Direct difference methods have been used to solve the simultaneous non-linear partial differential equations for melt spinning without recourse to linearisation or perturbation approximation. The stability of each difference schemes was studied by error analysis using the Taylor series, and by comparison of the results obtained from numerical simulation with the logical value in melt spinning. It is found that computation with 19 digit long double precision has significantly simplified the stability problem of difference equations. Using this method, the precise critical draw ratio of draw resonance in an isothermal and uniform tension spinning of Newtonian fluids can be obtained in between 20.218 and 21.219, a figure consistent with 20.218 which was obtained by a linear perturbation approximation method by Kase and Denn. It thus has paved the way to computation of full information for unsteady melt spinning processes using the difference method.