Efficient Simulation of a Reaction-Diffusion System with a Fast Reaction in the Asymptotic Limit

Abstract

We study a reaction-diffusion system of three chemical species, where two chemicals react with a much faster reaction rate than the other reaction in the model. We are interested in the asymptotic limit as the fast reaction rate becomes infinite. This forces the reaction interface to have an asymptotically small width with asymptotically large height. This interface is moving in time and causes interior layers that are progressively more challenging and costly for numerical simulations of the three species model, as the singularity becomes sharper with larger reaction rates. But in the asymptotic limit, an equivalent two component model can be defined that is significantly cheaper computationally and allows for effective studies for the model. The equivalence is demonstrated by the analytical definition of the two component model and by comparing numerical results to ones for the three species model with progressively larger reaction rates, which also demonstrate the computational efficiency. The state-of-the-art finite element package COMSOL Multiphysics is used for the simulations, thus also showing a practical way how to handle and visualize moving interior layers in reaction-diffusion systems. COMSOL is popular in many areas of engineering and the sciences and thus the mathematical example here can provide guidance to a wide range of users with models consisting of partial differential equations.