A non‐standard finite difference scheme for an advection‐diffusion‐reaction equation

In this note, a non-standard finite difference (NSFD) scheme is proposed for an advection-diffusion-reaction equation with nonlinear reaction term. We first study the diffusion-free case of this equation, that is, an advection-reaction equation. Two exact finite difference schemes are constructed for the advection-reaction equation by the method of characteristics. As these exact schemes are complicated and are not convenient to use, an NSFD scheme is derived from the exact scheme. Then, the NSFD scheme for the advection-reaction equation is combined with a finite difference space-approximation of the diffusion term to provide a NSFD scheme for the advection-diffusion-reaction equation. This new scheme could preserve the fixed points, the positivity, and the boundedness of the solution of the original equation. Numerical experiments verify the validity of our analytical results. Copyright (C) 2014 John Wiley & Sons, Ltd.