Uniformly convergent additive finite difference schemes for singularly perturbed parabolic reaction–diffusion systems

In this paper 1D parabolic systems of two singularly perturbed equations of reaction–diffusion type are examined. For the time discretization we consider two additive (or splitting) schemes defined on a uniform mesh and for the space discretization we use the classical central difference approximation defined on a Shishkin mesh. The uniform convergence of both the semidiscrete and the fully discrete problems is proved. The additive schemes are used to solve a test problem, and the results obtained with these schemes and the standard discretization using the backward Euler method are compared. Also, numerical results are presented in the case of systems of three equations. All the numerical results show the advantage in computational cost of the additive schemes compared to the standard discretization.