High order methods on Shishkin meshes for singular perturbation problems of convection-diffusion type

In this paper we construct and analyze two compact monotone finite difference methods to solve singularly perturbed problems of convection-diffusion type. They are defined as HODIE methods of order two and three, i.e., the coefficients are determined by imposing that the local error be null on a polynomial space. For arbitrary meshes, these methods are not adequate for singularly perturbed problems, but using a Shishkin mesh we can prove that the methods are uniformly convergent of order two and three except for a logarithmic factor. Numerical examples support the theoretical results.