On the uniform in small parameter convergence of a weighted scheme for the one-dimensional time-dependent convection-diffusion equation. (Q1569382)

In this paper a weighted two-layer difference scheme for the one-dimensional time-dependent convection-diffusion equation is examined. In this scheme, the first spatial derivative is approximated by the central divided difference. It is shown that, on the piecewise uniform grid condensing in the boundary layer, the scheme is uniformly convergent for \(\sigma\geq0.5\) with respect to small parameter in the sense of the grid norm with an \(O(N^{-2}\ln^{2}N+(\sigma-0.5)\tau+\tau^{2})\) convergence rate, where \(\sigma\) is the scheme parameter, \(N\) is the number of grid points, and \(\tau\) is the time step.