In this paper, the differences in the development of vortex structure with increasing elasticity between creeping and inertial flows in both planar and axisymmetric contraction configurations are studied for an Oldroyd B fluid. The numerical calculations are performed using a semi-Lagrangian finite volume method described in a paper by Phillips and Williams [J. Non-Newtonian Fluid Mech. 87 (1999) 214]. This method combines the advantages of fixed grids inherent in Eulerian methods with modifications to treat the convective terms using particle tracking methods. The method alleviates some of the difficulties encountered when discretizing differential constitutive equations of hyperbolic type for viscoelastic models. The semi-Lagrangian technique is applied to the convective terms in the momentum and constitutive equations. It is a natural way of treating these terms without resorting to upwinding techniques.