Numerical integration of reaction-diffusion systems

Approximating numerically the solutions of a reaction-diffusion system in an efficient manner requires the application of implicit methods, since the Courant-Friedrichs-Lewy condition on explicit methods imposes a time step of the order of the square of the space step. In this article, we review two types of strategies which are expected to yield reasonably precise solutions within a reasonable computing time. The first examines methods for solving the linear step necessary in any resolution procedure; estimates of CPU time in terms of the error are given in the non preconditioned and in the preconditioned case-provided that it is possible to define an efficient preconditioner. The second strategy is based on splitting, with or without extrapolation. The respective faults and qualities of both strategies are examined; they lead to a list of difficult analytical and numerical problems with possible hints as to their solution.