A Two-grid Method with Richardson Extrapolation For a Semilinear Convection-diffusion Problem

A boundary value problem for a second-order semilinear singularly perturbed ordinary differential equation is considered. We use Newton and Picard iterations for a linearization. To solve the problem at each iteration we apply the difference scheme with the property of uniform with respect to the singular perturbation parameter convergence. A modified Samarskii and central difference schemes on Shishkin mesh are considered. It is known that these schemes are almost second order accuracy uniformly with respect to the singular perturbation parameter. To decrease the required number of arithmetical operations for resolving the difference scheme, a two-grid method is proposed. To increase the accuracy of difference scheme, we investigate the possibility to apply Richardson extrapolation using known solutions of the difference scheme on both meshes. The comparison of modified Samarskii and central difference schemes is carried out. The results of some numerical experiments are discussed.