Introduction to numerical relativity through examples

F.S. Guzmán

Instituto de Física y Matemáticas, Universidad Michoacana de San Nicolás de Hidalgo, Edificio C–3, Cd. Universitaria, 58040 Morelia, Michoacán, México, e–mail: guzman@ifm.umich.mx

Recibido el 1 de mayo de 2006
Aceptado el 1 de noviembre de 2006

Abstract

In these notes some examples of how to apply finite differencing to the solution of partial differential equations are presented and analyzed. The aim of this manuscript is to offer the reader a first step toward the numerical solution of sufficiently complicated and interesting problems within general relativity. The topics include the solution of the wave equation in one spatial dimension and the solution of real and complex self–gravitating scalar fields with spherical symmetry.

Keywords: Numerical methods; numerical relativity; self–gravitating systems.

Resumen

En estas notas se presentan y analizan algunos ejemplos de aplicación del método de diferencias finitas a la solución de ecuaciones diferenciales parciales. La motivación de este manuscrito es ofrecer al lector un primer paso en la solución numérica de problemas suficientemente complicados e interesantes en relatividad general, o sea, dentro del area llamada relatividad numérica. Los temas incluyen la solución de la ecuación de onda en una dimensión espacial y la solución de campos escalares auto–gravitantes, tanto reales como complejos con simetría esférica.

Descriptores: Métodos numéricos; relatividad numérica; sistemas auto–gravitantes.

PACS: 04.25.Dm

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Acknowledgments

This work is partly supported by projects CIC–UMSNH–4.9 and PROMEP–UMICH–PTC–121 and PROMEP–UMSNH–CA–22.

References

1. M. Campanelli, C.O. Lousto, P. Marronetti, and Y. Zlochower, Phys. Rev. Lett. 96 (2006) 111101.        [ Links ]

2. J.G. Baker, J. Centrella, Dae–Il Choi, M. Koppitz, and J. van Meter, Phys. Rev. Lett. 96 (2006) 111102.        [ Links ]

3. M.W. Choptuik et al., Phys. Rev. D 68 (2003) 044001.        [ Links ]

4. O. Sarbach and L. Lehner, Phys. Rev. D 71 (2005) 026002.        [ Links ]

5. J. Thornburg, gr–qc/9906022        [ Links ]

6. I. Hawke, S. Husa, and B. Szilagyi, http://numrel.aei.mpg.de/Education/Tutorials/School04/tov.ps        [ Links ]

7. C. Bona and C. Palenzuela–Luque, Elements of Numerical Relativity, Lect. Notes Phys. 673 (Springer, Berlin Heidelberg, 2005).        [ Links ]

8. P. Diener and F. S. Guzmán, http://numrel.aei.mpg.de/Education/Tutorials/trackI/tutorial1.pdf        [ Links ]

9. B. Gustafsson, H–O. Kreiss, and J. Oliger, Time Dependent Problems and Difference Methods (Wiley–Interscience, 1996).        [ Links ]

10. R.J. LeVeque, in Numerical methods for conservation laws (Birkhauser, Basel 1992).        [ Links ]

11. W.H. Press, S.A. Teukolsky, W.T. Watterling, and B.P Flannery, Numerical Recipes in Fortran (Cambridge University Press, 1992).        [ Links ]

12. M. Alcubierre et al., Class. Quantum Grav. 20 (2003) 2883.        [ Links ]

13. M. Alcubierre et al., Class. Quantum Grav. 19 (2002) 5017.        [ Links ]

14. Gregory B. Cook, Living Reviews in Relativity. 2005–5. http://www.livingreviews.org/Articles/Volume3/2000-5cook        [ Links ]

15. M.W. Choptuik, Phys. Rev. Lett. 70 (1993) 9.        [ Links ]

16. E. Seidel and W–M. Suen, Phys. Rev. Lett. 66 (1991) 1659.        [ Links ]

17. R. Ruffini and S. Bonazolla, Phys. Rev. 187 (1969) 1767.        [ Links ]

18. E. Seidel and W–M. Suen, Phys. Rev. D 42 (1990) 384.        [ Links ]

19. J. Balakrishna, E. Seidel, and W–M. Suen, Phys. Rev. D 58 (1998) 104004.        [ Links ]

20. M. Gleiser, Phys. Rev. D 38 (1988) 2376.        [ Links ]

21. S.H. Hawley and M.W. Choptuik, Phys. Rev. D 62 (2000) 104024.        [ Links ]

22. F.S. Guzmán, Phys. Rev. D 70 (2004) 044033.        [ Links ]

23. F.E. Schunck and D.F. Torres, Int. J. Mod. Phys. D 9 (2000) 601.        [ Links ]

24. M. Colpi, S.L. Shapiro, and I. Wasserman, Phys. Rev. Lett. 57 (1986) 2485.        [ Links ]

25. F.S. Guzmán and L.A. Ureña–López, Ap J 645 (2006) 814; astro–ph/0603613.        [ Links ]

26. D.F. Torres, S. Capozziello, and G. Lambiase, Phys. Rev. D 62 (2000) 104012;         [ Links ] F.S Guzmán, Phys. Rev. D 73 (2006) 021501(R).        [ Links ]

27. D.F. Torres, Nucl. Phys. B 26 (2002) 377.        [ Links ]

28. Y–F. Tuan, R. Narayan, and M.J. Rees, Ap J 606 (2004) 1112.        [ Links ]

29. J. Balakrishna, R. Bondarescu, G. Daues, F.S. Guzmán, and E. Seidel, Class. Quantum Grav. 23 (2006) 2631.        [ Links ]

30. M. Alcubierre, J.A. González, and M. Salgado, Phys. Rev. D 70 (2004) 064016.        [ Links ]

31. J.A. Font, "Numerical Hydrodynamics in General Relativity", Living Rev. Relativity 6, (2003), http://www.livingreviews.org/1rr-2003-4        [ Links ]

32. Gregory B. Cook, "Initial Data for Numerical Relativity", Living Rev. Relativity 3, (2000), http://www.livingreviews.org/lrr-2000-5        [ Links ]

33. J.M. Martí and E. Muller, "Numerical Hydrodynamics in Special Relativity", Living Rev. Relativity 6, (2003), http://www.livingreviews.org/lrr-2003-7        [ Links ]

34. Peter Anninos, "Computational Cosmology: from the Early Universe to the Large Scale Structure", Living Rev. Relativity 1, (1998), http://www.livingreviews.org/lrr-1998-2        [ Links ]

35. L. Lehner, Class. Quant. Grav. 18 (2001) R25.        [ Links ]

36. M. Alcubierre; gr–qc/0412019.        [ Links ]