Stable schemes for partial differential equations: the one-dimensional reaction–diffusion equation

Reaction–diffusion equations are fundamental in modelling several natural phenomena. The numerical schemes used to solve these equations often suffer from numerical stability problems. In this paper, a new type of algorithm to solve the diffusion equation in a stable and explicit manner is extended to the reaction–diffusion equation. The new scheme imposes a fixed value for the stability coefficient below the stability limit, and uses this information in order to determine a new grid. The values of the variables at this grid are then obtained by interpolation from the original grid. The scheme is applied to the linear single kinetic reaction–diffusion equation and to the classical Fisher equation. Different possibilities of extending the new scheme to the reaction–diffusion equations are discussed. It is shown that, for the linear case, including both terms (reaction and diffusion) in the computation of the new grid gives more accurate results and is more correct than just including the diffusion term. To solve the non-linear Fisher equation, a fractional-step method, where the reaction and diffusion terms are solved separately, is chosen. The new scheme provides realistic results when compared with analytic solutions.