An efficient numerical technique for two-parameter singularly perturbed problems having discontinuity in convection coefficient and source term

We devise a spline-based numerical technique for a class of two-parameter singularly perturbed problems having discontinuous convection and source terms. The problem is discretized using the Crank-Nicolson formula in the temporal direction, and the trigonometric B-spline basis functions are used in the spatial direction. The presence of perturbation parameters and the discontinuous convection/source terms result in the interior and boundary layers in the solution to the problem. Our primary focus is to resolve these layers and develop a uniformly convergent scheme. Initially, the proposed method gives almost first and second order convergence in the spatial and temporal directions, respectively. Then, to improve the accuracy in the spatial direction, we have used the Richardson extrapolation technique. Two numerical examples are taken to demonstrate the layer phenomenon and confirm the theoretical proofs. It is evident from the tables that the Richardson extrapolation technique increases the accuracy from one to two in the spatial direction.