A computational code is developed for the numerical solution of one-dimensional transient gas-liquid flows using drift-flux models, in isothermal and also with phase change situations. For these two-phase models, classical upwind schemes such as Roe- and Godunov-type schemes are generally difficult to derive and expensive to use, since there are no treatable analytic expressions for the Jacobian matrix, eigenvalues and eigenvectors of the system of equations. On the other hand, the high-order compact finite difference scheme becomes an attractive alternative on these occasions, as it does not make use of any wave propagation information from the system of equations. The present paper extends the localized artificial diffusivity method for high-order compact finite difference schemes to solve two-phase flows with discontinuities. The numerical method has simple formulation, straightforward implementation, low computational cost and, most importantly, high-accuracy. The numerical methodology proposed is validated by solving several numerical examples given in the literature. The simulations are sixth-order accurate and it is shown that the proposed numerical method provides accurate approximations of shock waves and contact discontinuities. This is an essential property for simulations of realistic mass transport problems relevant to operations in the petroleum industry.