A study of difference schemes with the first derivative approximated by a central difference ratio

For an ordinary second-order differential equation in which the coefficient of the highest derivative is a small parameter, the classical difference scheme which uses a central difference ratio to approximate the first derivative is investigated. By means of a detailed analysis of Green's function of the grid problem, it is established that the scheme is solvable on Shishkin's piecewise-uniform grid which clusters in the boundary layer and has uniform accuracy O(N^-2 In² N) with respect to the small parameter, where N is the number of grid nodes.