Long term black hole evolution with the BSSN system by pseudospectral methods

We present long term evolutions of a single black hole of mass $M$ with the BSSN system using pseudospectral methods. For our simulations we use the SGRID code where the BSSN system is implemented in its standard second order in space form. Previously we found that such simulations are quite unstable. The main goal of this paper is to present two improvements which now allow us to evolve for longer times. The first improvement is related to the boundary conditions at the excised black hole interior. We now use a gauge condition that ensures that all modes are going into the black hole, so that no boundary conditions are needed at the excision surface. The second more significant improvement has to do with our particular numerical method and involves filters based on projecting the double Fourier expansions used for the angular dependence onto Spherical Harmonics. With these two improvements it is now easily possible to evolve for several thousand $M$. The only remaining limitation seems to be the radiative outer boundary conditions used here. Yet this problem can be ameliorated by pushing out the location of the outer boundary, which leads to even longer run-times.