Solving singularly perturbed advection-reaction equations via non-standard finite difference methods

We design and implement two non-standard finite difference methods (NSFDMs) to solve singularly perturbed advection-reaction equations (SPARE). Our methods constitute a big plus to the class of those "rare" fitted operator methods, which can be extended to singularly perturbed partial differential equations. Unlike the standard finite difference methods (SFDMs), the NSFDMs designed in this paper allow the time and the space step sizes to vary independently of one another and of the parameter (epsilon)in the SPARE under consideration. The NSFDMs replicate the linear stability properties of the fixed points of the continuous problem. Furthermore, these methods preserve the positivity and boundedness properties of the exact solution. Numerical simulations that confirm the theoretical results are presented.