  We present a new pseudospectral code, bamps, for numerical relativity written
with the evolution of collapsing gravitational waves in mind. We employ the
first order generalized harmonic gauge formulation. The relevant theory is
reviewed and the numerical method is critically examined and specialized for
the task at hand. In particular we investigate formulation parameters, gauge
and constraint preserving boundary conditions well-suited to non-vanishing
gauge source functions. Different types of axisymmetric twist-free moment of
time symmetry gravitational wave initial data are discussed. A treatment of the
axisymmetric apparent horizon condition is presented with careful attention to
regularity on axis. Our apparent horizon finder is then evaluated in a number
of test cases. Moving on to evolutions, we investigate modifications to the
generalized harmonic gauge constraint damping scheme to improve conservation in
the strong field regime. We demonstrate strong-scaling of our pseudospectral
penalty code. We employ the Cartoon method to efficiently evolve axisymmetric
data in our 3+1 dimensional code. We perform test evolutions of Schwarzschild
perturbed by gravitational waves and by gauge pulses, both to demonstrate the
use of our blackhole excision scheme and for comparison with earlier results.
Finally numerical evolutions of supercritical Brill waves are presented to
demonstrate durability of the excision scheme for the dynamical formation of a
blackhole.
