Non-negativity-preserving and maximum-principle-satisfying finite difference methods for Fisher’s equation with delay

Little attention has been devoted to the numerical studies on maximum-principle-satisfying FDMs for Fisher’s equation with delay. Monotone difference schemes can preserve the maximum principle of the continuous problem. However, it is difficult to develop monotone difference schemes for Fisher’s equation with delay because of delay term. The main novelties of this study are to develop the maximum-principle-satisfying FDMs for it by using cut-off technique to adjust the numerical solutions obtained applying non-negativity-preserving FDMs. Firstly, by using a new weighted difference formula with parameter θ, new numerical formula and explicit Euler method to discrete the diffusion term, delay term and temporal variable, respectively, a class of new explicit non-negativity-preserving FDMs are established for one-dimensional problem. Then, by applying cut-off technique to adjust their numerical solutions, a kind of new explicit maximum-principle-satisfying FDMs are developed. Secondly, based on previous work, by using implicit Euler method for the approximation to the temporal variable, an implicit non-negativity-preserving FDM is designed. Likewise, by applying cut-off technique to adjust the obtained numerical solutions, an implicit maximum-principle-satisfying FDM is devised. By convergent analyses, cut-off techniques do not reduce convergent rates. Thirdly, the extensions of our methods to two-dimensional problems are discussed. Finally, numerical results confirm the correctness of theoretical findings and the efficiency of our methods for long-term simulations.