Analysis of finite difference methods for convection-diffusion problem

Abstract

We consider finite difference methods for one dimensional convection diffusion problem. An error analysis shows that the solution of the upwind scheme is not uniformly convergent in the discrete maximum norm due to its behavior in the layer. Then, we introduce and analyze a numerical method, Il.inAllen-Southwell scheme, that is first-order uniformly convergent in the discrete maximum norm throughout the domain. Finally, we present numerical results that confirm theoretical findings.