  We present a new pseudo-spectral code for the simulation of evolution systems
that are second order in space. We test this code by evolving a non-linear
scalar wave equation. These non-linear waves can be stably evolved using very
simple constant or radiative boundary conditions, which we show to be
well-posed in the scalar wave case. The main motivation for this work, however,
is to evolve black holes for the first time with the BSSN system by means of a
spectral method. We use our new code to simulate the evolution of a single
black hole using all applicable methods that are usually employed when the BSSN
system is used together with finite differencing methods. In particular, we use
black hole excision and test standard radiative and also constant outer
boundary conditions. Furthermore, we study different gauge choices such as
$1+\log$ and constant densitized lapse. We find that these methods in principle
do work also with our spectral method. However, our simulations fail after
about $100M$ due to unstable exponentially growing modes. The reason for this
failure may be that we evolve the black hole on a full grid without imposing
any symmetries. Such full grid instabilities have also been observed when
finite differencing methods are used to evolve excised black holes with the
BSSN system.
