Decay to spatially homogeneous states for the numerical solution of reaction-diffusion systems

The aim of this paper is to discuss the behaviour of the numerical solution of systems of nonlinear reaction-diffusion equations with homogeneous Neumann boundary conditions. We show that, under certain conditions, the solution obtained using known finite difference methods reproduces the behaviour of the exact solution. In particular we prove that the numerical solution decays as time increases to a spatially homogeneous vector, which is a suitably «weighted» mean value of the numerical solution itself.