Large velocity overshoots with very strong velocity gradients appear near the side walls of a smooth contraction (contraction ratio 2.86:1) following a square duct and preceding a plane sudden expansion of large expansion ratio (Dld=1.43), when viscoelastic fluids flow in the laminar regime, as observed experimentally by Poole et al. [R.J. Poole, M.P. Escudier, A. Afonso, F.T. Pinho, Laminar viscoelastic flow over a backward-facing step, in: Proceedings of the XIV International Congress on Rheology, Paper NF-06, Seoul, South Korea, 2004; R.J. Poole, M.P. Escudier, P.J. Oliveira, Laminar flow of a viscoelastic shear-thinning liquid through a plane sudden expansion preceded by a gradual contraction, Proc. R. Soc. London Series A 40 (2005) 3827-3845; R.J. Poole, M.P. Escudier, A. Afonso, F.T. Pinho, Laminar flow of a viscoelastic shear-thinning liquid over a backward-facing step preceded by a gradual contraction, Phys. Fluids, submitted for publication]. To help understand this phenomenon an extensive numerical investigation was undertaken in the five parameter space of a fluid described by the Phan-Thien-Tanner (PTT) constitutive equation with non-zero second normal-stress difference (N-2 not equal 0 due to xi not equal 0) and a solvent contribution. The numerical simulations were able to qualitatively capture the velocity overshoots on the XZ-centreline profiles, and observed also overshoots on the XY-mid-plane and the so far unreported enhanced velocity peak at the duct corners. For this particular geometry the overshoots took place for particular combinations of the independent parameters, in particular they required large Weissenberg numbers, large second normal-stress differences. strain-hardening of the extensional viscosity and intense shear-thinning, together with non-negligible inertia. The parametric investigation also reports the variations of the three-dimensional velocity profiles which progressed from being rather fiat shaped at low elasticity and high Reynolds numbers (typical of purely viscous behaviour) towards a slim almost triangular shape at high elasticity numbers. (c) 2006 Elsevier B.V. All rights reserved.