A computational study of three numerical methods for some advection-diffusion problems

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 [5], fourth order upwind scheme [5] and non-standard finite difference scheme (NSFD) [10]. 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 [16]. Many methods suffer from computational noise when modeling the second test problem. We compute some errors, namely L2 and L∞ errors, dissipation and dispersion errors, total variation and the total mean square error for both problems. We then use an optimization technique [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 numerical experiments.