  In order to perform accurate and stable long-time numerical integration of
the Einstein equation, several hyperbolic systems have been proposed. We here
present numerical comparisons between weakly hyperbolic, strongly hyperbolic,
and symmetric hyperbolic systems based on Ashtekar's connection variables. The
primary advantage for using this connection formulation in this experiment is
that we can keep using the same dynamical variables for all levels of
hyperbolicity. Our numerical code demonstrates gravitational wave propagation
in plane symmetric spacetimes, and we compare the accuracy of the simulation by
monitoring the violation of the constraints. By comparing with results obtained
from the weakly hyperbolic system, we observe the strongly and symmetric
hyperbolic system show better numerical performance (yield less constraint
violation), but not so much difference between the latter two. Rather, we find
that the symmetric hyperbolic system is not always the best in numerical
performances.
  This study is the premier to present full numerical simulations using
Ashtekar's variables. We also describe our procedures in detail.
