General Relativity and Gravitation

, Volume 40, Issue 1, pp 159–182 | Cite as

Regularization of spherical and axisymmetric evolution codes in numerical relativity

  • Milton Ruiz
  • Miguel Alcubierre
  • Darío Núñez
Research Article


Several interesting astrophysical phenomena are symmetric with respect to the rotation axis, like the head-on collision of compact bodies, the collapse and/or accretion of fields with a large variety of geometries, or some forms of gravitational waves. Most current numerical relativity codes, however, cannot take advantage of these symmetries due to the fact that singularities in the adapted coordinates, either at the origin or at the axis of symmetry, rapidly cause the simulation to crash. Because of this regularity problem it has become common practice to use full-blown Cartesian three-dimensional codes to simulate axi-symmetric systems. In this work we follow a recent idea of Rinne and Stewart and present a simple procedure to regularize the equations both in spherical and axi-symmetric spaces. We explicitly show the regularity of the evolution equations, describe the corresponding numerical code, and present several examples clearly showing the regularity of our evolutions.


Spherical and axial evolutions Regularization Numerical relativity 


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  1. 1.
    Abrahams A.M. and Evans C.R. (1993). Critical behavior and scaling in vacuum axisymmetric gravitational collapse. Phys. Rev. Lett. 70: 2980–2983 CrossRefADSGoogle Scholar
  2. 2.
    Alcubierre M., Brandt S., Brügmann B., Holz D., Seidel E., Takahashi R. and Thornburg J. (2001). Symmetry without symmetry: numerical simulation of axisymmetric systems using cartesian grids. Int. J. Mod. Phys. D 10: 273–289, Gr-qc/9908012 CrossRefADSGoogle Scholar
  3. 3.
    Alcubierre M., Corichi A., González J., Nuñez D., Reimann B. and Salgado M. (2005). Generalized harmonic spatial coordinates and hyperbolic shift conditions. Phys. Rev. D 72: 124–018, Gr-qc/0507007 Google Scholar
  4. 4.
    Alcubierre M. and González J. (2005). Regularization of spherically symmetric evolution codes in numerical relativity. Comput. Phys. Commun. 167: 76, Gr-qc/0401113 CrossRefGoogle Scholar
  5. 5.
    Baker, J.G., Centrella, J., Choi, D.I., Koppitz, M., Meter, J.: Gravitational wave extraction from an inspiraling configuration of merging black holes. Phys. Rev. Lett. 96, 111,102 (2006)Google Scholar
  6. 6.
    Bardeen J. and Piran T. (1983). General relativistic axisymmetric rotating systems: coordinates and equations. Phys. Rep. 196: 205 CrossRefADSMathSciNetGoogle Scholar
  7. 7.
    Bona, C., Ledvinka, T., Palenzuela, C.: General-covariant evolution formalism for numerical relativity. Phys. Rev. D67, 104,005 (2003)Google Scholar
  8. 8.
    Bona, C., Ledvinka, T., Palenzuela, C.: A symmetry-breaking mechanism for the z4 general-covariant evolution system. Phys. Rev. D69, 064,036 (2004)Google Scholar
  9. 9.
    Bona C., Massó J., Seidel E. and Stela J. (1995). New formalism for numerical relativity. Phys. Rev. Lett. 75: 600–603, Gr-qc/9412071 CrossRefADSGoogle Scholar
  10. 10.
    Campanelli, M., Lousto, C.O., Marronetti, P., Zlochower, Y.: Accurate evolutions of orbiting black-hole binaries without excision. Phys. Rev. Lett. 96, 111,101 (2006)Google Scholar
  11. 11.
    Choptuik M.W., Hirschmann E.W., Liebling S.L. and Pretorius F. (2003). An axisymmetric gravitational collapse code. Class. Quantum Gravity 20: 1857–1878 MATHCrossRefADSMathSciNetGoogle Scholar
  12. 12.
    Eppley K. (1977). Evolution of time-symmetric gravitational waves: Initial data and apparent horizons. Phys. Rev. D 16: 1609 CrossRefADSMathSciNetGoogle Scholar
  13. 13.
    Evans C.R. (1986). An approach for calculating axisymmetric gravitational collapse. In: Centrella, J.M. (eds) Dynamical spacetimes and numerical relativity, pp 3–39. Cambridge University Press, London Google Scholar
  14. 14.
    Garfinkle, D., Duncan, G.C.: Numerical evolution of brill waves. Phys. Rev. D63, 044,011 (2001)Google Scholar
  15. 15.
    Gustafsson B., Kreiss H. and Oliger J. (1995). Time Dependent Problems and Difference Methods. Wiley, New York MATHGoogle Scholar
  16. 16.
    Holz, D., Miller, W., Wakano, M., Wheeler, J.: Directions in general relativity. In: Hu, B., Jacobson,~T. (eds.) Proceedings of the 1993 international symposium, maryland; papers in honor of di eter brill. Directions in General Relativity: Proceedings of the 1993 International Symposium, Maryland; Papers in honor of Di eter Brill. Cambridge University Press, Cambridge (1993)Google Scholar
  17. 17.
    Nagy, G., Ortiz, O.E., Reula, O.A.: Strongly hyperbolic second order Einstein’s evolution equations. Phys. Rev. D70, 044,012 (2004)Google Scholar
  18. 18.
    Pretorius, F.: Evolution of binary black hole spacetimes. Phys. Rev. Lett. 95, 121,101 (2005)Google Scholar
  19. 19.
    Rinne, O.: Axisymmetric numerical relativity. PhD thesis, University of Cambridge (2005)Google Scholar
  20. 20.
    Rinne O. and Stewart J.M. (2005). A strongly hyperbolic and regular reduction of Einstein’s equations for axisymmetric spacetimes. Class. Quantum Gravity 22: 1143–1166 MATHCrossRefADSMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  • Milton Ruiz
    • 1
  • Miguel Alcubierre
    • 1
  • Darío Núñez
    • 1
  1. 1.Instituto de Ciencias NuclearesUniversidad Nacional Autónoma de MéxicoMexico D.F.Mexico

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