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Detecting ill posed boundary conditions in general relativity

Detecting ill posed boundary conditions in general relativity
Detecting ill posed boundary conditions in general relativity
A persistent challenge in numerical relativity is the correct specification of boundary conditions. In this work we consider a many parameter family of symmetric hyperbolic initial-boundary value formulations for the linearized Einstein equations and analyze its well posedness using the Laplace-Fourier technique. By using this technique ill posed modes can be detected and thus a necessary condition for well posedness is provided. We focus on the following types of boundary conditions: i) Boundary conditions that have been shown to preserve the constraints, ii) boundary conditions that result from setting the ingoing constraint characteristic fields to zero and iii) boundary conditions that result from considering the projection of Einstein's equations along the normal to the boundary surface. While we show that in case i) there are no ill posed modes, our analysis reveals that, unless the parameters in the formulation are chosen with care, there exist ill posed constraint violating modes in the remaining cases.
0022-2488
3888-3899
Calabrese, Gioel
b6d18b27-64cd-426f-b86e-1b3a848f03ed
Sarbach, Olivier
9b1b946c-5ff3-4eeb-b685-915c662edff8
Calabrese, Gioel
b6d18b27-64cd-426f-b86e-1b3a848f03ed
Sarbach, Olivier
9b1b946c-5ff3-4eeb-b685-915c662edff8

Calabrese, Gioel and Sarbach, Olivier (2003) Detecting ill posed boundary conditions in general relativity. Journal of Mathematical Physics, 44 (9), 3888-3899. (doi:10.1063/1.1599056).

Record type: Article

Abstract

A persistent challenge in numerical relativity is the correct specification of boundary conditions. In this work we consider a many parameter family of symmetric hyperbolic initial-boundary value formulations for the linearized Einstein equations and analyze its well posedness using the Laplace-Fourier technique. By using this technique ill posed modes can be detected and thus a necessary condition for well posedness is provided. We focus on the following types of boundary conditions: i) Boundary conditions that have been shown to preserve the constraints, ii) boundary conditions that result from setting the ingoing constraint characteristic fields to zero and iii) boundary conditions that result from considering the projection of Einstein's equations along the normal to the boundary surface. While we show that in case i) there are no ill posed modes, our analysis reveals that, unless the parameters in the formulation are chosen with care, there exist ill posed constraint violating modes in the remaining cases.

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Published date: 2003

Identifiers

Local EPrints ID: 40856
URI: http://eprints.soton.ac.uk/id/eprint/40856
ISSN: 0022-2488
PURE UUID: 058c8d4f-e7cc-429b-a6f0-1a64911fadbf

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Date deposited: 11 Jul 2006
Last modified: 07 Jan 2022 22:26

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Author: Gioel Calabrese
Author: Olivier Sarbach

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