Improved $\leps$-uniform Accurate Schemes for Singularly Perturbed Parabolic Convection-Diffusion Problems. Defect-Correction Technique

We consider 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 $\eps$, $\eps \in (0,1]$. Solutions of well-known classical numerical schemes for such problems do not converge $\eps$-uniformly (the errors of such schemes depend on the value of the parameter $\eps$ and are comparable with the solution itself for small values of~$\eps$). The convergence order of the existing $\eps$-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 $\eps$-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 $\eps$-uniform convergence of the defect-correction schemes with a rate of order $N^{-k}\ln^k N + K^{-k_0}}$, $k=1,2$, $k_0=1,2,3$, 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 with numerical experiments.