Analysis of a class of stabilized and structure-preserving finite difference methods for Fisher-Kolmogorov-Petrovsky-Piscounov equation

Abstract

In this study, using implicit Euler method and second-order centered difference methods to approximate the first-order temporal and second-order spatial derivatives, respectively, introducing a stabilized term and applying $u(x_i,y_j,t_k)-{\left[u\right(x_i,y_j,t_k\left)\right]}^{p}u(x_i,y_j,t_{k+1})$ to approximate the nonlinear term $u(x_i,y_j,t_{k+1})-{\left[u\right(x_i,y_j,t_{k+1}\left)\right]}^{p+1}$ at $(x_i,y_j,t_{k+1})$, a class of stabilized, non-negativity- and boundedness-preserving finite difference methods (FDMs) are derived for Fisher-Kolmogorov-Petrovsky-Piscounov (Fisher-KPP) equation. Here, $u(x_i,y_j,t_k)$ denotes the exact solution of the original problem at $(x_i,y_j,t_k)$. In comparison with the existent maximum-principle-satisfying FDMs, our methods can further preserve the energy-dissipation property of the continuous problem with $p=1$ or $p=2$, and homogeneous Dirichlet boundary conditions. What's more, our methods can unconditionally inherit these properties as the coefficient of the stabilized term satisfies certain requirement. Secondly, as the proposed methods are applied to solve Allen-Cahn equation, the obtained solutions can unconditionally inherit the maximum value principle and energy-dissipation property of the Allen-Cahn equations as long as the coefficient of the stabilized term satisfies certain condition. Thirdly, error estimations in maximum norm are derived by using the energy method combined with the boundedness of the exact and numerical solutions. Finally, numerical results confirm the correctness of theoretical findings.