Non-standard finite-differences schemes for generalized reaction–diffusion equations

Abstract

Reaction–diffusion equations are commonly used in different science and engineering fields to describe spatial patterns arising from the interaction of chemical or biochemical reactions and diffusive transport mechanisms. In this work we design, in a systematic way, non-standard finite-differences (FD) schemes for a class of reaction–diffusion equations of the form 1 x σ d d x ( x σ d u d x ) = f ( u ( x ) ) , x ∈ [ 0 , 1 ] , where σ is the shape power that accounts for the complexity of the domain geometry. The proposed FD scheme, that is derived from a Green’s function formulation, replicates the underlying geometry and reduces to traditional FD schemes for sufficiently small values of the grid spacing. Numerical results show that the non-standard FD scheme offers smaller approximation errors with respect to traditional schemes, specially for coarse grids.