Damping characteristics of finite difference methods for one-dimensional reaction–diffusion equations

摘要

The exact solution of finite difference methods for the one-dimensional, linear reaction–diffusion equation in infinite spatial domains is obtained analytically and compared with the exact solution of the corresponding partial differential equation by considering a fixed time interval and a limit process. It is shown that the exact solution of the difference equation differs from the exact one only in a temporal damping term which is asymptotically identical to the exact one when the grid spacing is much smaller than the wave length. The analysis presented in the paper also shows and emphasizes that the linear stability analysis of finite difference equations for the one-dimensional linear reaction–diffusion equation can be performed without any recourse whatsoever to the norms and spectrum of the amplification matrix.