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In this paper we are considering a semilinear singular perturbation reaction-- diffusion boundary value problem, which contains a small perturbationparameter that acts on the highest order derivative. We construct a differencescheme on an arbitrary nonequidistant mesh using a collocation method andGreen's function. We show that the constructed difference scheme has a uniquesolution and that the scheme is stable. The central result of the paper is$\epsilon$-uniform convergence of almost second order for the discreteapproximate solution on a modified Shishkin mesh. We finally provide twonumerical examples which illustrate the theoretical results on the uniformaccuracy of the discrete problem, as well as the robustness of the method.

A uniformly convergent difference scheme on a modified Shishkin mesh for the singular perturbation boundary value problem