In the first part of this work, we apply finite difference methods, specially mesh refinement techniques, in order to numerically evolve a single black hole, which is represented by the puncture initial data. We use standard second order finite differences, and the second order Iterated Crank-Nicholson integrator. We observe that, in order to obtain a second order accurate evolution we must impose second order accurate interface conditions at the refinement boundaries. We test our evolution with both the geodesic and the 1+log slicing conditions, and observe the expected results. We conclude that our mesh refinement technique generates convergent evolutions, and the puncture method behaves very well with it.

The second part of this work deals with a modification of the hybrid Lazarus'' method for wave extraction. This method is divided in three parts: an early evolution, a set of transformations to produce perturbations over a Kerr background from the numerical data, and Teukolsky evolution. By using our evolution code (with mesh refinement) and gauges (1+log, gamma-driver, shifting-shift), we deviate from the original Lazarus approach. We used an independent implementation of the Lazarus transformations, validating the original results, and of the Teukolsky equation. We obtained results similar to the original Lazarus, both on the waveforms as well as on the negative results at later times. For instance, strong pulses that contaminate some gauge transformations, which may be explained in part by the propagating gauge modes of the 1+log slicing. Increasing the accuracy of the initial black hole evolution we seem to obtain better final results for the Kerr test case. Because of the gauge problems, we develop an approximated embedding method which approximates location of the numerical slice into the Kerr spacetime. This method is much less sensitive to the gauge perturbations. Given the difficulties of the Lazarus procedure, we decide to use the Lazarus method as a wave extraction tool. Using this embedding technique we developed the spacelike'' wave extraction method. Our preliminary result is consistent with the numerical waveforms for at least three cycles. Although we see some differences, it is too early to claim physical reality on them.