A parameter uniform method for two-parameter singularly perturbed boundary value problems with discontinuous data.

Abstract

We consider two-parameter singularly perturbed problems of reaction-convection-diffusion type in one dimension. The convection coefficient and source term are discontinuous at a point in the domain. The problem is numerically solved using the upwind difference method on an appropriately defined Shishkin-Bakhvalov mesh. At the point of discontinuity, a three-point difference scheme is used. A convergence analysis is given and the method is shown to be first-order uniformly convergent with respect to the perturbation parameters. The numerical results presented in the paper confirm our theoretical results of first-order convergence. Summing up: • The Shishkin-Bakhvalov mesh is graded in the layer region and uniform in the outer region as shown in the graphical abstract. • The method presented here has uniform convergence of order one in the supremum norm. • The numerical orders of convergence obtained in numerical examples with Shishkin- Bakhvalov mesh are better than those for Shishkin mesh.