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