© 2015 Elsevier Inc.Three numerical methods have been used to solve two problems described by advection-diffusion equations with specified initial and boundary conditions. The methods used are the third order upwind scheme , fourth order upwind scheme  and non-standard finite difference scheme (NSFD) . We considered two test problems. The first test problem we considered has steep boundary layers near x = 1 and this is challenging problem as many schemes are plagued by non-physical oscillation near steep boundaries . Many methods suffer from computational noise when modeling the second test problem. We compute some errors, namely L 2 and L ∞ errors, dissipation and dispersion errors, total variation and the total mean square error for both problems. We then use an optimization technique  to find the optimal value of the time step at a given value of the spatial step which minimizes the dispersion error and this is validated by numerical experiments.