Stability and Convergence of Finite Difference Methods for Systems of Nonlinear Reaction-Diffusion Equations

Abstract

We consider the initial-boundary value problem for the general reaction-diffusion system (*) $U_t = D\nabla ^2 U + \Sigma _j M_j U_{xj} + F$, where U is a vector, $D,M_j $, and F depend on $(x,t,U)$, and D and $M_j$ are diagonal matrices with $D \geqq 0$. It is known that if there is a box S in U-space such that F does not point out of S, then S is invariant for (*). This invariance gives rise to an a priori estimate which in turn yields the existence of a smooth solution. In this paper we construct a family of locally second-order accurate finite difference schemes for (*). We prove that, under certain conditions on the mesh, S is also invariant for the difference equations. This enables us to derive an error bound which is $O(h)$ for t fixed, and which tends to diam S as $t \to \infty $. Next, we obtain a time-independent error bound by imposing a monotonicity condition on F, and we examine the implications of this condition for the solution of (*). Finally we derive a time-independent error bound for a special case of (*) in which F is “slowly varying.”