Scaling effects and front propagation in a class of reaction-diffusion equations: From classic to anomalous diffusion

A simulation study is proposed where a reaction-diffusion equation in a semi-infinite medium is numerically solved by a second-order approximate finite-difference method. The non-linearities present both in the diffusivity and in the kinetic term required a proper iterative method that proved to be satisfactory in robustness and efficiency. The dynamics of the moving front were investigated in the absence and presence of a chemical reaction. In the former case, a non-linear diffusivity led to correlations between the speed of the diffusion front and the strength of interactions between diffusing species, though preserving the classical time scaling. In the latter, a genuine non-classical trend of the travelling front was detected for long times, in agreement with the recent studies concerning anomalous diffusion. An analytical approach based on scaling considerations motivated the numerical results. This study may explain the onset of a subdiffusive trend in reaction-diffusion front dynamics observed in permeation experiments through layers of different substrates, such as catalytic beds and building materials.