A Study of Numerical Methods for Reaction-Diffusion Equations

Abstract

Numerical methods are applied to one-dimensional unsteady reaction-diffusion equations to seek traveling wave solutions. These equations describe flame propagation in certain combustion systems. Model scalar reaction-diffusion equations which admit traveling waves as exact solutions are formulated, and one of these is solved with twelve different numerical integration schemes for a test problem. The diffusion terms are differenced using the explicit method introduced by Saul’yev [Integration of Equations of Parabolic Type by the Method of Nets, Pergamon, New York, 1964], which, it is shown, can be formally accurate to $O({{\Delta t^3} / {\Delta x^2}} )$. Both implicit and explicit techniques for the reaction terms are tested. The results of the study with these schemes indicates that numerical diffusion can reduce accuracy significantly, that numerical dispersion truncation errors can reduce accuracy if they are sufficiently large, and that an accurate representation of the reaction terms in the difference equations is important to retain overall accuracy. In addition to the above test problem, results are given using one of the explicit methods for the computation of a propagating ozone decomposition flame. The results show good agreement with the fourth-order accurate results of Margolis [J. Comput. Phys., 27 (1978), pp. 410–427] which indicates that more efficient lower order numerical methods can be sufficiently accurate for practical computations.