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Revisiting spherically symmetric relativistic hydrodynamics

 

F.S. Guzmán, F.D. Lora-Clavijo, and M.D. Morales

 

Instituto de Física y Matemáticas, Universidad Michoacana de San Nicolás de Hidalgo, Edificio C-3, Cd. Universitaria, 58040 Morelia, Michoacán, México.

 

Recibido el 19 de abril de 2012;
Aceptado el 25 de julio de 2012.

 

Abstract

In this paper we revise two classical examples of Relativistic Hydrodynamics in order to illustrate in detail the numerical methods commonly used in fluid dynamics, specifically those designed to deal with shocks, which are based on a finite volume approximation. The two cases we consider are the relativistic blast wave problem and the evolution of a Tolman-Oppenheimer-Volkoff star model, in spherical symmetry. In the first case we illustrate the implementation of relativistic Euler's equations on a fixed background space-time, whereas in the second case we also show how to couple the evolution of the fluid to the evolution of the space-time.

Keywords: Hydrodynamics astrophysical applications; numerical methods (mathematics); Einstein equation; general relativity.

 

PACS: 95.30.Lz; 02.60.-x; 04.20.-q

 

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Acknowledgments

This work is supported by grants CIC-UMSNH-4.9 and CONACyT 106466. FDLC and MDMA acknowledge support from CONACyT.

 

References

1. T. W. Baumgarte and S. L. Shapiro, Numerical relativity: solving Einstein's equations on the computer (Cambridge University Press 2010).         [ Links ]

2. B. F. Schutz, A first course in general relativity (Cambridge University Press, 1985).         [ Links ]

3. Banyouls et al., ApJ 476 (1997)221-231.         [ Links ]

4. J. A. Font, M. Miller, W-M. Suen, and M. Tobias, Phys. Rev. D 61 (2000) 044011.         [ Links ]

5. M. Alcubierre, Introduction to 3+1 Numerical Relativity (Oxford Science Publications, 2008).         [ Links ]

6. R. J. LeVeque, Numerical Methods for Conservation Laws (Lectures in Mathematics, ETH ZŸrich, second edition).         [ Links ]

7. http://www.ifm.umich.mx/guzman/Grupo/RMF/Maple_Leaf_evolution_equations_geometry.pdf        [ Links ]

8. S. T. Millmore and I. Hawke, Class. Quantum Grav. 27 (2010) 015007.         [ Links ]

9. D. W. Neilsen and M. W. Choptuik, Class. Quantum Grav. 17 (2000) 733-759.         [ Links ]

10. S. L. Shapiro and S. A. Teukolsky, Black Holes, White Dwarfs and Neutron Stars: The Physics of Compact Objects (Wiley -VCH, 1983).         [ Links ]

11. S. K. Godunov, Mat. Sb 47 (1959) 271.         [ Links ]

12. E. F. Toro, Riemann Solvers and Numerical Methods for Fluid Dynamics: A Practical Introduction. (Springer, erd edition).         [ Links ]

13. A. Harten, P. D. Lax & B. van Leer, SIAM Rev. 25 (1983) 35. B. Einfeldt, SIAM, J. Num. Anal. 25 (1988) 294.         [ Links ]

14. http://www.ifm.umich.mx/guzman/Grupo/RMF/Maple_Leaf_IVP-TOV.pdf.         [ Links ]

15. F. S. Guzmán, Rev. Mex. Fis. E 56 (2010) 51-68.         [ Links ]

16. J. Font et al., Phys. Rev. D 65 (2002) 08402.         [ Links ]

17. S. W. Hawking and G. F. R. Ellis, The large scale structure of space-time, (Cambridge monographs on mathematical physics, 1973).         [ Links ]

18. M. Yokosawa, Astrophys. Space Sci. 107 (1984) 109-123.         [ Links ]