The use of positive matrices for the analysis of the large time behavior of the numerical solution of reaction-diffusion systems

Abstract

In this paper we study the numerical solution of nonlinear reaction-diffusion systems with homogeneous Neumann boundary conditions, via the known θ \theta -method. We show that if conditions for the positivity of solutions are imposed, then the study of the asymptotic behavior of the numerical solution can be done by means of the theory of stochastic matrices. In this way it is shown that the numerical solution reproduces the asymptotic behavior of the corresponding theoretical one. In particular, we obtain the decay of the solution to its mean value. An analysis of the asymptotic stability of the equilibrium points and the convergence of the numerical scheme is given based on the use of M-matrices. Finally we consider the case in which the nonlinear term satisfies a condition of quasimonotonicity.