Explicit, Monotone and Structure-Preserving Finite Difference Methods for Fisher-Kolmogorov-Petrovsky-Piscounov Equation and Allen-Cahn Equation

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 $L^\infty$-norm is proven to be $O\left(\tau + h_x^2 + h_y^2 + \left(\frac{h_x}{\tau}\right)^2 + \left(\frac{h_y}{\tau}\right)^2\right)$ as some suitable conditions are satisfied. Here, $\tau$, $h_x$ 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.