High-order time-accurate schemes for parabolic singular perturbation convection-diffusion problems with Robin boundary conditions

The Robin boundary value problem for a singularly perturbed parabolic PDE with convection is considered on an interval. The highest space derivatives in the equation and in the boundary condition contain the perturbation parameter e. For such problems the errors of well-known numerical methods increase unboundedly as $\varepsilon\ll N^{-1}$, where $N$ is the number of mesh points over the 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 give an e-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 $\varepsilon$-uniformly convergent schemes by a defect correction technique. The efficiency of the new defect-correction schemes is confirmed with numerical experiments.