Calculating initial data for the conformal Einstein equations by pseudo-spectral methods

✍ Scribed by Jörg Frauendiener

• Book ID: 104339031
• Publisher: Elsevier Science
• Year: 1999
• Tongue: English
• Weight: 283 KB
• Volume: 109
• Category: Article
• ISSN: 0377-0427
• DOI: 10.1016/s0377-0427(99)00168-5

✦ Synopsis

We present a numerical scheme for determining hyperboloidal initial data sets for the conformal ÿeld equations by using pseudo-spectral methods. This problem is to split into two parts. The ÿrst step is the determination of a suitable conformal factor which transforms from an initial data set in physical space-time to a hyperboloidal hypersurface in the ambient conformal manifold. This is achieved by solving the Yamabe equation, a nonlinear second-order equation. The second step is a division by the conformal factor of certain ÿelds which vanish on I, the zero set of the conformal factor. The challenge there is to numerically obtain a smooth quotient. Both parts are treated by pseudo-spectral methods. The nonlinear equation is solved iteratively while the division problem is treated by transforming the problem to the coe cient space, solving it there by the QR-factorisation of a suitable matrix, and then transforming back. These hyperboloidal initial data can be used to generate general relativistic space-times by evolution with the conformal ÿeld equations.