Numerical Studies of the Asymptotic Behavior of a Reaction-Diffusion System with a Fast Reaction

Abstract

A model system of reaction-diffusion equations exhibiting a fast reaction is studied with an initial condition defining three interfaces between regions of dominance of two main reactants within the spatial domain. Two cases varying the reactant coefficient of the model problem are studied, the second case using reactant coefficient three orders of magnitude larger than the first. A numerical method consisting of spatial discretization by finite differences in space and a time discretization employing Numerical Differentiation Formulas is used to obtain numerical solutions for both cases. Comparing the numerical interface solutions between the two cases shows that the asymptotic limit of the model problem has been nearly approached for both reaction coefficients considered. Therefore, the qualitative features of the internal reactant boundaries can be reliably studied using the smaller reaction coefficient, saving substantial computation time. This report also includes a systematic study of several numerical parameters including the absolute and relative ODE tolerances that guarantee the efficiency and reliability of the numerical solutions used to draw the conclusions about the model solutions.