A spectral approach to numerical simulations of the ADM equations
We present a numerical study of the Einstein equations, according to the Arnowitt-Deser-Misner (ADM) formalism, in order to simulate the dynamics of gravitational fields. We took in consideration the original $3+1$ decomposition of the ADM equations, in vacuum conditions, in simplified geometries. The numerical code is based on spectral methods, making use of filtering (de-aliasing) techniques. The algorithm has been stabilized via an adaptive time-refinement, based on a procedure that checks self-consistently the regularity of the solutions. The accuracy of our numerical model has been validated through a series of standard tests. Finally, we present also a new kind of initial data that can be used for testing numerical codes.