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We study the stability of three-dimensional numerical evolutions of theEinstein equations, comparing the standard ADM formulation to variations on afamily of formulations that separate out the conformal and traceless parts ofthe system. We develop an implementation of the conformal-traceless (CT)approach that has improved stability properties in evolving weak and stronggravitational fields, and for both vacuum and spacetimes with active couplingto matter sources. Cases studied include weak and strong gravitational wavepackets, black holes, boson stars and neutron stars. We show under whatconditions the CT approach gives better results in 3D numerical evolutionscompared to the ADM formulation. In particular, we show that our implementationof the CT approach gives more long term stable evolutions than ADM in all thecases studied, but is less accurate in the short term for the range ofresolutions used in our 3D simulations.Comment: 17 pages, 15 figures. Small changes in the text, and a change in the  list of authors. One new reference adde

Towards a stable numerical evolution of strongly gravitating systems in general relativity: The conformal treatments