A model for two-phase pipeline flow is presented, with evaporation and condensation modelled using a relaxation source term based on statistical rate theory. The model is solved numerically using a Godunov splitting scheme, making it possible to solve the hyperbolic fluid-mechanic equation system and the relaxation term separately. The hyperbolic equation system is solved using the multi-stage (MUSTA) finite volume scheme. The stiff relaxation term is solved using two approaches: one based on the Backward Euler method, and one using a time-asymptotic scheme. The results from these two methods are presented and compared for a CO2 pipeline depressurisation case. Copyright © 2013 Inderscience Enterprises Ltd.