Abstract :An explicit numerical scheme developed by Von Rosenberg for the convection-diffusion equation in one spatial dimension is reviewed and analyzed. The convergence of this scheme is outlined and a comparative study was established with an explicit standard finite difference scheme. The results show that Von Rosenberg scheme considerably improves the stability condition and accuracy in difficult problems associated with very small diffusion coefficients. This study extends and corrects the original work presented by Von Rosenberg, which represents an original contribution.