Nonlinear diffusion in spinodal decomposition: a numerical solution

The numerical solution of the one-dimensional nonlinear diffusion equation with a negative diffusion coefficient (up-hill diffusion) by a five-point approximation central difference scheme is considered. The stability criteria are discussed in detail and a numerical solution is provided for a specific case in which the time evolution of a periodic composition wave is presented with growth eventually leading to a stationary configuration. A critical comparison of the numerical solution with existing analytical solutions is shown. This leads to a simple semi-empirical growth law for studying the kinetics of spinodal decomposition in alloys.