This article deals with numerical simulations of draw resonance of polymer fluids by employing direct difference methods to solving the governing equations in melt spinning. The stability of each difference method was studied by a comparison of the results obtained from simulations with the theoretical solutions or values. The numerical simulation confirms that the critical draw ratio of draw resonance in an isothermal and uniform tension spinning of a Newtonian fluid is between 20 and 21. The cross-sectional area of a spinline in draw resonance was found to decrease monotonically from a spinneret toward a take-up bobbin, although the taken-up filament shows periodical variation. This study has also illustrated the mechanism of draw resonance previously proposed by the author. © 1993 John Wiley & Sons, Inc.