Department of Mathematics and Applied Mathematics, University of Pretoria, Pretoria 0002, South Africa
Copyright © 2013 A. R. Appadu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Three numerical methods have been used to solve the one-dimensional advection-diffusion
equation with constant coefficients. This partial differential equation is dissipative but not dispersive.
We consider the Lax-Wendroff scheme which is explicit, the Crank-Nicolson scheme which is implicit, and a nonstandard finite difference scheme (Mickens 1991). We solve a 1D numerical experiment with
specified initial and boundary conditions, for which the exact solution is known using all these three
schemes using some different values for the space and time step sizes denoted by and , respectively,
for which the Reynolds number is 2 or 4. Some errors are computed, namely, the error rate with respect
to the norm, dispersion, and dissipation errors. We have both dissipative and dispersive errors, and
this indicates that the methods generate artificial dispersion, though the partial differential considered
is not dispersive. It is seen that the Lax-Wendroff and NSFD are quite good methods to approximate
the 1D advection-diffusion equation at some values of and . Two optimisation techniques are then
implemented to find the optimal values of when for the Lax-Wendroff and NSFD schemes, and this is validated by numerical experiments.