  Relativistic jets reside in high-energy astrophysical systems of all scales.
Their interaction with the surrounding media is critical as it determines the
jet evolution, observable signature, and feedback on the environment. During
its motion the interaction of the jet with the ambient media inflates a highly
pressurized cocoon, which under certain conditions collimates the jet and
strongly affects its propagation. Recently, Bromberg et al. (2011) derived a
general simplified (semi)analytic solution for the evolution of the jet and the
cocoon in case of an unmagnetized jet that propagates in a medium with a range
of density profiles. In this work we use a large suite of 2D and 3D
relativistic hydrodynamic simulations in order to test the validity and
accuracy of this model. We discuss the similarities and differences between the
analytic model and numerical simulations and also, to some extent, between 2D
and 3D simulations. Our main finding is that although the analytic model is
highly simplified, it properly predicts the evolution of the main ingredients
of the jet-cocoon system, including its temporal evolution and the transition
between various regimes (e.g., collimated to uncollimated). The analytic
solution predicts a jet head velocity that is faster by a factor of about 3
compared to the simulations, as long as the head velocity is Newtonian. We use
the results of the simulations to calibrate the analytic model which
significantly increases its accuracy. We provide an applet that calculates
semi-analytically the propagation of a jet in an arbitrary density profile
defined by the user at http://www.astro.tau.ac.il/~ore/propagation.html.
