This paper is concerned with the design of a very simple and efficient Godunov-type method for the so-called two-fluid two-pressure compress-ible model for two-phase flows. It is a contribution to the proceedings of a workshop organized by the EDF R&D French utility company and devoted to the verification of numerical schemes for two-phase flows. The present study focuses on the convective part of the two-fluid two-pressure model and is a short form of a longer paper [1] where the numerical strategy also takes into account additional terms associated with sources, pressure relaxation and drag forces. Numerical simulations and comparisons with other strategies are proposed in the last section.