A fourth-order compact difference scheme for solving 2D EFK equation

In this article, a fourth order compact difference scheme for solving the two-dimensional extended Fisher–Kolmogorov (2D EFK) equation is proposed and analyzed. This scheme is three-level implicit, based on a novel time discretization idea of $u(x_i,y_j,t_n)\approx \frac{1}{4}(U_{i,j}^{n+1}+2U_{i,j}^n+U_{i,j}^{n-1})$. The discrete energy functional method is used to obtain prior estimates of numerical solutions in the maximum norm. Furthermore, the convergence of the difference solutions in the maximum norm is analyzed, and the convergence rate is obtained as $O({\tau }^2+h^4)$, which without any restriction on the grid ratio with time step $\tau$ and mesh size $h$. Finally, numerical examples are given to support the theoretical analysis.