×

Pointwise error estimate in difference setting for the two-dimensional nonlinear fractional complex Ginzburg-Landau equation. (English) Zbl 1480.65230

Summary: In this paper, we propose a three-level linearized implicit difference scheme for the two-dimensional spatial fractional nonlinear complex Ginzburg-Landau equation. We prove that the difference scheme is stable and convergent under mild conditions. The optimal convergence order \(\mathcal{O}(\tau^2+{h_x^2}+{h_y^2})\) is obtained in the pointwise sense by developing a new two-dimensional fractional Sobolev imbedding inequality based on the work in [K. Kirkpatrick et al., Commun. Math. Phys. 317, No. 3, 563–591 (2013; Zbl 1258.35182)], an energy argument and careful attention to the nonlinear term. Numerical examples are presented to verify the validity of the theoretical results for different choices of the fractional orders \(\alpha\) and \(\beta\).

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
65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
26A33 Fractional derivatives and integrals
35R11 Fractional partial differential equations
35Q56 Ginzburg-Landau equations

Citations:

Zbl 1258.35182
PDFBibTeX XMLCite
Full Text: DOI Link

References:

[1] Akhmediev, NN; Ankiewicz, A.; Soto-Crespo, JM, Multisoliton solutions of the complex Ginzburg-Landau equation, Phys. Rev. Lett., 79, 21, 4047-4051 (1997) · Zbl 0947.35151 · doi:10.1103/PhysRevLett.79.4047
[2] Aranson, IS; Kramer, L., The world of the complex Ginzburg-Landau equation, Rev. Modern Phys., 74, 99-143 (2002) · Zbl 1205.35299 · doi:10.1103/RevModPhys.74.99
[3] Arshed, S., Soliton solutions of fractional complex Ginzburg-Landau equation with Kerr law and non-Kerr law media, Optik, 160, 322-332 (2018) · doi:10.1016/j.ijleo.2018.02.022
[4] Bao, W.; Tang, Q., Numerical study of quantized vortex interaction in the Ginzburg-Landau equation on bounded domains, Commun. Comput. Phys., 14, 3, 819-850 (2013) · Zbl 1388.65046 · doi:10.4208/cicp.250112.061212a
[5] Bartels, S., A posteriori error analysis for time-dependent Ginzburg-Landau type equations, Numer. Math., 99, 557-583 (2005) · Zbl 1073.65089 · doi:10.1007/s00211-004-0560-7
[6] Çelik, C.; Duman, M., Crank-nicolson method for the fractional diffusion equation with the Riesz fractional derivative, J. Comput. Phys., 231, 1743-1750 (2012) · Zbl 1242.65157 · doi:10.1016/j.jcp.2011.11.008
[7] Defterli, O.; D’Elia, M.; Du, Q.; Gunzburger, M.; Lehoucq, R.; Meerschaert, MM, Fractional diffusion on bounded domains, Fract. Calc. Appl. Anal., 18, 342-360 (2015) · Zbl 1488.35557 · doi:10.1515/fca-2015-0023
[8] Degong, P.; Jin, S.; Tang, M., On the time splitting spectral method for the complex Ginzburg-Landau equation in the large time and space scale limit, SIAM J. Sci. Comput., 30, 5, 2466-2487 (2008) · Zbl 1176.35170 · doi:10.1137/070700711
[9] Du, Q.; Gunzburger, M.; Lehoucq, RB; Zhou, K., Analysis and approximation of nonlocal diffusion problems with volume constraints, SIAM Rev., 54, 667-696 (2012) · Zbl 1422.76168 · doi:10.1137/110833294
[10] Du, Q.; Gnnzburger, M.; Peterson, J., Finite element approximation of a periodic Ginzburg-Landau model for type-II superconductors, Numer. Math., 64, 85-114 (1993) · Zbl 0792.65095 · doi:10.1007/BF01388682
[11] Fei, M.; Huang, C.; Wang, N.; Zhang, G., Galerkin-Legendre spectral method for the nonlinear Ginzburg-Landau equation with the Riesz fractional derivative, Math. Meth. Appl. Sci., 15, 2711-2730 (2021) · Zbl 1486.65192 · doi:10.1002/mma.5852
[12] Feynman, RP; Hibbs, AR, Quantum mechanics and path integrals (1965), New York: McGraw-Hill, New York · Zbl 0176.54902
[13] Gorenflo, R.; Mainardi, F., Random walk models for space-fractional diffusion processes, Fract. Calc. Appl. Anal., 1, 2, 167-191 (1998) · Zbl 0946.60039
[14] Gu, X-M; Shi, L.; Liu, T., Well-posedness of the fractional Ginzburg-Landau equation, Appl. Anal., 98, 14, 2545-2558 (2019) · Zbl 1422.35111 · doi:10.1080/00036811.2018.1466281
[15] Guo, B.; Huo, Z., Global well-posedness for the fractional nonlinear Schrödinger equation, Commun. Partial Differ. Equ., 36, 247-255 (2011) · Zbl 1211.35268 · doi:10.1080/03605302.2010.503769
[16] Guo, B.; Huo, Z., Well-posedness for the nonlinear fractional schrödinger equation and inviscid limit behavior of solution for the fractional Ginzburg-Landau equation, Fract. Calc. Appl. Anal., 16, 1, 226-242 (2013) · Zbl 1312.35180
[17] Hao, ZP; Sun, ZZ, A linearized high-order difference scheme for the fractional Ginzburg-Landau equation, Numer. Meth. Part. D. E., 33, 1, 105-124 (2017) · Zbl 1359.65150 · doi:10.1002/num.22076
[18] He, D.; Pan, K., An unconditionally stable linearized difference scheme for the fractional Ginzburg-Landau equation, Numer. Algorithms, 79, 3, 899-925 (2018) · Zbl 1402.65088 · doi:10.1007/s11075-017-0466-y
[19] Heydari, MH; Hosseininia, M.; Atangana, A.; Avazzadeh, Z., A meshless approach for solving nonlinear variable-order time fractional 2D Ginzburg-Landau equation, Eng. Anal. Bound. Elem., 120, 166-179 (2020) · Zbl 1464.65143 · doi:10.1016/j.enganabound.2020.08.015
[20] Hu, H.L., Jin, X., He, D., Pan, K., Zhang, Q.: A conservative difference scheme with optimal pointwise error estimates for two-dimensional space fractional nonlinear Schrödinger equations. arXiv:1910.08311
[21] Kirkpatrick, K.; Lenzmann, E.; Staffilani, G., On the continuum limit for discrete NLS with long-range lattice interactions, Commun. Math. Phys., 317, 563-591 (2013) · Zbl 1258.35182 · doi:10.1007/s00220-012-1621-x
[22] Kitzhofer, G.; Koch, O.; Weinmüller, EB, Pathfollowing for essentially singular boundary value problems with application to the complex Ginzburg-Landau equation, BIT Numer. Math., 49, 141-160 (2009) · Zbl 1162.65372 · doi:10.1007/s10543-008-0208-6
[23] Laskin, N., Fractional quantum mechanics, Phys. Rev. E, 62, 3135-3145 (2000) · Zbl 0948.81595 · doi:10.1103/PhysRevE.62.3135
[24] Laskin, N., Fractional quantum mechanics and lévy path integrals, Phys. Lett. A, 268, 298-305 (2000) · Zbl 0948.81595 · doi:10.1016/S0375-9601(00)00201-2
[25] Li, J.; Xia, L., Well-posedness of fractional Ginzburg-Landau equation in Sobolev spaces, Appl. Anal., 92, 5, 1074-1084 (2013) · Zbl 1291.35423 · doi:10.1080/00036811.2011.649733
[26] Li, M.; Huang, C., An efficient difference scheme for the coupled nonlinear fractional Ginzburg-Landau equations with the fractional Laplacian, Numer. Meth. Part. D. E., 35, 1, 394-421 (2019) · Zbl 1419.65024 · doi:10.1002/num.22305
[27] Li, M.; Huang, C.; Wang, N., Galerkin element method for the nonlinear fractional Ginzburg-Landau equation, Appl. Numer. Math., 118, 131-149 (2017) · Zbl 1367.65144 · doi:10.1016/j.apnum.2017.03.003
[28] Lu, H.; Bates, PW; Lü, SJ; Zhang, MJ, Dynamics of the 3-D fractional complex Ginzburg-Landau equation, J. Differ. Equations, 259, 5276-5301 (2015) · Zbl 1328.35279 · doi:10.1016/j.jde.2015.06.028
[29] Lu, H.; Lü, SJ; Zhang, MJ, Fourier spectral approximations to the dynamics of 3D fractional complex Ginzburg-Landau equation, Discrete Cont. Dyn. S.-A, 37, 5, 2539-2564 (2017) · Zbl 1357.65196 · doi:10.3934/dcds.2017109
[30] Millot, V.; Sire, Y., On a fractional Ginzburg-Landau equation and 1/2-Harmonic maps into spheres, Arch. Ration. Mech. Anal., 215, 125-210 (2015) · Zbl 1372.35291 · doi:10.1007/s00205-014-0776-3
[31] Mohebbi, A., Fast and high-order numerical algorithms for the solution of multidimensional nonlinear fractional Ginzburg-Landau equation, Eur. Phys. J. Plus, 133, 2, 67 (2018) · doi:10.1140/epjp/i2018-11846-x
[32] Owolabi, KM; Pindza, E., Numerical simulation of multidimensional nonlinear fractional Ginzburg-Landau equations, Discret. Contin. Dyn. Syst.-Ser. S, 13, 3, 835-851 (2020) · Zbl 1442.65321 · doi:10.3934/dcdss.2020048
[33] Peng, L.; Zhou, Y.; Ahmad, B., The well-posedness for fractional nonlinear schrödinger equations, Comput. Math. Appl., 77, 1998-2005 (2019) · Zbl 1442.35521 · doi:10.1016/j.camwa.2018.11.037
[34] Pu, X.; Guo, B., Well-posedness and dynamics for the fractional Ginzburg-Landau equation, Appl. Anal., 92, 2, 31-33 (2013) · Zbl 1293.35310 · doi:10.1080/00036811.2011.614601
[35] Sakaguchi, H.; Malomed, BA, Stable solitons in coupled Ginzburg-Landau equations describing Bose-Einstein condensates and nonlinear optical waveguides and cavities, Physica D, 183, 282-292 (2003) · Zbl 1038.35127 · doi:10.1016/S0167-2789(03)00181-7
[36] Shu, J.; Li, P.; Zhang, J.; Liao, O., Random attractors for the stochastic coupled fractional Ginzburg-Landau equation with additive noise, J. Math. Phys., 56, 102702 (2015) · Zbl 1341.35153 · doi:10.1063/1.4934724
[37] Sun, ZZ, A second-order accurate linearized difference scheme for the two-dimensional Cahn-Hilliard equation, Math. Comp., 64, 1463-1471 (1995) · Zbl 0847.65056
[38] Takac, P.; Jungel, A., A nonstiff Euler discretization of the complex Ginzburg-Landau equation in one space dimension, SIAM J. Numer. Anal., 38, 1, 292-328 (2000) · Zbl 0965.65111 · doi:10.1137/S0036142998332852
[39] Tarasov, V.; Zaslavsky, G., Fractional dynamics of coupled oscillators with long-range interaction, Chaos, 16, 023110 (2006) · Zbl 1152.37345 · doi:10.1063/1.2197167
[40] Tarasov, V.; Zaslavsky, G., Fractional Ginzburg-Landau equation for fractal media, Physica A, 354, 249-261 (2005) · doi:10.1016/j.physa.2005.02.047
[41] Wang, N.; Huang, C., An efficient split-step quasi-compact finite difference method for the nonlinear fractional Ginzburg-Landau equations, Comput. Math. Appl., 75, 2223-2242 (2018) · Zbl 1409.65057 · doi:10.1016/j.camwa.2017.12.005
[42] Wang, P.; Huang, C., An efficient fourth-order in space difference scheme for the nonlinear fractional Ginzburg-Landau equation, BIT Numer. Math., 58, 3, 783-805 (2018) · Zbl 1412.65095 · doi:10.1007/s10543-018-0698-9
[43] Wang, P.; Huang, C., An implicit midpoint difference scheme for the fractional Ginzburg-Landau equation, J. Comput. Phys., 312, 31-49 (2016) · Zbl 1351.76191 · doi:10.1016/j.jcp.2016.02.018
[44] Wang, P.; Huang, C., Split-step alternating direction implicit difference scheme for the fractional schrödinger equation in two dimensions, Comput. Math. Appl., 71, 5, 1114-1128 (2016) · Zbl 1443.65145 · doi:10.1016/j.camwa.2016.01.022
[45] Wang, P.; Huang, C.; Zhao, L., Point-wise error estimate of a conservative difference scheme for the fractional schrödinger equation, J. Comput. Appl. Math., 306, 231-247 (2016) · Zbl 1382.65260 · doi:10.1016/j.cam.2016.04.017
[46] Wu, YS, Multiparticle quantum mechanics obeying fractional statistics, Phys. Rev. Lett., 53, 111-114 (1984) · doi:10.1103/PhysRevLett.53.111
[47] Zaky, M.A., Hendy, A.S., Macas-Daz, J.E.: High-order finite difference/spectral-Galerkin approximations for the nonlinear time-space fractional Ginzburg-Landau equation. Eng. Anal. Bound Elem. doi:10.1002/num.22630 (2020)
[48] Zeng, F.; Liu, F.; Li, C.; Burrage, K.; Turner, I.; Anh, V., A Crank-Nicolson ADI spectral method for a two-dimensional Riesz space fractional nonlinear reaction-diffusion equation, SIAM J. Numer. Anal., 52, 6, 2599-2622 (2014) · Zbl 1382.65349 · doi:10.1137/130934192
[49] Zeng, W.; Xiao, A.; Li, X., Error estimate of Fourier pseudo-spectral method for multidimensional nonlinear complex fractional Ginzburg-Landau, Equations. Appl. Math. Lett., 93, 40-45 (2019) · Zbl 1414.65026 · doi:10.1016/j.aml.2019.01.041
[50] Zhang, L., Zhang, Q., Sun, H. -W.: Exponential Runge-Kutta method for two-dimensional nonlinear fractional complex Ginzburg-Landau equations. J. Sci. Comput 83. Article 59, doi:10.1007/s10915-020-01240-x (2020) · Zbl 1442.65340
[51] Zhang, M.; Zhang, G-F; Liao, L-D, Fast iterative solvers and simulation for the space fractional Ginzburg-Landau equations Ginzburg-Landau equations, Comput. Math. Appl., 78, 5, 1793-1800 (2019) · Zbl 1442.65187 · doi:10.1016/j.camwa.2019.01.026
[52] Zhang, Q.; Lin, X.; Pan, K.; Ren, Y., Linearized ADI schemes for two-dimensional space-fractional nonlinear Ginzburg-Landau equation, Comput. Math. Appl., 80, 1201-1220 (2020) · Zbl 1447.65042 · doi:10.1016/j.camwa.2020.05.027
[53] Zhang, Q., Zhang, L., Sun, H.: A three-level finite difference method with preconditioning technique for two-dimensional nonlinear fractional complex Ginzburg-Landau equations. J. Comput. Appl. Math. 389doi:10.1016/j.cam.2020.113355 (2021) · Zbl 1462.65119
[54] Zhang, Z.; Li, M.; Wang, Z., A linearized Crank-Nicolson Galerkin FEMs for the nonlinear fractional Ginzburg-Landau equation, Appl. Anal., 98, 15, 2648-2667 (2019) · Zbl 1432.65151 · doi:10.1080/00036811.2018.1469008
[55] Zhao, X.; Sun, Z.; Hao, Z., A fourth-order compact ADI scheme for two-dimensional nonlinear space fractional schrödinger equation, SIAM J. Sci. Comput., 36, 6, 2865-2886 (2014) · Zbl 1328.65187 · doi:10.1137/140961560
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.