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

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

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 \in (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 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.

Keywords

Time-fractional Zakharov-Kuznetsov (ZK) equation / Existence / Stability / Convergence / 65M06 / 65M12 / 35R11

References

[1] Benito J, Garcia A, Negreanu M, Urena F, Vargas A. Two finite difference methods for solving the Zakharov-Kuznetsov-Modified Equal-Width equation. Eng. Anal. Bound. Elem.. 2023, 153: 213-225.

[2] Biswas A, Zerrad E. Solitary wave solution of the Zakharov-Kuznetsov equation in plasmas with power law nonlinearity. Nonlinear. Anal. Real.. 2010, 11: 3272-3274.

[3] Cao Y, Nikan O, Avazzadeh Z. A localized meshless technique for solving 2D nonlinear integro-differential equation with multi-term kernels. Appl. Numer. Math.. 2023, 183: 140-156.

[4] Cen D, Wang Z. Time two-grid technique combined with temporal second order difference method for two-dimensional semilinear fractional sub-diffusion equations. Appl. Math. Lett.. 2022, 129. 107919

[5] Cen D, Wang Z, Mo Y. Second order difference schemes for time-fractional KdV-Burgers’ equation with initial singularity. Appl. Math. Lett.. 2021, 112. 106829

[6] Chen H, Xu D, Zhou J. A second-order accurate numerical method with graded meshes for an evolution equation with a weakly singular kernel. J. Comput. Appl. Math.. 2019, 356: 152-163.

[7] Faminskii A. Regular solutions to initial-boundary value problems in a half-strip for two-dimensional Zakharov-Kuznetsov equation. Nonlinear. Anal. Real.. 2020, 51. 102959

[8] Guan K, Ou C, Wang Z. Mathematical analysis and a second-order compact scheme for nonlinear Caputo-Hadamard fractional sub-diffusion equations. Mediterr. J. Math.. 2024, 21: 77.

[9] Huang J, Liu Y, Liu Y, Tao Z, Cheng Y. A class of adaptive multiresolution ultra-weak discontinuous Galerkin methods for some nonlinear dispersive wave equations. SIAM J. Sci. Comput.. 2022, 44A745-A769.

[10] Kilbas, A., Srivastava, H., Trujillo, J.: Theory and Applications of Fractional Differential Equations. Elsevier Science, Amsterdam (2006)

[11] Kopteva N. Error analysis of the L1 method on graded and uniform meshes for a fractional-derivative problem in two and three dimensions. Math. Comput.. 2019, 88: 2135-2155.

[12] Li C, Zhang J. Symmetry analysis and exact solutions of generalized fractional Zakharov-Kuznetsov equations. Symmetry. Basel.. 2019, 11: 601.

[13] Li D, Sun W, Wu C. A novel numerical approach to time-fractional parabolic equations with nonsmooth solutions. Numer. Math. Theor. Meth. Appl.. 2021, 14355-376.

[14] Liao H, Mclean W, Zhang J. A discrete Grönwall inequality with applications to numerical schemes for subdiffusion problems. SIAM J. Numer. Anal.. 2019, 57: 218-237.

[15] Liao, H., Mclean, W., Zhang, J.: A second-order scheme with nonuniform time steps for a linear reaction-sudiffusion problem. Commun. Comput. Phys. 30, 567–601 (2021)

[16] Lyu P, Vong S. A high-order method with a temporal nonuniform mesh for a time-fractional Benjamin-Bona-Mahony equation. J. Sci. Comput.. 2019, 80: 1607-1628.

[17] Ma, H., Yu, Y., Ge, D.: The auxiliary equation method for solving the Zakharov-Kuznetsov (ZK) equation. Comput. Math. Appl. 58, 2523–2527 (2009)

[18] Nishiyama H, Noi T, Oharu S. Conservative finite difference schemes for the generalized Zakharov-Kuznetsov equations. J. Comput. Appl. Math.. 2012, 236: 2998-3006.

[19] Ou C, Cen D, Wang Z, Vong S. Fitted schemes for Caputo-Hadamard fractional differential equations. Numer. Algorithms. 2023.

[20] Ou C, Wang Z, Vong S. A second-order fitted scheme combined with time two-grid technique for two-dimensional nonlinear time fractional telegraph equations involving initial singularity. J. Comput. Appl. Math.. 2024, 448. 115936

[21] Ren Z, Wei L, He Y, Wang S. Numerical analysis of an implicit fully discrete local discontinuous Galerkin method for the fractional Zakharov-Kuznetsov equation. Math. Model. Anal.. 2012, 17: 558-570.

[22] Samko, S., Kilbas, A., Marichev, O.: Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach, Yverdon (1993)

[23] Seadawy A. Stability analysis for two-dimensional ion-acoustic waves in quantum plasmas. Phys. Plasmas.. 2014, 21: 52107.

[24] Shen J, Sun Z, Cao W. A finite difference scheme on graded meshes for time-fractional nonlinear Korteweg-de Vries equation. Appl. Math. Comput.. 2019, 361752-765

[25] Stynes M, O’Riordan E, Gracia J. Error analysis of a finite difference method on graded meshes for a time-fractional diffusion equation. SIAM J. Numer. Anal.. 2017, 55: 1057-1079.

[26] Sun Y, Tian Z. An efficient fourth-order three-point scheme for solving some nonlinear dispersive wave equations. Commun. Nonlinear. Sci.. 2023, 125. 107366

[27] Sun, Z.: Finite Difference Method for Nonlinear Evolution Equations. Science Press, Beijing (2018)

[28] Tang T. A finite difference scheme for partial integro-differential equations with a weakly singular kernel. Appl. Numer. Math.. 1993, 11: 309-319.

[29] Veeresha, P., Prakasha, G.: Solution for fractional Zakharov-Kuznetsov equations by using two reliable techniques (in Chinese). J. Phys. 60, 313–330 (2019)

[30] Wang, K., Yao, S.: Numerical method for fractional Zakharov-Kuznetsov equations with He’s fractional derivative. Therm. Sci. 23, 2163–2170 (2019)

[31] Wang, X., Liu, Y.: All single travelling wave patterns to fractional Jimbo-Miwa equation and Zakharov-Kuznetsov equation. Pramana 92, 31 (2019)

[32] Wang, X., Sun, Z.: A second order convergent difference scheme for the initial-boundary value problem of Korteweg-de Vires equation. Numer. Methods Partial Differential Equations 37, 2873–2894 (2021)

[33] Wang Z, Ou C, Vong S. A second-order scheme with nonuniform time grids for Caputo-Hadamard fractional sub-diffusion equations. J. Comput. Appl. Math.. 2022, 414. 114448

[34] Wang, Z., Xiao, M., Mo, Y.: Time two-grid fitted scheme for the nonlinear time fractional Schrödinger equation with nonsmooth solutions. Commun. Nonlinear Sci. Numer. Simul. 137, 108119 (2024)

[35] Wazwaz, A.: The extended tanh method for the Zakharov-Kuznetsov (ZK) equation, the modified ZK equation, and its generalized forms. Commun. Nonlinear. Sci. 13, 1039–1047 (2008)

[36] Wei L, He Y, Zhang X. Analysis of an implicit fully discrete local discontinuous Galerkin method for the time-fractional KDV equation. Adv. Appl. Math. Mech.. 2015, 7: 510-527.

[37] Xiao M, Wang Z, Mo Y. An implicit nonlinear difference scheme for two-dimensional time-fractional Burgers equation with time delay. J. Appl. Math. Comput.. 2023, 69: 2919-2934.

[38] Zakharov, V., Kuznetsov, E.: On three dimensional solitons. Sov. Phys. JETP. 39, 285–286 (1974)

[39] Zhou M, Kanth A, Aruna K. Numerical solutions of time fractional Zakharov-Kuznetsov equation via natural transform decomposition method with nonsingular kernel derivatives. J. Funct. Space.. 2021, 23: 1-7

Funding

National Natural Science Foundation of China(11701103)

Natural Science Foundation of Guangdong Province(2023A1515011504)

RIGHTS & PERMISSIONS

Shanghai University