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

Abstract

We present a numerical scheme for determining hyperboloidal initial data sets for the conformal field equations by using pseudo-spectral methods. This problem is to split into two parts. The first 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 fields 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 coefficient 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 field equations.