  We study the stability of three-dimensional numerical evolutions of the
Einstein equations, comparing the standard ADM formulation to variations on a
family of formulations that separate out the conformal and traceless parts of
the system. We develop an implementation of the conformal-traceless (CT)
approach that has improved stability properties in evolving weak and strong
gravitational fields, and for both vacuum and spacetimes with active coupling
to matter sources. Cases studied include weak and strong gravitational wave
packets, black holes, boson stars and neutron stars. We show under what
conditions the CT approach gives better results in 3D numerical evolutions
compared to the ADM formulation. In particular, we show that our implementation
of the CT approach gives more long term stable evolutions than ADM in all the
cases studied, but is less accurate in the short term for the range of
resolutions used in our 3D simulations.
