A robust fitted numerical method for singularly perturbed turning point problems whose solution exhibits an interior layer

Abstract

In this paper we considerer singularly perturbed convection-diffusion-reaction problems with a turning point whose solution exhibits an interior layer. We establish bounds on the solution to these problems and their derivatives. We construct a fitted mesh finite difference method (FMFDM). The method consists of an upwind scheme on an appropriately designed piece wise uniform mesh of Shishkin type. This mesh is fine near the turning point and coarse elsewhere. A rigorous error analysis shows that the developed method is uniformly convergent of order almost one. In order to improve the accuracy of the proposed FMFDM, we apply Richardson extrapolation. Two numerical examples are considered to illustrate the theoretical findings. Results show that this method is reliable and competitive.

Mathematics Subject Classification (2010): 34B05, 65B05, 65L10, 65L11, 65L12, 65L70.

Keywords: Finite difference method, singularly perturbed problems, turning point problems, interior layer, uniform convergence