A Second-Order Scheme with Nonuniform Time Grids for the Two-Dimensional Time-Fractional Zakharov-Kuznetsov Equation

In this paper, we investigate the numerical method for the two-dimensional time-fractional Zakharov-Kuznetsov (ZK) equation. By the method of order reduction, the model is first transformed into an equivalent system. A nonlinear difference scheme is then proposed to solve the equivalent model with min{2,r alpha}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\min \{2, r\alpha \}$$\end{document}-th order accuracy in time and second-order accuracy in space, where alpha is an element of(0,1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha \in (0,1)$$\end{document} is the fractional order and the grading parameter r >= 1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r\geqslant 1$$\end{document}. The existence of the numerical solution is carefully studied by the renowned Browder fixed point theorem. With the help of the Gr & ouml;nwall inequality and some crucial skills, we analyze the unconditional stability and convergence of the proposed scheme based on the energy method. Finally, numerical experiments are given to illustrate the correctness of our theoretical analysis.