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

doi:10.1007/s42967-024-00449-z

Communications on Applied Mathematics and Computation ›› 2026, Vol. 8 ›› Issue (2): 456-471.doi: 10.1007/s42967-024-00449-z

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

Lisha Chen¹, Zhibo Wang²

1. School of Mathematics and Statistics, Guangdong University of Technology, Guangzhou, 510006, Guangdong, China;
2. Center for Mathematics and Interdisciplinary Sciences, School of Mathematics and Statistics, Guangdong University of Technology, Guangzhou, 510006, Guangdong, China

收稿日期:2024-05-02 修回日期:2024-07-03 出版日期:2026-04-07 发布日期:2026-04-07
通讯作者: Zhibo Wang,E-mail:wzbmath@gdut.edu.cn E-mail:wzbmath@gdut.edu.cn
基金资助:
This research is supported by Guangdong Basic and Applied Basic Research Foundation, China (No. 2023A1515011504).

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

Lisha Chen¹, Zhibo Wang²

1. School of Mathematics and Statistics, Guangdong University of Technology, Guangzhou, 510006, Guangdong, China;
2. Center for Mathematics and Interdisciplinary Sciences, School of Mathematics and Statistics, Guangdong University of Technology, Guangzhou, 510006, Guangdong, China

Received:2024-05-02 Revised:2024-07-03 Online:2026-04-07 Published:2026-04-07
Contact: Zhibo Wang,E-mail:wzbmath@gdut.edu.cn E-mail:wzbmath@gdut.edu.cn
Supported by:
This research is supported by Guangdong Basic and Applied Basic Research Foundation, China (No. 2023A1515011504).

摘要/Abstract

摘要： 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 \begin{document}$ \min \{2, r\alpha \} $\end{document}-th order accuracy in time and second-order accuracy in space, where \begin{document}$ \alpha \in (0,1) $\end{document} is the fractional order and the grading parameter \begin{document}$ 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 Grönwall 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.

关键词: Time-fractional Zakharov-Kuznetsov (ZK) equation, Existence, Stability, Convergence

Abstract: 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 \begin{document}$ \min \{2, r\alpha \} $\end{document}-th order accuracy in time and second-order accuracy in space, where \begin{document}$ \alpha \in (0,1) $\end{document} is the fractional order and the grading parameter \begin{document}$ 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 Grönwall 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.

Key words: Time-fractional Zakharov-Kuznetsov (ZK) equation, Existence, Stability, Convergence

中图分类号:

Lisha Chen, Zhibo Wang. A Second-Order Scheme with Nonuniform Time Grids for the Two-Dimensional Time-Fractional Zakharov-Kuznetsov Equation[J]. Communications on Applied Mathematics and Computation, 2026, 8(2): 456-471.

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

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

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