Numerical simulation of free surface flow phenomena is presented for the system where a polymeric melt flows into a vertical rectangular cavity which has an inlet gate. The numerical technique combines a modified version of marker and cell method with DuFort- Frankel finite difference approximation to the equations of motion. The Leonov model is introduced for the viscoelastic fluid and compared with the Carreau model analysis for inelastic fluid. The low density polyethylene is used as sample material and the free surface profiles of the flow are simulated with time. The jet-like behavior in the flow front of the polymeric melt appears when both the Gravity number and the Weissenberg number are small. The ratio of gravitational to viscous force plays prevailing role than the elastic force itself. Thus the overall flow pattern is mainly a function of the Gravity number than the Froude number as speculated. The numerical predictions of surface profiles and the velocities of the advancing front are in good agreement with the experimental data.