A uniformly convergent scheme on a nonuniform mesh for convection–diffusion parabolic problems

In this paper we construct a numerical method to solve one-dimensional time-dependent convection–diffusion problem with dominating convection term. We use the classical Euler implicit method for the time discretization and the simple upwind scheme on a special nonuniform mesh for the spatial discretization. We show that the resulting method is uniformly convergent with respect to the diffusion parameter. The main lines for the analysis of the uniform convergence carried out here can be used for the study of more general singular perturbation problems and also for more complicated numerical schemes. The numerical results show that, in practice, some of the theoretical compatibility conditions seem not necessary.