A second-order nonstandard finite difference scheme and application to a model of biological and chemical processes

Abstract

We consider a dynamical system defined by a system of autonomous differential equations on a set Ω ⊂ R n \Omega \subset \mathbb {R}^n . By using Mickens’ rule on the nonlocal approximation of nonlinear terms, we construct an implicit Nonstandard Finite Difference (NSFD) scheme which is an explicit time reversible scheme. Apart from being elementary stable, we show that the NSFD scheme is of second-order and domain-preserving, thereby solving a pending problem on the construction of higher-order nonstandard schemes without spurious solutions. It is shown that the new scheme applies directly for mass action-based models of biological and chemical processes.