Nonstandard finite difference schemes based on Green's function formulations for reaction–diffusion–convection systems

Nonstandard finite differences (NSFD) schemes can improve the accuracy and reduce computational costs of traditional finite-difference schemes. However, the development of these schemes is based on analytical solutions and ad hoc rules. In this paper, we derive a NSFD scheme based on Green's function formulations for reaction–diffusion–convection systems. The formulation of the NSFD scheme is divided into three stages. In the first one, the domain of the original boundary value problem is decomposed into N   subdomains. In the second stage, for each subdomain, integral formulations based on Green's functions are derived for each subdomain. Finally, the resulting integrals are approximated by means of quadrature rules. The proposed NSFD scheme based on Green's function formulation incorporates, in a natural way, the effects of boundary nodes in the discretization approximation (i.e., avoiding the use of heuristic rules), it also exhibits global approximation orders of O(h2)O(h2) and leads to smaller approximation errors with respect to standard FD schemes. Numerical simulations on a catalytic particle model and a benchmark tubular reactor model are used for illustrating the accuracy and performance of the proposed NSFD scheme as compared to standard finite differences schemes.

► Numerical solutions of RDC systems are given using an integral formulation approach.
► Nonstandard finite differences are a particular case of integral formulations.
► The scheme proposed exhibit global approximation orders of O(h2)O(h2).
► Proposed scheme exhibit better numerical approximation than traditional schemes.