### A FAST COMPACT DIFFERENCE METHOD FOR TWO-DIMENSIONAL NONLINEAR SPACE-FRACTIONAL COMPLEX GINZBURG-LANDAU EQUATIONS

Lu Zhang1, Qifeng Zhang2, Hai-wei Sun3

1. 1. School of Mathematics and Statistics, Xuzhou University of Technology, Xuzhou 221018, China;
2. Department of Mathematics, Zhejiang Sci-Tech University, Hangzhou 310018, China;
3. Department of Mathematics, University of Macau, Macao, China
• Received:2020-02-05 Revised:2020-05-04 Online:2021-09-15 Published:2021-10-15
• Supported by:
Q. Zhang was partially supported by Natural Science Foundation of Zhejiang Province (Grant No. LY19A010026), Zhejiang Province “Yucai” Project (2019), Natural Science Foundation of China (Grant No. 11501514) and Fundamental Research Funds of Zhejiang Sci-Tech University (Grant 2019Q072). L. Zhang was partially supported by research from Xuzhou University of Technology (Grant XKY201530) and the “Peiyu” Project from Xuzhou University of Technology (Grant XKY2019104), H. Sun was supported in part by research grants of the Science and Technology Development Fund, Macau SAR (File no. 0118/2018/A3), and MYRG2018-00015-FST from the University of Macau.

Lu Zhang, Qifeng Zhang, Hai-wei Sun. A FAST COMPACT DIFFERENCE METHOD FOR TWO-DIMENSIONAL NONLINEAR SPACE-FRACTIONAL COMPLEX GINZBURG-LANDAU EQUATIONS[J]. Journal of Computational Mathematics, 2021, 39(5): 708-732.

This paper focuses on a fast and high-order finite difference method for two-dimensional space-fractional complex Ginzburg-Landau equations. We firstly establish a three-level finite difference scheme for the time variable followed by the linearized technique of the nonlinear term. Then the fourth-order compact finite difference method is employed to discretize the spatial variables. Hence the accuracy of the discretization is $\mathcal{O}$(τ2 + $h_1^4$ + $h_2^4$) in L2-norm, where τ is the temporal step-size, both h1 and h2 denote spatial mesh sizes in x- and y- directions, respectively. The rigorous theoretical analysis, including the uniqueness, the almost unconditional stability, and the convergence, is studied via the energy argument. Practically, the discretized system holds the block Toeplitz structure. Therefore, the coefficient Toeplitz-like matrix only requires $\mathcal{O}$(M1M2) memory storage, and the matrix-vector multiplication can be carried out in $\mathcal{O}$(M1M2(log M1 + log M2)) computational complexity by the fast Fourier transformation, where M1 and M2 denote the numbers of the spatial grids in two different directions. In order to solve the resulting Toeplitz-like system quickly, an efficient preconditioner with the Krylov subspace method is proposed to speed up the iteration rate. Numerical results are given to demonstrate the well performance of the proposed method.

CLC Number:

 [1] I.S. Aranson, L. Kramer, The world of the complex Ginzburg-Landau equation, Reviews of Modern Physics, 74(2002), 99-143.[2] S. Arshed, Soliton solutions of fractional complex Ginzburg-Landau equation with Kerr law and non-Kerr law media, Optik, 160(2018), 322-332.[3] C. Çelik, M. Duman, Crank-Nicolson method for the fractional diffusion equation with the Riesz fractional derivative, Journal of Computational Physics, 231:4(2012), 1743-1750.[4] R. Chan, X.Q. Jin, An Introduction to Iterative Toeplitz Solvers, SIAM, Philadelphia, 2007.[5] R. Chan, M. Ng, Conjugate gradient methods for Toeplitz systems, SIAM Review, 38(1996), 427-482.[6] X. Cheng, H. Qin, J. Zhang, A compact ADI scheme for two-dimensional fractional sub-diffusion equation with Neumann boundary condition, Applied Numerical Mathematics, 156(2020), 50-62.[7] R.P. Feynman, A.R. Hibbs, Quantum Mechanics and Path Integrals, McGraw-Hill, New York, 1965.[8] R. Gorenflo, F. Mainardi, Random walk models for space-fractional diffusion processes, Fractional Calculus and Applied Analysis, 1:2(1998), 167-191.[9] X.M. Gu, L. Shi, T.H. Liu, Well-posedness of the fractional Ginzburg-Landau equation, Applicable Analysis, 98:14(2018), 2545-2558.[10] B.L. Guo, Z.H. Huo, Well-posedness for the nonlinear fractional Schrödinger equation and inviscid limit behavior of solution for the fractional Ginzburg-Landau equation, Fractional Calculus and Applied Analysis, 16:1(2013), 226-242.[11] D.D. He, K.J. Pan, An unconditionally stable linearized difference scheme for the fractional Ginzburg-Landau equation, Numerical Algorithms, 79:3(2018), 899-925.[12] X.Q. Jin, Developments and Applications of Block Toeplitz Iterative Solvers. Science Press & Kluwer Academic Publishers, Beijing/Dordrecht, The Netherlands, 2002.[13] N. Laskin, Fractional quantum mechanics, Physical Review E, 62(2000), 3135-3145.[14] N. Laskin, Fractional quantum mechanics and Lévy path integrals, Physics Letters A, 268(2000), 298-305.[15] M. Li, C.M. Huang, An efficient difference scheme for the coupled nonlinear fractional GinzburgLandau equations with the fractional Laplacian, Numerical Methods for Partial Differential Equations, 35:1(2019), 394-421.[16] M. Li, C.M. Huang, N. Wang, Galerkin element method for the nonlinear fractional GinzburgLandau equation, Applied Numerical Mathematics, 118(2017), 131-149.[17] H.L. Liao, Z.Z. Sun, H.S. Shi, Error estimate of fourth-ordercompact scheme for linear Schrödinger equations, SIAM Journal on Numerical Analysis, 47(6) (2010), 4381-4401.[18] W.J. Liu, W.T. Yu, C.Y. Yang, M.L. Liu, Y.J. Zhang, M. Lei, Analytic solutions for the generalized complex Ginzburg-Landau equation in fiber lasers, Nonlinear Dynamics, 89(2017), 2933-2939.[19] H. Lu, P.W. Bates, S.J. Lü, M.J. Zhang, Dynamics of the 3-D fractional complex Ginzburg-Landau equation, Journal Differential Equations, 259(2015), 5276-5301.[20] H. Lu, S.J. Lü, Asymptotic dynamics of 2D fractional complex Ginzburg-Landau equation, International Journal of Bifurcation and Chaos, 23:2(2013), 1350202.[21] H. Lu, S.J. Lü, Random attractor for fractional Ginzburg-Landau equation with multiplicative noise, Taiwanese Journal of Mathematics, 18:2(2014), 435-450.[22] H. Lu, S.J. Lü, M.J. Zhang, Fourier spectral approximations to the dynamics of 3D fractional complex Ginzburg-Landau equation, Discrete and Continuous Dynamical Systems, 37:5(2017), 2539-2564.[23] V. Millot, Y. Sire, On a fractional Ginzburg-Landau equation and 1/2-Harmonic maps into spheres, Archive for Rational Mechanics and Analysis, 215(2015), 125-210.[24] A. Mohebbi, Fast and high-order numerical algorithms for the solution of multidimensional nonlinear fractional Ginzburg-Landau equation, The European Physical Journal Plus, 133:2(2018), 67, https://doi.org/10.1140/epjp/i2018-11846-x[25] I. Podlubny, Fractional Differential Equations, Academic Press, New York, 1999.[26] X.K. Pu, B.L. Guo, Well-posedness and dynamics for the fractional Ginzburg-Landau equation, Applicable Analysis, 92:2(2013), 31-33.[27] Y. Saad, Iterative Methods for Sparse Linear Systems, SIAM, Philadelphia, 2003.[28] J. Shu, P. Li, J. Zhang, O. Liao, Random attractors for the stochastic coupled fractional GinzburgLandau equation with additive noise, Journal of Mathematical Physics, 56(2015), 102702.[29] Z.Z. Sun, The Method of Order Reduction and Its Application to the Numerical Solutions of Partial Differential Equations, Science Press, Beijing, 2009.[30] V. Tarasov, G. Zaslavsky, Fractional dynamics of coupled oscillators with long-range interaction, Chaos, 16(2006), 023110.[31] V. Tarasov, G. Zaslavsky, Fractional Ginzburg-Landau equation for fractal media, Physica A, 354(2005), 249-261.[32] N. Wang, C.M. Huang, An efficient split-step quasi-compact finite difference method for the nonlinear fractional Ginzburg-Landau equations, Computers and Mathematics with Applications, 75(2018), 2223-2242.[33] P.D. Wang, C.M. Huang, An efficient fourth-order in space difference scheme for the nonlinear fractional Ginzburg-Landau equation, BIT Numerical Mathematics, 58:3(2018), 783-805.[34] P.D. Wang, C.M. Huang, An implicit midpoint difference scheme for the fractional GinzburgLandau equation, Journal of Computational Physics, 312(2016), 31-49.[35] Y.S. Wu, Multiparticle quantum mechanics obeying fractional statistics, Physical Review Letters, 53(1984), 111-114.[36] J.W. Zhang, H.D. Han, H. Brunner, Numerical blow-up of semilinear parabolic PDEs on unbounded domains in $\mathbb{R}$2, Journal of Scientific Computing, 49(3) (2019), 367-382.[37] J.W. Zhang, Z.L. Xu, X.N. Wu, Unified approach to split absorbing boundary conditions for nonlinear Schrödinger equations:Two-dimensional case, Physical Review E, 79(4) (2009), 046711.[38] M. Zhang, G.F. Zhang, L.D. Liao, Fast iterative solvers and simulation for the space fractional Ginzburg-Landau equations Ginzburg-Landau equations, Computers and Mathematics with Applications, 78:5(2019), 1793-1800.[39] Q. Zhang, T. Li, Asymptotic stability of compact and linear θ-methods for space fractional delay generalized diffusion equation, Journal of Scientific Computing, 81(2019), 2413-2446.[40] Q. Zhang, X. Lin, K. Pan, Y. Ren, Linearized ADI schemes for two-dimensional space-fractional nonlinear Ginzburg-Landau equation, Computers and Mathematics with Applications, 80(2020), 1201-1220.[41] Q. Zhang, Y. Ren, X. Lin, Y. Xu, Uniform convergence of compact and BDF methods for the space fractional semilinear delay reaction-diffusion equations, Applied Mathematics and Computation, 358(2019), 91-110.[42] X. Zhao, Z.Z. Sun, Z.P. Hao, A fourth-order compact ADI scheme for two-dimensional nonlinear space fractional Schrödinger equation, SIAM Journal on Scientific Computing, 36:6(2014), 2865-2886.
 [1] Baiying Dong, Xiufeng Feng, Zhilin Li. AN L∞ SECOND ORDER CARTESIAN METHOD FOR 3D ANISOTROPIC INTERFACE PROBLEMS [J]. Journal of Computational Mathematics, 2022, 40(6): 882-912. [2] Darko Volkov. A STOCHASTIC ALGORITHM FOR FAULT INVERSE PROBLEMS IN ELASTIC HALF SPACE WITH PROOF OF CONVERGENCE [J]. Journal of Computational Mathematics, 2022, 40(6): 955-976. [3] Yanping Chen, Qiling Gu, Qingfeng Li, Yunqing Huang. A TWO-GRID FINITE ELEMENT APPROXIMATION FOR NONLINEAR TIME FRACTIONAL TWO-TERM MIXED SUB-DIFFUSION AND DIFFUSION WAVE EQUATIONS [J]. Journal of Computational Mathematics, 2022, 40(6): 936-954. [4] Mingming Zhao, Yongfeng Li, Zaiwen Wen. A STOCHASTIC TRUST-REGION FRAMEWORK FOR POLICY OPTIMIZATION [J]. Journal of Computational Mathematics, 2022, 40(6): 1004-1030. [5] Mei Yang, Ren-Cang Li. HEAVY BALL FLEXIBLE GMRES METHOD FOR NONSYMMETRIC LINEAR SYSTEMS [J]. Journal of Computational Mathematics, 2022, 40(5): 711-727. [6] Rong Zhang, Hongqi Yang. A DISCRETIZING LEVENBERG-MARQUARDT SCHEME FOR SOLVING NONLIEAR ILL-POSED INTEGRAL EQUATIONS [J]. Journal of Computational Mathematics, 2022, 40(5): 686-710. [7] Wei Zhang. STRONG CONVERGENCE OF THE EULER-MARUYAMA METHOD FOR A CLASS OF STOCHASTIC VOLTERRA INTEGRAL EQUATIONS [J]. Journal of Computational Mathematics, 2022, 40(4): 607-623. [8] Yaolin Jiang, Zhen Miao, Yi Lu. WAVEFORM RELAXATION METHODS FOR LIE-GROUP EQUATIONS* [J]. Journal of Computational Mathematics, 2022, 40(4): 649-666. [9] Xiaonian Long, Qianqian Ding. A SECOND ORDER UNCONDITIONALLY CONVERGENT FINITE ELEMENT METHOD FOR THE THERMAL EQUATION WITH JOULE HEATING PROBLEM [J]. Journal of Computational Mathematics, 2022, 40(3): 354-372. [10] Siyuan Qi, Guangqiang Lan. STRONG CONVERGENCE OF THE EULER-MARUYAMA METHOD FOR NONLINEAR STOCHASTIC VOLTERRA INTEGRAL EQUATIONS WITH TIME-DEPENDENT DELAY [J]. Journal of Computational Mathematics, 2022, 40(3): 437-452. [11] Xiaoyu Wang, Ya-xiang Yuan. STOCHASTIC TRUST-REGION METHODS WITH TRUST-REGION RADIUS DEPENDING ON PROBABILISTIC MODELS [J]. Journal of Computational Mathematics, 2022, 40(2): 294-334. [12] Mohammed Harunor Rashid. METRICALLY REGULAR MAPPING AND ITS UTILIZATION TO CONVERGENCE ANALYSIS OF A RESTRICTED INEXACT NEWTON-TYPE METHOD [J]. Journal of Computational Mathematics, 2022, 40(1): 44-69. [13] Yang Chen, Chunlin Wu. DATA-DRIVEN TIGHT FRAME CONSTRUCTION FOR IMPULSIVE NOISE REMOVAL [J]. Journal of Computational Mathematics, 2022, 40(1): 89-107. [14] Qianqian Chu, Guanghui Jin, Jihong Shen, Yuanfeng Jin. NUMERICAL ANALYSIS OF CRANK-NICOLSON SCHEME FOR THE ALLEN-CAHN EQUATION [J]. Journal of Computational Mathematics, 2021, 39(5): 655-665. [15] Yifen Ke, Changfeng Ma. MODIFIED ALTERNATING POSITIVE SEMIDEFINITE SPLITTING PRECONDITIONER FOR TIME-HARMONIC EDDY CURRENT MODELS [J]. Journal of Computational Mathematics, 2021, 39(5): 733-754.
Viewed
Full text

Abstract