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Finite differencing second order systems describing black hole spacetimes

Finite differencing second order systems describing black hole spacetimes
Finite differencing second order systems describing black hole spacetimes
Keeping Einstein's equations in second order form can be appealing for computational efficiency, because of the reduced number of variables and constraints. Stability issues emerge, however, which are not present in first order formulations. We show that a standard discretization of the second order "shifted'' wave equation leads to an unstable semi-discrete scheme if the shift parameter is too large. This implies that discretizations obtained using integrators such as Runge-Kutta, Crank-Nicholson, leap-frog are unstable for any fixed value of the Courant factor. We argue that this situation arises in numerical relativity, particularly in simulations of spacetimes containing black holes, and discuss several ways of circumventing this problem. We find that the first order reduction in time based on "ADM'' type variables is very effective.
1550-7998
1-4
Calabrese, G.
ffefb9b7-fca3-41ac-89ea-1a7a85dbd7d9
Calabrese, G.
ffefb9b7-fca3-41ac-89ea-1a7a85dbd7d9

Calabrese, G. (2005) Finite differencing second order systems describing black hole spacetimes. Physical Review D, 71 (2), 1-4. (doi:10.1103/PhysRevD.71.027501).

Record type: Article

Abstract

Keeping Einstein's equations in second order form can be appealing for computational efficiency, because of the reduced number of variables and constraints. Stability issues emerge, however, which are not present in first order formulations. We show that a standard discretization of the second order "shifted'' wave equation leads to an unstable semi-discrete scheme if the shift parameter is too large. This implies that discretizations obtained using integrators such as Runge-Kutta, Crank-Nicholson, leap-frog are unstable for any fixed value of the Courant factor. We argue that this situation arises in numerical relativity, particularly in simulations of spacetimes containing black holes, and discuss several ways of circumventing this problem. We find that the first order reduction in time based on "ADM'' type variables is very effective.

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Published date: 2005

Identifiers

Local EPrints ID: 40851
URI: http://eprints.soton.ac.uk/id/eprint/40851
ISSN: 1550-7998
PURE UUID: 8cfb6609-2c3a-4499-8b78-d24e8556709a

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Date deposited: 11 Jul 2006
Last modified: 15 Jul 2019 18:58

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