HODIE finite difference schemes on generalized Shishkin meshes

In this work we study a class of HODIE finite difference schemes to solve linear one-dimensional convection–diffusion problems of singular perturbation type. The numerical method is constructed on nonuniform Shishkin type meshes, defined by a generating function, including classical Shishkin meshes and Shishkin–Bakhvalov meshes. We will prove the uniform convergence, with respect to the singular perturbation parameter, of the HODIE scheme on this type of meshes, having order bigger than one. We show some numerical examples confirming in practice the theoretical results and also we see numerically that an appropriate extrapolation will be useful to improve the errors and the order of convergence, when the singular perturbation parameter is sufficiently small.