Fountain flow is the phenomenon of deceleration and outward motion of fluid particles as they approach a slower moving interface. Numerical simulations have been undertaken for the flow of viscoelastic fluids, obeying an integral constitutive equation of the K-BKZ type, capable of describing the behavior of polymer melts. Two polyethylene melts are considered, a branched LDPE and a linear HDPE. Their rheology is well captured by the integral model. The flow simulations are performed for planar and axisymmetric geometries and show the shape and extent of the free surface, as well as the stresses and pressures in the system. The semicircle is a good rough approximation for the free surface of fountain flow, but detailed computations show the effect of elasticity on the free surface, which is non-monotonic for the LDPE as the elasticity level (or apparent shear rate) increases. The less elastic HDPE shows a monotonic decrease in the extent of the flow front as elasticity increases. In all cases, the excess pressure losses (front pressure correction) increase with increasing flow rate. The effect of a nonzero second normal stress difference is to extend the flow front and increase the pressure losses.