### Abstract:

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 [4], fourth order
upwind scheme [4] and Non-Standard Finite Difference scheme (NSFD) [9]. We considered two test problems.
The first test problem has steep boundary layers near x = 1 and this is challenging problem as many schemes are
plagued by non-physical oscillation near steep boundaries [15]. Many methods suffer from computational noise
when modelling the second test problem especially when the coefficient of diffusivity is very small for instance
0.01. We compute some errors, namely L2 and L1 errors, dissipation and dispersion errors, total variation and
the total mean square error for both problems and compare the computational time when the codes are run on
a matlab platform. We then use the optimization technique devised by Appadu [1] 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
some numerical experiments.