High-order time-accurate schemes for parabolic singular perturbation problems with convection

We consider the first boundary value problem for a singularly perturbed parabolic PDE with convection on an interval. For the case of sufficiently smooth data, it is easy to construct a standard finite difference operator and a piecewise uniform mesh, condensing in the boundary layer, which gives an ε-uniformly convergent difference scheme. The order of convergence for such a scheme is exactly one and up to a small logarithmic factor one with respect to the time and space variables, respectively. In this paper we construct high-order time-accurate ε-uniformly convergent schemes by a defect correction technique. The efficiency of the new defect-correction scheme is confirmed by numerical experiments. 2000 Mathematics Subject Classification: 65M06, 65M12, 65M15