 , March 2014 , pp. 827 - 852
Copyright
Copyright © Global Science Press Limited 2014

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References

[1]Alcubierre, M., Introduction to 3+1 numerical relativity, Inter. Series Mono. Physics, Vol. 140, Oxford Univ. Press, 2008.Google Scholar
[2]Amorim, P., Ben-Artzi, M. and LeFloch, P.G., Hyperbolic conservation laws on manifolds: Total variation estimates and finite volume method, Meth. Appl. Analysis 12 (2005), 291–324.Google Scholar
[3]Amorim, P., LeFloch, P.G., and Okutmustur, B., Finite volume schemes on Lorentzian manifolds, Comm. Math. Sc. 6 (2008), 1059–1086.Google Scholar
[4]Ben-Artzi, M., Falcovitz, J., and LeFloch, P.G., Hyperbolic conservation laws on the sphere. A geometry-compatible finite volume scheme, J. Comput. Phys. 228 (2009), 5650–5668.Google Scholar
[5]Ben-Artzi, M. and LeFloch, P.G., The well posedness theory for geometry compatible hyperbolic conservation laws on manifolds, Ann. Inst. H. Poincaré Anal. Nonlinéaire 24 (2007), 989–1008.Google Scholar
[6]Bouchut, F., Nonlinear stability of finite volume methods for hyperbolic conservation laws and well balanced schemes for sources, Birkhäuser, Verlag, Bäsel, 2004.Google Scholar
[7]Banyuls, F., Font, J.A., Ibánez, J.M., Martí, J.M., and Miralles, J.A., Numerical 3+1 general relativistic hydrodynamics: a local characteristic approach, Astrophys. J. 476 (1997), 221–231.Google Scholar
[8]Dafermos, C.M., Hyperbolic conservation laws in continuum physics, Grundlehren der Mathematischen Wissenschaften, Vol. 325, Springer Verlag, Berlin, 2000.Google Scholar
[9]Dubois, F. and LeFloch, P.G., Boundary conditions for nonlinear hyperbolic systems of conservation laws, J. Diff. Equa. 71 (1988), 93–122.Google Scholar
[10]Dubois, F. and LeFloch, P.G., Boundary conditions for nonlinear hyperbolic systems of conservation laws, Proc. Inter. Conf. on Hyperbolic problems, Aachen (Germany), March 1988, Notes on Numer. Fluid Mech., Vol. 24, Vieweg, 1989, pp. 96–106.Google Scholar
[11]Font, J.A., Numerical hydrodynamics and magnetohydrodynamics in general relativity, Living Rev. Relativity 11 (2008), 7.Google Scholar
[12]Hawley, J.F., Smarr, L.L., and Wilson, J.R., A numerical study of nonspherical black hole accretion. I. Equations and test problems, Astrophys. J. 277 (1984), 296–311.Google Scholar
[13]LeFloch, P.G., Hyperbolic systems of conservation laws, Lectures in Mathematics, ETH Zürich, Birkhäuser, 2002.Google Scholar
[14]LeFloch, P.G., Makhlof, H., and Okutmustur, B., Relativistic Burgers equations on curved spacetimes. Derivation and finite volume approximation, SIAM J. Numer. Anal. 50 (2012), 2136–2158.CrossRefGoogle Scholar
[15]LeFloch, P.G. and Okutmustur, B., Hyperbolic conservation laws on spacetimes. A finite volume scheme based on differential forms, Far East J. Math. Sci. 31 (2008), 49–83.Google Scholar
[16]LeVeque, R.J., Balancing source terms and flux gradients in high-resolution Godunov methods: The quasi-steady wave-propagation algorithm, J. Comput. Phys (1998), 346–365.Google Scholar
[17]LeVeque, R.J., Finite volume methods for hyperbolic problems, Cambridge University Press, Cambridge, 2002.Google Scholar
[18]Martí, J.M. and Müller, E., Numerical hydrodynamics in special relativity”, Living Rev. Relativity 6 (2003), 7.Google Scholar
[19]Nessyahu, H. and Tadmor, E., Non-oscillatory central differencing for hyperbolic conservation laws, J. Comput. Phys. 87 (1990), 408–463.Google Scholar
[20]Novak, J. and Ibáńez, J. M., Gravitational waves from the collapse and bounce of a stellar core in tensor-scalar gravity, Astrophys. J. 533 (2000), 392–405.CrossRefGoogle Scholar
[21]Papadopoulos, P. and Font, J.A., Relativistic hydrodynamics round black holes and horizon adapted coordinate systems, Phys. Rev. D. 58 (1999), 024005.Google Scholar
[22]Papadopoulos, P. and Font, J.A., Relativistic hydrodynamics on spacelike and null surfaces: formalism and computations of spherically symmetric spacetimes, Phys. Rev. D 61 (2000), 024015.Google Scholar
[23]Puppo, G. and Russo, G., Staggered finite difference schemes for conservation laws, J. Sci. Comput. 27 (2006), 403–418.CrossRefGoogle Scholar
[24]Radice, D. and Rezzolla, L., Discontinuous Galerkin methods for general-relativistic hydrodynamics: Formulation and application to spherically symmetric spacetimes, Phys. Rev. D 84 (2011), 024010.Google Scholar
[25]Russo, G., Central schemes for conservation laws with application to shallow water equations, Rionero, S., Romano, G. Ed., Trends and Applications of Mathematics to Mechanics: STAMM 2002, Springer Verlag, Italy, 2005, pp. 225–246.CrossRefGoogle Scholar
[26]Russo, G., High-order shock-capturing schemes for balance laws, in “Numerical solutions of partial differential equations”, Adv. Courses Math. CRM Barcelona, Birkhäuser, Basel, 2009, pp. 59–147.Google Scholar
[27]Wald, R.M., General Relativity, University of Chicago Press, 1984.Google Scholar