Invariant Regions and asymptotic behaviour for the numerical solution of reaction-diffusion systems by a class of alternating direction methods

In this paper we study the asymptotic behaviour of the numerical solution of systems of nonlinear reaction-diffusion equations, with homogeneous Dirichlet boundary conditions. We construct a class of alternating direction methods. In order to obtain a good simulation of the analytical solution, we require the difference schemes to be of positive type; this fact enables us to prove that, if an invariant setS exists for the analytical solutions,S is also invariant for the numerical solution and, moreover, to find a time-independent error estimate, if the nonlinear termF satisfies a monotonicity condition.