Consistency of finite-difference solutions of Einstein’s equations

M. W. Choptuik

doi:10.1103/PhysRevD.44.3124

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Consistency of finite-difference solutions of Einstein’s equations

M. W. Choptuik

Canadian Institute for Theoretical Astrophysics, 60 St. George Street, Toronto, Ontario, Canada M5S 1A1

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Phys. Rev. D 44, 3124 – Published 15 November, 1991

DOI: https://doi.org/10.1103/PhysRevD.44.3124

Abstract

In the past, arguments have been advanced suggesting that certain finite-difference solutions of the 3+1 form of Einstein’s equations suffer from a fundamental inconsistency. Specifically, it has been claimed that freely evolved solutions, where the constraint equations are not explicitly imposed after the initial time, will generally satisfy discrete versions of the constraints to a lower order in the basic mesh spacing h than the truncation order of the discretized evolution equations. This issue is reexamined here, and using the key observation, originally due to Richardson, that a numerical differentiation need not produce an O( $h^{p - 1}$ ) quantity from an O( $h^{p}$ ) one, it is argued that there should be no such inconsistency for convergent difference schemes. Numerical results from a study of spherically symmetric solutions of a massless scalar field minimally coupled to the gravitational field are presented in support of this claim. These results show that the expected convergence of various residual quantities can be achieved in practice.

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[c2] Gravitational Radiation, edited by N. Deruelle and T. Piran (North-Holland, Amsterdam, 1983).

[c3] R. Arnowitt, S. Deser, and C. W. Misner, in Gravitation—An Introduction to Current Research, edited by L. Witten (Wiley, New York, 1962).

[c4] J. Centrella, Astrophys. J. 241, 875 (1980).

[c5] J. Centrella, in The Origin and Evolution of Galaxies, edited by B. J. T. Jones and J. E. Jones (D. Reidel, Boston, 1983), pp. 41–64.

[c6] T. Nakamura, Prog. Theor. Phys. 65, 1876 (1981).

[c7] T. Nakamura, Suppl. Prog. Theor. Phys. 70, 202 (1981).

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[c9] T. Nakamura, K. -I. Maeda, S. Miyama, and M. Sasaki, in Proceedings of the Second Marcel Grossmann Meeting on Recent Developments in General Relativity, Trieste, Italy, 1979, edited by R. Ruffini (North-Holland, Amsterdam, 1981), pp. 675–677.

[c10] T. Nakamura and H. Sato, Prog. Theor. Phys. 67, 1396 (1982).

[c11] T. Piran, J. Comput. Phys. 35, 254 (1980).

[c12] K. R. Eppley, Ph.D. thesis, Princeton University, 1975.

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[c16] S. L. Shapiro and S. A. Teukolsky, Astrophys. J. 298, 35 (1986).

[c17] T. Piran, in Gravitational Radiation [2], pp. 203–256.

[c18] J. M. Stewart, in Classical General Relativity, edited by W. B. Bonnor, J. N. Isham, and M. A. H. MacCallum (Cambridge University Press, Cambridge, England, 1984).

[c19] L. F. Richardson, Philos. Trans. R. Soc. 210, 307 (1910).

[c20] R. D. Richtmyer and K. W. Morton, Difference Methods for Initial-Value Problems, 2nd Ed. (Wiley, New York, 1957).

[c21] H.-O. Kreiss and J. Oliger, Methods for the Approximate Solution of Time Dependent Problems, Garp Publications Series No. 10 (Unipub, Lanham, MD, 1973).

[c22] G. I. Marchuk, Methods of Numerical Mathematics (Springer-Verlag, New York, 1982).

[c23] M. Choptuik, Ph.D. thesis, University of British Columbia, 1986.

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[c25] D. S. Goldwirth and T. Piran, Phys. Rev. D 36, 3575 (1987).

[c26] C. W. Misner, K. S. Thorne, and J. A. Wheeler, Gravitation (Freeman, San Francisco, 1973).

[c27] B. K. Berger, D. M. Chitre, V. E. Moncrief and Y. Nutku, Phys. Rev. D 5, 2467 (1972).

[c28] W. G. Unruh, Phys. Rev. D 14, 870 (1976).

[c29] T. Piran, in Proceedings of the Second Marcel Grossmann Meeting on Recent Developments in General Relativity [9].

[c30] J. R. Wilson, in Sources of Gravitational Radiation [1].

[c31] J. M. Bardeen and T. Piran, Phys. Rep. 96, 205 (1983).

[c32] R. F. Stark and T. Piran, Phys. Rev. Lett. 55, 891 (1985).

[c33] L. Smarr, Ann. N.Y. Acad. Sci. 302, 569 (1977).

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