From the summary: We describe a numerical algorithm for the steady flow of viscoelastic fluid. The conservative and constitutive equations are solved using the finite volume method with a hybrid scheme for velocities and first-order upwind approximation for the viscoelastic stress. A non-uniform staggered grid system is used. The iterative SIMPLE algorithm is employed to relax the coupled momentum and continuity equations. The nonlinear algebraic equations over the flow domain are solved iteratively by the symmetric coupled Gauss-Seidel method. In both, the full approximation storage multigrid algorithm is used. An Oldroyd-B fluid model was selected for the calculation. Results are reported for planar \(4:1\) abrupt contraction at various Weissenberg numbers.