A finite difference method for singularly perturbedconvection-diffusion problems withnear-second-order discretization error is presented. Such equationsarise in a number of different areas. Local singularities can arise inthe finite difference solution. These singularities necessitaterefinement of the mesh.

The local refinement of the mesh results in the need for values atnonmesh points, or slave points. Values at these slave points must beestimated using local interpolation. The interpolation results in adeterioration in the truncation error estimates.

The main thrust of this paper is to introduce a method for avoidingslave points by changing the orientation of the mesh used at suchsingularities. By switching from the conventional grid (using stepsparallel to the coordinate axes) to a diagonal grid, the localdifference approximations require only the use of points in the regularmesh.

Strategies for mesh refinement avoiding slave points are discussed,and truncation error estimates are obtained. The efficiency of themethod is then discussed using numerical experiments. These experimentsshow the use of the method as a black box with the refinement and changein difference schemes in use.

The paper has a good balance of detail for the expert, whileallowing the less expert reader to gain insight into the problem and itssolution.