Stability of invariant regions for an alternating direction method for systems of nonlinear reaction-diffusion equations

Abstract

In this paper, we consider an alternating direction method for the numerical solution of systems of nonlinear reaction-diffusion equations, with homogeneous Dirichlet boundary conditions, and we consider some properties of the difference scheme. If the system admits a bounded invariant regionS of the phase space, we prove that, under certain conditions on the mesh,S is also invariant for the difference equations. Moreover we find an error bound which tends to diamS ast→+∞ and which is 0 (h) fort fixed. Finally, we derive a time-independent error bound for a special case of mild nonlinearity.