A high order uniformly convergent alternating direction scheme for time dependent reaction-diffusion singularly perturbed problems

In this work we design and analyze a numerical method to solve efficiently two dimensional initial-boundary value reaction-diffusion problems, where the diffusion parameter can be very small in comparison with the reaction term. The method is defined by combining the Peaceman & Rachford alternating direction method to discretize in time with a finite difference scheme of HODIE type, which is defined on an appropriate piecewise uniform mesh. We prove that the resulting scheme is ε-uniformly convergent having order two in time variable and order three in spatial variables respectively. Some numerical examples illustrate in practice the efficiency and the orders of uniform convergence theoretically proved. We also show how it is easy to avoid the well-known order reduction phenomenon which is produced in the time semidiscretization process when the boundary conditions are time dependent.