(untitled) (Advances in Applied Mathematics and Mechanics, 2025)
In this study, a class of explicit structure-preserving Du Fort-Frankel-type FDMs are firstly developed for Fisher-Kolmogorov-Petrovsky-Piscounov (Fisher-KPP) equation. They inherit some properties of the continuous problems, such as non-negativity, maximum principle and monotonicity. Besides, by using the discrete maximum principle, the error estimate in L^∞-norm is proven to be O(τ + h_x^2 + h_y^2 + (τh_x)^2 + (τh_y)^2) as some suitable conditions are satisfied. Here, τ, h_x and h_y are time step and spatial meshsizes in x- and y- directions, respectively. Then, as the current FDMs are used to solve Allen-Cahn equation, the obtained numerical solutions satisfy the discrete maximum principle and the discrete energy-dissipation law. Our methods are easy to be implemented because of explicitness. Finally, numerical results confirm theoretical findings and the efficiency of our methods.