Numerical methods for evolutionary reaction–diffusion problems with nonlinear reaction terms

We have developed new methods to integrate efficiently reaction–diffusion parabolic problems with nonlinear reaction terms. In order to obtain uniform and unconditional convergence, such methods combine the advantages of alternating direction methods, the additive Runge–Kutta methods designed by Cooper and Sayfy for nonlinear Stiff problems as well as the use of Shishkin meshes in the singularly perturbed case. The resulting algorithms are only linearly implicit and they have the same order of computational complexity, per time step, that any explicit method. We show some numerical experiences which illustrate the good properties of our schemes predicted by the theoretical results.