On non-standard finite difference models of reaction–diffusion equations

Reaction–diffusion equations arise in many fields of science and engineering. Often, their solutions enjoy a number of physical properties. We design, in a systematic way, new non-standard finite difference schemes, which replicate three of these properties. The first property is the stability/instability of the fixed points of the associated space independent equation. This property is preserved by non-standard one- and two-stage theta methods, presented in the general setting of stiff or non-stiff systems of differential equations. Schemes, which preserve the principle of conservation of energy for the corresponding stationary equation (second property) are constructed by non-local approximation of nonlinear reactions. Assembling of theta-methods in the time variable with energy-preserving schemes in the space variable yields non-standard schemes which, under suitable functional relation between step sizes, display the boundedness and positivity of the solution (third property). A spectral method in the space variable coupled with a suitable non-standard scheme in the time variable is also presented. Numerical experiments are provided.