High-order time-accurate schemes for singularly perturbed parabolic convection-diffusion problems with Robin boundary conditions

The boundary value problem for a singularly perturbed parabolic PDE with convection is considered on an interval in the case of the singularly perturbed Robin boundary condition; the highest space derivatives in the equation and in the boundary condition are multiplied by the perturbation parameter $eps$.In contrast to a Dirichlet boundary value problem, for the problem under consideration the errors of well known classical methods, generally speaking, grow without bound as $eps ll N^{-1}$ where $N$ defines the number of mesh points with respect to $x$.The order of convergence for known $eps$-uniformly convergent schemes does not exceed $1$., In this paper, using a defect correction technique we construct$eps$-uniformly convergent schemes of high-order time-accuracy.The efficiency of the new defect-correction schemes is confirmed with numerical experiments. An original technique for experimental studying of convergence orders is developed for cases when the orders of convergence in the$x$-direction and in the $t$-direction can be essentially different.