Unconditional and optimal pointwise error estimates of finite difference methods for the two-dimensional complex Ginzburg-Landau equation. (English) Zbl 07892705
Summary: In this paper, we give improved error estimates for linearized and nonlinear Crank-Nicolson type finite difference schemes of Ginzburg-Landau equation in two dimensions. For linearized Crank-Nicolson scheme, we use mathematical induction to get unconditional error estimates in discrete \(L^2\) and \(H^1\) norm. However, it is not applicable for the nonlinear scheme. Thus, based on a “cut-off” function and energy analysis method, we get unconditional \(L^2\) and \(H^1\) error estimates for the nonlinear scheme, as well as boundedness of numerical solutions. In addition, if the assumption for exact solutions is improved compared to before, unconditional and optimal pointwise error estimates can be obtained by energy analysis method and several Sobolev inequalities. Finally, some numerical examples are given to verify our theoretical analysis.
MSC:
| 65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |
| 65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |
Keywords:
complex Ginzburg-Landau equation; finite difference method; unconditional convergence; optimal estimates; pointwise error estimatesReferences:
| [1] | I. S. ARANSON, L. KRAMER. The world of the complex Ginzburg-Landau equation. Rev. Modern Phys., 2002, 74(1): 99-143. · Zbl 1205.35299 |
| [2] | Boling GUO, Bixiang WANG. Finite dimensional behaviour for the derivative Ginzburg-Landau equation in two spatial dimensions. Phys. D, 1995, 89(1-2): 83-99. · Zbl 0884.34062 |
| [3] | Yongsheng LI, Boling GUO. Global existence of solutions to the derivative 2D Ginzburg-Landau equation. J. Math. Anal. Appl., 2000, 249(2): 412-432. · Zbl 0966.35119 |
| [4] | Zhenchao CAO, Boling GUO, Bixiang WANG. Global existence theory for the two-dimensional derivative Ginzburg-Landau equation. Chinese. Sci. Bull., 1998, 43(5): 393-395. · Zbl 0995.35063 |
| [5] | Jinqiao DUAN, E. S. TITI, P. HOLMES. Regularity, approximation and asymptotic dynamics for a gener-alized Ginzburg-Landau equation. Nonlinearity, 1993, 6(6): 915-933. · Zbl 0808.35133 |
| [6] | Zhaohui HUO, Yueling JIA. Global well-posedness for the generalized 2D Ginzburg-Landau equation. J. Differential Equations, 2009, 247(1): 260-276. · Zbl 1173.35305 |
| [7] | Zhaopeng HAO, Zhizhong SUN. A linearized high-order difference scheme for the fractional Ginzburg-Landau equation. Numer. Methods Partial Differential Equations, 2017, 33(1): 105-124. · Zbl 1359.65150 |
| [8] | Meng LI, Chengming HUANG. An efficient difference scheme for the coupled nonlinear fractional Ginzburg-Landau equations with the fractional Laplacian. Numer. Methods Partial Differential Equations, 2019, 35(1): 394-421. · Zbl 1419.65024 |
| [9] | Zhizhong SUN, Qiding ZHU. On Tsertsvadze’s difference scheme for the Kuramoto-Tsuzuki equation. J. Comput. Appl. Math., 1998, 98(2): 289-304. · Zbl 0933.65104 |
| [10] | Zhizhong SUN, Qiding ZHU, Wanrong CAO. A three-level linearized compact difference scheme for the Ginzburg-Landau equation. Numer. Methods Partial Differ. Equ., 2015, 31(3): 876-899. · Zbl 1320.65116 |
| [11] | Pengde WANG, Chengming HUANG. An implicit midpoint difference scheme for the fractional Ginzburg-Landau equation. J. Comput. Phys., 2016, 312: 31-49. · Zbl 1351.76191 |
| [12] | Tingchun WANG, Boling GUO. Analysis of some finite difference schemes for two-dimensional Ginzburg-Landau equation. Numer. Methods Partial Differ. Equ., 2011, 27(5): 1340-1363. · Zbl 1233.65062 |
| [13] | Yun YAN, F. I. MOXLEY III, Weizhong DAI. A new compact finite difference scheme for solving the complex Ginzburg-Landau equation. Appl. Math. Comput, 2015, 260: 269-287. · Zbl 1410.65332 |
| [14] | Qifeng ZHANG, Lu ZHANG, Haiwei SUN. A three-level finite difference method with preconditioning tech-nique for two-dimensional nonlinear fractional complex Ginzburg-Landau equations. J. Comput. Appl. Math., 2021, 389: Paper No. 113355, 19 pp. · Zbl 1462.65119 |
| [15] | Chaoxia YANG. A linearized Crank-Nicolson-Galerkin FEM for the time-dependent Ginzburg-Landau equa-tions under the temporal gauge. Numer. Methods Partial Differ. Equ., 2014, 30(4): 1279-1290. · Zbl 1297.65121 |
| [16] | Zhiming CHEN, Shibing DAI. Adaptive Galerkin methods with error control for a dynamical Ginzburg-Landau model in superconductivity. SIAM J. Numer. Anal., 2001, 38(6): 1961-1985. · Zbl 0987.65096 |
| [17] | Qiang DU, M. D. GUNZBURGER, J. S. PETERSON. Analysis and approximation of the Ginzburg-Landau model of superconductivity. SIAM Rev., 1992, 34(2): 54-81. · Zbl 0787.65091 |
| [18] | Huadong GAO, Weiwei SUN. An efficient fully linearized semi-implicit Galerkin-mixed FEM for the dynam-ical Ginzburg-Landau equations of superconductivity. J. Comput. Phys., 2015, 294: 329-345. · Zbl 1349.65450 |
| [19] | Meng LI, Chengming HUANG, Nan WANG. Galerkin finite element method for the nonlinear fractional Ginzburg-Landau equation. Appl. Numer. Math., 2017, 118: 131-149. · Zbl 1367.65144 |
| [20] | Meng LI, Dongyang SHI, Junjun WANG. Unconditional superconvergence analysis of a linearized Crank-Nicolson Galerkin FEM for generalized Ginzburg-Landau equation. Comput. Math. Appl., 2020, 79(8): 2411-2425. · Zbl 1437.65197 |
| [21] | Mo MU. A linearized Crank-Nicolson-Galerkin method for the Ginzburg-Landau model. SIAM J. Sci. Com-put., 1997, 18(4): 1028-1039. · Zbl 0894.65046 |
| [22] | Dongyang SHI, Qian LIU. Unconditional superconvergent analysis of a new mixed finite element method for Ginzburg-Landau equation. Numer. Methods Partial Differ. Equ., 2019, 35(1): 422-439. · Zbl 1419.65042 |
| [23] | P. DEGOND, Shi JIN, Min TANG. On the time splitting spectral method for the complex Ginzburg-Landau equation in the large time and space scale limit. SIAM J. Sci. Comput., 2008, 30: 2466-2487. · Zbl 1176.35170 |
| [24] | M. H. HEYDARI, A. ATANGANA, Z. AVAZZADEH. Chebyshev polynomials for the numerical solution of fractal-fractional model of nonlinear Ginzburg-Landau equation. Eng. Comput., 2021, 37(2): 1377-1388. |
| [25] | M. GANESH, T. THOMPSON. A spectrally accurate algorithm and analysis for a Ginzburg-Landau model on superconducting surfaces. Multiscale. Model. Sim., 2018, 16(1): 78-105. · Zbl 1422.65451 |
| [26] | Zhiping MAO, Jie SHEN. Hermite spectral methods for fractional PDEs in unbounded domains. SIAM J. Sci. Comput., 2017, 39(5): A1928-A1950. · Zbl 1373.65075 |
| [27] | Tao TANG, Huifang YUAN, Tao ZHOU. Hermite spectral collocation methods for fractional PDEs in un-bounded domains. Commun. Comput. Phys., 2018, 24(4): 1143-1168. · Zbl 1475.65148 |
| [28] | Pengde WANG. Fast exponential time differencing/spectral-Galerkin method for the nonlinear fractional Ginzburg-Landau equation with fractional Laplacian in unbounded domain. Appl. Math. Lett., 2021, 112: Paper No. 106710, 7 pp. · Zbl 1453.65365 |
| [29] | Wei ZENG, Aiguo XIAO, Xueyang LI. Error estimate of Fourier pseudo-spectral method for multidimensional nonlinear complex fractional Ginzburg-Landau equations. Appl. Math. Lett., 2019, 93: 40-45. · Zbl 1414.65026 |
| [30] | M. ABBASZADEH, M. DEHGHAN. The fourth-order time-discrete scheme and split-step direct meshless finite volume method for solving cubic-quintic complex Ginzburg-Landau equations on complicated geome-tries. Eng. Comput., 2022, 38: 1543-1557. |
| [31] | A. SHOKRI, E. BAHMANI. Direct meshless local petrov-galerkin (DMLPG) method for 2D complex Ginzburg-Landau equation. Eng. Anal. Bound. Elem., 2019, 100: 195-203. · Zbl 1464.65129 |
| [32] | Xiaolin LI, Shuling LI. A linearized element-free Galerkin method for the complex Ginzburg-Landau equation. Comput. Math. Appl., 2021, 90: 135-147. · Zbl 1524.65572 |
| [33] | F. ZABIHI, M. SAFFARIAN. A meshless method using radial basis functions for the numerical solution of two-dimensional ZK-BBM equation. Int. J. Appl. Math. Comput. Sci., 2017, 3(4): 4001-4013. · Zbl 1397.65212 |
| [34] | Guodong ZHENG, Yumin CHENG. The improved element-free Galerkin method for diffusional drug release problems. International J. Appl. Mech., 2020, 12(8): 2050096. |
| [35] | G. Z. TSERTSVADZE. On the convergence of difference schemes for the Kuramoto-Tsuzuki equation and for systems of reaction-diffusion type. Zh. Vychisl Mat. Mat. Fiz., 1991, 31: 698-707. · Zbl 0736.65071 |
| [36] | Weizhu BAO, Yongyong CAI. Uniform error estimates of finite difference methods for the nonlinear Schrödinger equation with wave operator. SIAM J. Numer. Anal., 2012, 50(2): 492-521. · Zbl 1246.35188 |
| [37] | Tingchun WANG, Jiaping JIANG, Xiang XUE. Unconditional and optimal H 1 error estimate of a Crank-Nicolson finite difference scheme for the Gross-Pitaevskii equation with an angular momentum rotation term. J. Math. Anal. Appl., 2018, 459(2): 945-958. · Zbl 1379.65066 |
| [38] | Yanan ZHANG, Zhizhong SUN, Tingchun WANG. Convergence analysis of a linearized Crank-Nicolson scheme for the two-dimensional complex Ginzburg-Landau equation. Numer. Methods Partial Differ. Equ., 2013, 29(5): 1487-1503. · Zbl 1274.65262 |
| [39] | Yulin ZHOU. Application of Discrete Functional Analysis to the Finite Difference Methods. International Academic Publishers, Bejing, 1991. · Zbl 0732.65080 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
