Four Lectures on Numerical Relativity

  • Carles Bona


The first lecture is devoted to the causal structure of Einstein’s evo lution equations. They are written as a first-order system of balance laws, which is shown to be hyperbolic when the time coordinate is chosen in an invariant algebraic way (maximal slicing is recovered as a limiting case). The second lecture deals with first-order flux-conservative systems. The propagation of characteristic fields in an inhomogeneous background is also considered, with a view on relativity applications.

In the third lecture, explicit finite-difference numerical methods are reviewed, with an accent on flux-conservative second-order methods. Stability conditions are derived in each case. Finally, in the last lecture, total-variation-diminishing (TVD) methods are considered. The case of an inhomogeneous characteristic speed is illustrated with the evolution of a spherically symmetric (1D) black hole.


Black Hole Causal Structure Contact Discontinuity Advection Equation Characteristic Speed 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Lichnerowicz, A. (1944): L’intégration des équations de la gravitation relativiste et le problème des N corps. J. Math. Pures Appl. 23, 37–63MathSciNetMATHGoogle Scholar
  2. 2.
    Choquet-Bruhat, Y. (1962): The Cauchy problem. In Witten, L. (ed.): Gravitation: An Introduction to Current Research ,pp. 130–168. Wiley, New YorkGoogle Scholar
  3. 3.
    Arnowitt, R., Deser, S., Misner, C.W. (1962): The dynamics of general relativity. In Witten, L. (ed.): Gravitation: An Introduction to Current Research ,pp. 227–265. Wiley, New YorkGoogle Scholar
  4. 4.
    Bona, C, Massó, J., Seidel, E., Stela, J. (1995): New formalisms for numerical relativity. Phys. Rev. Lett. 75, 600–603ADSCrossRefGoogle Scholar
  5. 5.
    York Jr., J.W. (1972): Role of conformal three-geometry in the dynamics of gravitation. Phys. Rev. Lett. 28, 1082–1085ADSCrossRefGoogle Scholar
  6. 6.
    Choquet-Bruhat, Y., Ruggeri, T. (1983): Hyperbolicity of 3 + 1 Einstein equations. Comm. Math. Phys. 89, 269–275MathSciNetADSMATHCrossRefGoogle Scholar
  7. 7.
    Bona, C, Massó, J. (1988): Harmonic synchronisations of spacetime. Phys. Rev. D38, 2419–2422ADSGoogle Scholar
  8. 8.
    Bernstein, D. (1993): A numerical study of the black hole plus Brill wave space-time. PhD thesis, University of Illinois Urbana-Champaign.Google Scholar
  9. 9.
    Bona, C., Massó, J. (1992): Hyperbolic evlution system for numerical relativity. Phys. Rev. Lett. 68, 1097–1099MathSciNetADSMATHCrossRefGoogle Scholar
  10. 10.
    Li Ta-tsien (1994): Global classical solutions for quasilinear hyperbolic systems. Wiley, ChichesterMATHGoogle Scholar
  11. 11.
    LeVeque, R.J. (1992): Numerical methods for conservation laws. Birkhäuser, BaselMATHCrossRefGoogle Scholar
  12. 12.
    Richtmeyer, R.D., Morton, K.W. (1967): Difference methods for initial value problems. (2nd edition). Wiley-Interscience, New YorkGoogle Scholar
  13. 13.
    Press, W.H., Flannery, B.P., Teukolsky, S.A., Vetterling, W.T. (1992): Numerical recipes (2nd edition). Cambridge University Press, CambridgeGoogle Scholar
  14. 14.
    Sweby, P.K. (1984): High resolution schemes using flux limiters for hyperbolic conservation laws. SIAM J. Num. Anal. 5, 995–1011MathSciNetCrossRefGoogle Scholar
  15. 15.
    Harten, A., Hymann, J.M. (1983): Self adjusting grid methods for one-dimensional hyperbolic conservation laws. J. Comp. Phys. 50, 235–269ADSMATHCrossRefGoogle Scholar
  16. 16.
    Bernstein, D., Hobill, D., Smarr, L. (1989): Black hole spacetimes: Testing numerical relativity. In Evans, C, Finn, L., Hobill, D. (eds.): Frontiers in Numerical Relativity ,pp. 57–73. Cambridge University Press, CambridgeGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1996

Authors and Affiliations

  • Carles Bona
    • 1
  1. 1.Departament de FisicaUniversitat de les Illes BalearsSpain

Personalised recommendations