In the collapsar model of long GRBs the jet is formed at the center of the progenitorstar, propagates in its interior, and produces the observed gamma rays much after itsbreakout from the star. The loss of pressure support during breakout induces a strongrarefaction wave that propagates inside the jet and causes its bulk acceleration. Thismechanism has been already studied using axisymmetric magnetohydrodynamic (MHD)simulations assuming a prescribed shape for the surface between the jet and itsenvironment, as well as using simple rarefaction waves in planar geometry. Trying toimprove over these works, we solve the steady-state, axisymmetric, relativistic MHDequations using the method of characteristics. In this way the jet boundary is foundself-consistently and the rarefaction wave is studied in the axisymmetric geometry. Inthis poster we present our first results and a comparison with previous works.