Uniform convergence and order reduction of the fractional implicit Euler method to solve singularly perturbed 2D reaction-diffusion problems

In this paper we analyze the uniform convergence of a numerical method designed to approximate efficiently the solution of 2D parabolic singularly perturbed problems of reaction diffusion type. The method combines a modified fractional implicit Euler method to discretize in time, and the classical central finite difference scheme, on a special nonuniform mesh, to discretize in space. The resulting fully discrete scheme is uniformly convergent with respect to the diffusion parameter. The analysis of the convergence is made by using a two step technique, which discretizes first in time and later on in space. We show the order reduction phenomenon associated to the fractional implicit Euler method, which typically appears if the boundary conditions are time dependent and a natural evaluation of them is done. An appropriate choice for the boundary conditions is proposed and analyzed in detail, proving that the order reduction can be removed. Some numerical tests show the practical effects of our method; as well, we compare it with the classical choice for the boundary data in terms of the uniform consistency and the order of uniform convergence of the numerical scheme.