Advantages of the Samarskii-type schemes on the Shishkin mesh

The schemes of the Samarskii type are simple modifications of the upwind scheme. We use them on the Shishkin mesh and discuss their advantages over the upwind scheme when applied to the linear one-dimensional singularly perturbed convection–diffusion problem. One of the advantages is that the Samarskii-type schemes have exact first-order accuracy uniform in the perturbation parameter, as opposed to the upwind scheme which is almost first-order uniformly accurate because its accuracy is diminished by logarithmic factors. Although this is not a new result, we re-emphasize it in the paper. We also demonstrate another advantage, that the Samarskii-type schemes are almost second-order uniformly accurate on the layer component of the solution. Motivated by this fact, we present a further improvement of the numerical method.