Initial Data for Numerical Relativity: the Use of Spectral Methods

The determination of physical initial data is an important task in numerical relativity. In this direction we have applied the Galerkin and collocation methods to solve the Hamiltonian constraints resulting from the Cauchy formulation in the cases of spacetimes containing black holes as described by Ref. 1. We have shown that a considerable improvement in the accuracy is obtained if the basis functions are chosen such that the boundary conditions are satisfied. We have also introduced a new approach to solve numerically the constraint equations which consists in transforming them into parabolic equations after introducing fictitious diffusion terms. As a consequence, the application of Galerkin or collocation methods produces a dynamical system whose stationary solution corresponds to the initial data.