Parameter Study of a Reaction-Diffusion System Near the Reactant Coefficient Asymptotic Limit

Abstract

The speed of the fast reaction in a model system of transient reaction-diffusion equations is controlled by a reaction coefficient that has an asymptotic limit at infinity. The crucial feature of the problem is the appearance of sharp internal layers in the fast reaction rate as two species are consumed everywhere throughout the domain except along their interfaces. We extend past mathematical analysis of the associated stationary problem to a study of the behavior of the transient model as the reaction coefficient grows toward its asymptotic limit. The studies reveal for which value of the fast reaction coefficient the model has essentially reached its asymptotic limit. This result is useful in its own right, but particularly important for future numerical simulations for this model, because these are faster and more reliable for smaller values of the asymptotic parameter. The numerical solution of problems with asymptotic parameters poses the risk that the numerical method might not reliably capture the most crucial large or small features inherent to such problems. Therefore, systematic studies of several numerical parameters are presented to validate the reliability and accuracy of the numerical results.