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

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.