A robust fitted operator finite difference method for a two-parameter singular perturbation problem1

Abstract

We consider singularly perturbed two-point BVPs with two small parameters ϵ and μ multiplying the derivatives. It is pointed out in P.A. Farrel (Sufficient conditions for uniform convergence of a class of difference schemes for singular perturbation problems, IMA J. Numer. Anal. 7 (1987), pp. 459–472), that ‘in general, the exponentially fitted finite difference methods (EFFDMs) are more effective inside the layers. However, though these methods are uniformly convergent, they do not give fairly good approximations in the whole interval of interest’. In this paper, we study that the non-standard finite difference method (NSFDM) that we develop overcomes this weakness. Like EFFDMs, the NSFDM is also a method of fitted operator type. Secondly, unlike several earlier works (see, for example Gracia et al., Appl. Numer. Math. 56 (2006), pp. 962–980) where the authors use a combination of approaches in various regions, the method presented in this paper consists of just one scheme throughout the domain of interest. This is very important because it increases the possibilities of extending the approach both for higher dimensional and higher order problems. Combination of schemes usually suffers from the drawback that their selection is mostly based on the relative values of ϵ and μ, otherwise they fail to provide monotonic solutions. We also investigate a number of issues associated with a variety of NSFDMs and finally provide some comparative numerical results.