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A uniformly convergent hybrid difference scheme for a system of singularly perturbed initial value problems

ABSTRACT A system of second-order singularly perturbed initial value problems with weak hypotheses on the coefficients is considered. The equations have diffusion parameters of different magnitudes, which gives rise to overlapping boundary layers. The structure of these layers is analysed, and this leads to the construction of a Shishkin-type mesh. On this mesh a hybrid difference scheme is proposed, which is a combination of the second-order difference schemes on the fine mesh and the midpoint upwind scheme on the coarse mesh. By applying the truncation error estimate techniques and a difference analogue of Gronwall's inequality, it is proved that the scheme is almost second-order convergent, in the discrete maximum norm, independently of singular perturbation parameters. Numerical experiments are provided to validate the theoretical results.