This paper deals with the Aw--Rascle--Zhang [A. Aw and M. Rascle, SIAM J. Appl. Math., 60 (2000), pp. 916--938; H. M. Zhang, Transp. Res. B Methodol., 36 (2002), pp. 275--290] model for traffic flow on uni-directional road networks. We construct weak solutions to Riemann problems at the junctions, which conserve the mass and the generalized momentum. In particular, we introduce a new approach to approximate the homogenized pressure through an additional equation for the propagation of a reference pressure. The resulting system of coupled conservation laws is then solved using an appropriate numerical scheme of Godunov type. Numerical simulations show that the proposed approach enables us to approximate the homogenized pressure sufficiently well. The features of the new approach are illustrated through a comparative analysis with other methods proposed in the literature for the Aw--Rascle--Zhang second-order traffic model [2, 31] and the Lighthill--Whitham--Richards model [M. J. Lighthill and G. B. Whitman, Proc. A, 229 (1955), pp. 317--345; P. I. Richards, Oper. Res., 4 (1956), pp. 42--51].