Summary
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.
Keywords
Black Hole Causal Structure Contact Discontinuity Advection Equation Characteristic SpeedPreview
Unable to display preview. Download preview PDF.
References
- 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.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.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.Bona, C, Massó, J., Seidel, E., Stela, J. (1995): New formalisms for numerical relativity. Phys. Rev. Lett. 75, 600–603ADSCrossRefGoogle Scholar
- 5.York Jr., J.W. (1972): Role of conformal three-geometry in the dynamics of gravitation. Phys. Rev. Lett. 28, 1082–1085ADSCrossRefGoogle Scholar
- 6.Choquet-Bruhat, Y., Ruggeri, T. (1983): Hyperbolicity of 3 + 1 Einstein equations. Comm. Math. Phys. 89, 269–275MathSciNetADSMATHCrossRefGoogle Scholar
- 7.Bona, C, Massó, J. (1988): Harmonic synchronisations of spacetime. Phys. Rev. D38, 2419–2422ADSGoogle Scholar
- 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.Bona, C., Massó, J. (1992): Hyperbolic evlution system for numerical relativity. Phys. Rev. Lett. 68, 1097–1099MathSciNetADSMATHCrossRefGoogle Scholar
- 10.Li Ta-tsien (1994): Global classical solutions for quasilinear hyperbolic systems. Wiley, ChichesterMATHGoogle Scholar
- 11.LeVeque, R.J. (1992): Numerical methods for conservation laws. Birkhäuser, BaselMATHCrossRefGoogle Scholar
- 12.Richtmeyer, R.D., Morton, K.W. (1967): Difference methods for initial value problems. (2nd edition). Wiley-Interscience, New YorkGoogle Scholar
- 13.Press, W.H., Flannery, B.P., Teukolsky, S.A., Vetterling, W.T. (1992): Numerical recipes (2nd edition). Cambridge University Press, CambridgeGoogle Scholar
- 14.Sweby, P.K. (1984): High resolution schemes using flux limiters for hyperbolic conservation laws. SIAM J. Num. Anal. 5, 995–1011MathSciNetCrossRefGoogle Scholar
- 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.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