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

Abstract

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 parameterε. 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. © 2002, Institute of Mathematics, NAS of Belarus. All rights reserved.