  In this thesis we present the first numerical study of gravitational collapse
in braneworlds within the framework of the single brane model proposed by
Randall and Sundrum (RSII). We directly show that the evolutions of
sufficiently strong initial data configurations result in black holes (BHs)
with finite extension into the bulk. The extension changes from sphere to
pancake (or cigar, seen from a different perspective) as the size of BH
increases. We find preliminary evidences that BHs of the same size generated
from distinct initial data profiles are geometrically indistinguishable. As
such, a no-hair theorem of BH (uniqueness of BH) is suggested to hold in the
RSII spacetimes studied in this thesis---these spacetimes are axisymmetric
without angular momentum and non-gravitational charges. In particular, the BHs
we obtained as the results of the dynamical system, are consistent with the
ones previously obtained from a static vacuum system by Figueras and Wiseman.
We also obtained some results in closed form without numerical computation such
as the equality of ADM mass of the brane with the total mass of the braneworld.
  The calculation within the braneworld requires advances in the formalism of
numerical relativity (NR). The regularity problem in previous numerical
calculations in axisymmetric (and spherically symmetric) spacetimes, is
actually associated with neither coordinate systems nor the machine precision.
The fundamental variables (the unknown functions to be solved for in numerical
calculations) are more important to make the numerical calculations in
cylindrical coordinates regular. The generalized harmonic (GH) formalism and
the BSSN formalism for general relativity are developed to make them suitable
for calculations in non-Cartesian coordinates under non-flat background. A
conformal function of the metric is included into the GH formalism to simulate
the braneworld.
