Hostname: page-component-78c5997874-fbnjt Total loading time: 0 Render date: 2024-11-05T17:15:25.309Z Has data issue: false hasContentIssue false
  • English
  • Français

Numerical integration of the axisymmetric Robinson-Trautman equation by a spectral method

Published online by Cambridge University Press:  17 February 2009

D. A. Prager  and
A. W.-C. Lun
D. A. Prager
Affiliation:
Department of Mathematics, Monash University, Clayton, Victoria 3168, Australia.
A. W.-C. Lun
Affiliation:
Department of Mathematics, Monash University, Clayton, Victoria 3168, Australia.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We have adapted the Spectral Transform Method, a technique commonly used in non-linear meteorological problems, to the numerical integration of the Robinson-Trautman equation. This approach eliminates difficulties due to the S2 × R+ topology of the equation. The method is highly accurate for smooth data and is numerically robust. Under spectral decomposition the long-time equilibrium state takes a particularly simple form: all nonlinear (l ≥ 2) modes tend to zero. We discuss the interaction and eventual decay of these higher order modes, as well as the evolution of the Bondi mass and other derived quantities. A qualitative comparison between the Spectral Transform Method and two finite difference schemes is given.

Type
Research Article
Information
The ANZIAM Journal , Volume 41 , Issue 2 , October 1999 , pp. 271 - 280
Copyright
Copyright © Australian Mathematical Society 1999

References

[1]Browning, G. L., Hack, J. J. and Swarztrauber, P. N., “A comparison of three numerical methods for solving differential equations on the sphere”, Mon. Wea. Rev. 117 (1989) 1058–1075.2.0.CO;2>CrossRefGoogle Scholar
[2]Chruściel, P. T., “Semi-global existence and convergence of solutions of the Robinson-Trautman (2-dimensional Calabi) equation”, Commun. Math. Phys. 137 (1991) 289–313.CrossRefGoogle Scholar
[3]Lukács, B., Perjes, Z., Porter, J. and Sebastyen, A., “Lyapunov functional approach to radiative metrics”, Gen. Relativ. Gravit. 16 (1984) 691–701.CrossRefGoogle Scholar
[4]Prager, D. A., “Discretization of the two-sphere”, Honours Thesis, Monash University, Melbourne, 1992.Google Scholar
[5]Robinson, I. and Trautman, A., “Some spherical gravitational waves in general relativity”, Proc. Royal Soc. Load. A262 (1962) 463–473.Google Scholar
[6]Singleton, D., “Robinson-Trautman solution of Einstein's equations”, Ph. D. Thesis, Monash University, Melbourne, 1990.Google Scholar