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 ε. In contrast to the Dirichlet boundary-value problem, for the problem under consideration the errors of the well-known classical methods, generally speaking, grow without bound as ε ¿ N−1 where N defines the number of mesh points with respect to x. The order of convergence for the known ε-uniformly convergent schemes does not exceed 1. In this paper, using a defect correction technique, we construct ε-uniformly convergent schemes of highorder time-accuracy. The efficiency of the new defect-correction schemes is confirmed by numerical experiments. A new original technigue for experimental studying of convergence orders is developed for the cases where the orders of convergence in the x-direction and in the t-direction can be substantially different. 2000 Mathematics Subject Classification: 65M06; 65M12; 65M15.