Viscoelastic drag-reducing flows created by additives have been studied recently by using an artificial diffusion spectral method [Phys. Fluids 9 (1997) 743] or a local artificial diffusion finite difference scheme [A role of elastic energy in turbulent drag reduction by polymer additives, in: Proceedings of the 2nd International Turbulence and Shear Flow Phenomena Symposium, Stockholm, 27-29 June, 2001, pp. 35-40]. In this study, we employed the MINMOD scheme to approximate the stress derivatives in the constitutive equations and introduced no artificial diffusion term. Compared to the artificial diffusion methods, much higher drag-reduction rates are predicted by using the present method. A drag-reduction rate of 53% was obtained at Weissenberg number We(tau) = 45 with a Giesekus model. The numerical error of the artificial diffusion term was analyzed. In addition, the effects of the grid spacing and computational domain size on the numerical results were studied. (C) 2003 Elsevier B.V. All rights reserved.