From the summary: We construct an approximate Riemann solver for a class of two-phase models. These models describe gas-liquid flow in a long tube where the flow behaviour perpendicular to the tube axis is averaged, so that the model is essentially one-dimensional in the axis direction. The general idea is that the numerical algorithm for solving this model must be able to handle any valid slip relation which relates the velocities of the two phases. As the slip relation affects the Jacobian of the flux function, this means that flux vector or flux difference splittings cannot be based on algebraic manipulation of the Jacobian. We propose to use a first-order upwind scheme of Roe-type, where the construction of the approximate Riemann solver is fully numerical. This basic scheme, however, has the same limitation as the original Roe method, namely, that for systems the positivity of the solution is not guaranteed. The modification of the basic scheme to ensure positivity of the solution is based on the Harten-Lax-van Leer Riemann solver.