Novel Defect-correction High-order, in Space and Time, Accurate Schemes for Parabolic Singularly Perturbed Convection-diffusion Problems

New high-order accurate finite difference schemes based on defect correction are considered for an initial boundary-value problem on an interval for singularly perturbed parabolic PDEs with convection; the highest space derivative in the equation is multiplied by the perturbation parameter ε, ε ∈ (0, 1]. Solutions of the well-known classical numerical schemes for such problems do not converge ε-uniformly (the errors of such schemes depend on the value of the parameter ε and are comparable with the solution itself for small values of ε). The convergence order of the existing ε-uniformly convergent schemes does not exceed 1 in space and time. In this paper, using a defect correction technique, we construct a special difference scheme that converges ε-uniformly with the second (up to a logarithmic factor) order of accuracy with respect to x and with the second order of accuracy and higher with respect to t. The conditions are given which ensure the ε-uniform convergence of the defect-correction schemes with a rate of O(N−k ln N + K−k0), k = 1, 2, k0 = 1, 2, 3, where N + 1 and K + 1 denote the number of the mesh points in x and t, respectively. Theoretical results and the efficiency of the newly constructed schemes are confirmed by numerical experiments. 2000 Mathematics Subject Classification: 65M06; 65M12; 65M15.