Non-isothermal simulations have been undertaken for the flow of an IUPAC low-density polyethylene melt used previously in an international experimental study. First, extrusion flow from an infinitely long capillary die (L/R = infinity) is considered with isothermal walls and different thermal boundary conditions on the extrudate surface. Then the flow through flat dies is studied with the walls kept at different temperatures to create a non-isothermal flow. The viscoelasticity of the polymer melt is described by an integral constitutive equation of the K-BKZ type with a relaxation spectrum, which fits well experimental data for the shear and elongational viscosities and the normal stresses as measured in shear flow. The non-isothermal computations have been based on a pseudo-time method introduced earlier [X.-L. Luo and R. I. Tanner, Rheol. Acta, 26 (1987) 499-507] by applying the Morland-Lee hypothesis. The coupling between the momentum and energy equations is through the time-temperature shift factor by which the pseudo-time is defined. To avoid spurious oscillations in the temperature field due to high convection, a suitable upwind method has been used. The simulations have been performed for the full range of experimental measurements in the system before melt fracture occurs, that is, for apparent shear rates reaching 10 s(-1) at 150 degrees C, and showing a very strong viscoelastic character of the melt corresponding to a stress ratio (S-R) of about 2 and a Trouton ratio (T-R) of about 50. Stable solutions have been obtained for the whole range of experimental values. They show that the capillary die experiments were basically isothermal, due to small Nahme numbers (Na much less than 1), but that the Peclet numbers reach up to 44, having an influence on the cooling process occurring in the extrudate. The case of extrusion through flat dies with walls kept at different temperatures shows that the extrudate swell is strongly affected by the asymmetry in the temperature field. More importantly, bending of the extrudate occurs towards the wall with the cooler temperature. The present results aptly manifest the influence that viscoelasticity and thermal boundary conditions have on extrudate swell and are in qualitative agreement with previous experimental studies on the effect of temperature on extrusion flows.