A unified analysis for reaction–diffusion models with application to the spiral waves dynamics of the Barkley model

Abstract

Applying the gradient discretisation method (GDM), the paper develops a comprehensive numerical analysis for nonlinear equations called the reaction–diffusion model. Using only three properties, this analysis provides convergence results for several conforming and non-conforming numerical schemes that align with the GDM. As an application of this analysis, the hybrid mimetic mixed (HMM) method for the reaction–diffusion model is designed, and its convergence is established. Numerical experiments using the HMM method are presented to facilitate the study of the creation of spiral waves in the Barkley model and the ways in which the waves behave when interacting with the boundaries of their generating medium.