In this article, a finite volume method for the numerical solution of viscoelastic flows is presented. The flow of a differential upper-convected Maxwell (UCM) model fluid through an abrupt expansion has been chosen as a prototype example. The conservation and constitutive equations are solved using the finite volume method (FVM) in a staggered grid with an upwind scheme for the viscoelastic stresses and a hybrid scheme for the velocities. Numerical results show the viscoelasticity of polymer solutions is the main factor influencing sweep efficiency.