A Finite Volume Method Preserving the Invariant Region Property for the Quasimonotone Reaction-Diffusion Systems

Abstract

We present a finite volume method preserving the invariant region property (IRP) for the reaction-diffusion systems with quasimonotone functions, including nondecreasing, decreasing, and mixed quasimonotone systems. The diffusion terms and time derivatives are discretized using a finite volume method that satisfies the discrete maximum principle (DMP) and the backward Euler method, respectively. The discretization leads to an implicit and nonlinear scheme, and it is proved to preserve the invariant region property unconditionally. We construct an iterative algorithm and prove the invariant region property at each iteration step. Numerical examples are provided to confirm the accuracy and invariant region property of our scheme.