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Numerical simulation of viscoelastic flows is challenging because of the hyperbolic nature of viscoelastic constitutive equations. Despite their superior accuracy and efficiency, pseudo-spectral methods require the introduction of artificial diffusion (AD) for numerical stability in hyperbolic problems, which alters the physical nature of the system. This study presents a hybrid numerical procedure that integrates an upwind total variation diminishing (TVD) finite-difference scheme, which is known for its stability in hyperbolic problems, for the polymer stress convection term into an overall pseudo-spectral numerical framework. Numerically stable solutions are obtained for Weissenberg number well beyond O(100) without the need for either global or local AD. Side-by-side comparison with an existing pseudo-spectral code reveals the impact of AD, which is shown to differ drastically between flow regimes. Elastoinertial turbulence (EIT) becomes unphysically suppressed when AD, at any level necessary for stabilizing the pseudo-spectral method, is used. This is attributed to the importance of sharp stress shocks in its self-sustaining cycles. Nevertheless , in regimes dominated by the classical inertial mechanism for turbulence generation, there is still an acceptable range of AD that can be safely used to predict the statistics, dynamics, and structures of drag-reduced turbulence. Detailed numerical resolution analysis of the new hybrid method, especially for capturing the EIT states, is also presented.