The gradient discretisation method for the chemical reactions of biochemical systems

Abstract

Purpose – The main purpose of this paper is to introduce the gradient discretisation method (GDM) to a system of reaction diffusion equations subject to non-homogeneous Dirichlet boundary conditions. Then, the authors show that the GDM provides a comprehensive convergence analysis of several numerical methods for the considered model. The convergence is established without non-physical regularity assumptions on the solutions. Design/methodology/approach – In this paper, the authors use the GDM to discretise a system of reaction diffusion equations with non-homogeneous Dirichlet boundary conditions. Findings – The authors provide a generic convergence analysis of a system of reaction diffusion equations. The authors introduce a specific example of numerical scheme that fits in the gradient discretisation method. The authors conduct a numerical test to measure the efficiency of the proposed method. Originality/value – This work provides a unified convergence analysis of several numerical methods for a system of reaction diffusion equations. The generic convergence is proved under the classical assumptions on the solutions.

Keywords

• A gradient discretisation method (GDM)
• Gradient schemes
• Convergence analysis
• Existence of weak solutions
• Two-dimensional reaction–diffusion Brusselator system
• Dirichlet boundary conditions