An unconditionally stable linearized difference scheme for the fractional Ginzburg-Landau equation. (English) Zbl 1402.65088
Summary: In this paper, we propose a linearized implicit finite difference scheme for solving the fractional Ginzburg-Landau equation. The scheme, which involves three time levels, is unconditionally stable and second-order accurate in both time and space variables. Moreover, the unique solvability, the unconditional stability, and the convergence of the method in the \(L^{\infty}\)-norm are proved by the energy method and mathematical induction. Compared with the implicit midpoint difference scheme [P. Wang and C. Huang, J. Comput. Phys. 312, 31–49 (2016; Zbl 1351.76191)], current linearized method generally reduces the computational cost. Finally, numerical results are presented to confirm the theoretical results.
MSC:
| 65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |
| 35Q56 | Ginzburg-Landau equations |
| 35R11 | Fractional partial differential equations |
| 65M15 | Error bounds 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:
Ginzburg-Landau equation; fractional Laplacian; pointwise error estimate; unconditional stability; convergenceCitations:
Zbl 1351.76191References:
| [1] | Aranson, IS; Kramer, L., The world of the complex Ginzburg-Landau equation, Rev. Mod. Phys., 74, 99-143, (2002) · Zbl 1205.35299 · doi:10.1103/RevModPhys.74.99 |
| [2] | Duan, J.; Titi, E.; Holmes, P., Regularity approximation and asymptotic dynamics for a generalized Ginzburg-Landau equation, Nonlinearity, 6, 915-933, (1993) · Zbl 0808.35133 · doi:10.1088/0951-7715/6/6/005 |
| [3] | Duan, J.; Holmes, P.; Titi, E., Global existence theory for a generalized Ginzburg-Landau equation, Nonlinearity, 5, 1303-1314, (1992) · Zbl 0783.35070 · doi:10.1088/0951-7715/5/6/005 |
| [4] | Doering, CR; Gibbon, JD; Levermore, CD, Weak and strong solutions of the complex Ginzburg-Landau equation, Physica D, 71, 285-318, (1994) · Zbl 0810.35119 · doi:10.1016/0167-2789(94)90150-3 |
| [5] | Gao, H.; Lin, G.; Duan, J., Asymptotics for the generalized two-dimensional Ginzburg-Landau equation, J. Math. Anal. Appl., 247, 198-216, (2000) · Zbl 0952.35035 · doi:10.1006/jmaa.2000.6848 |
| [6] | Guo, B.; Wang, G.; Li, D., The attractor of the stochastic generalized Ginzburg-Landau equation, Sci. China Ser. A, 51, 955-964, (2008) · Zbl 1154.35318 · doi:10.1007/s11425-007-0181-6 |
| [7] | Huo, Z.; Jia, Y., Global well-posedness for the generalized 2D Ginzburg-Landau equation, J. Differ. Equations, 247, 260-276, (2009) · Zbl 1173.35305 · doi:10.1016/j.jde.2009.03.015 |
| [8] | 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 |
| [9] | 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 |
| [10] | Milovanov, A.; Rasmussen, J., Fractional generalization of the Ginzburg-Landau equation: an unconventional approach to critical phenomena in complex media, Phys. Lett. A, 337, 75-80, (2005) · Zbl 1135.82320 · doi:10.1016/j.physleta.2005.01.047 |
| [11] | Mvogo, A.; Tambue, A.; Ben-Bolie, G.; Kofane, T., Localized numerical impulse solutions in diffuse neural networks modeled by the complex fractional Ginzburg-Landau equation, Commun. Nonlinear Sci., 39, 396-410, (2016) · Zbl 1510.35320 · doi:10.1016/j.cnsns.2016.03.008 |
| [12] | Yang, Q.; Liu, F.; Turner, I., Numerical methods for fractional partial differential equations with Riesz space fractional derivatives, Appl. Math. Model., 34, 200-218, (2010) · Zbl 1185.65200 · doi:10.1016/j.apm.2009.04.006 |
| [13] | Zhuang, P.; Liu, F.; Anh, V.; Turner, I., Numerical methods for the variable-order fractional advection-diffusion equation with a nonlinear source term, SIAM J. Numer. Anal., 47, 1760-1781, (2009) · Zbl 1204.26013 · doi:10.1137/080730597 |
| [14] | Wang, D.; Xiao, A.; Yang, W., Crank-nicolson difference scheme for the coupled nonlinear Schrödinger equations with the Riesz space fractional derivative, J. Comput. Phys., 242, 670-681, (2013) · Zbl 1297.65100 · doi:10.1016/j.jcp.2013.02.037 |
| [15] | Podlubny, I.: Fractional differential equations. Academic Press, San Diego (1999) · Zbl 0924.34008 |
| [16] | Xu, Y.; Agrawal, O., Numerical solutions and analysis of diffusion for new generalized fractional Burgers equation, Fract. Calc. Appl Anal., 16, 709-736, (2013) · Zbl 1312.65141 · doi:10.2478/s13540-013-0045-4 |
| [17] | He, D.; Pan, K., An unconditionally stable linearized CCD-ADI method for generalized nonlinear Schrödinger equations with variable coefficients in two and three dimensions, Comput. Math. Appl., 73, 2360-2374, (2017) · Zbl 1373.65056 · doi:10.1016/j.camwa.2017.04.009 |
| [18] | Tarasov, V., Psi-series solution of fractional Ginzburg-Landau equation, J. Phys. A-Math. Gen., 39, 8395-8407, (2006) · Zbl 1122.34003 · doi:10.1088/0305-4470/39/26/008 |
| [19] | Pu, X.; Guo, B., Well-posedness and dynamics for the fractional Ginzburg-Landau equation, Appl. Anal., 92, 318-334, (2013) · Zbl 1293.35310 · doi:10.1080/00036811.2011.614601 |
| [20] | 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, 226-242, (2013) · Zbl 1312.35180 |
| [21] | Lu, H.; Lö, S.; Feng, Z., Asymptotic dynamics of 2D fractional complex Ginzburg-Landau equation, Int. J. Bifurcat. Chaos, 13, 1350202, (2013) · Zbl 1284.35456 · doi:10.1142/S0218127413502027 |
| [22] | Millot, V.; Sire, Y., On a fractional Ginzburg-Landau equation and 1/2-harmonic maps into spheres, Arch. Ration. Mech. An., 215, 125-210, (2015) · Zbl 1372.35291 · doi:10.1007/s00205-014-0776-3 |
| [23] | Lord, GJ, Attractors and inertial manifolds for finite-difference approximations of the complex Ginzburg-Landau equation, SIAM J. Numer. Anal., 34, 1483-1512, (1997) · Zbl 0888.65104 · doi:10.1137/S003614299528554X |
| [24] | Xu, Q.; Chang, Q., Difference methods for computing the Ginzburg-Landau equation in two dimensions, Numer. Meth. Part. D. E., 27, 507-528, (2011) · Zbl 1218.65089 · doi:10.1002/num.20535 |
| [25] | Wang, T.; Guo, B., Analysis of some finite difference schemes for two-dimensional Ginzburg-Landau equation, Numer. Meth. Part. D. E., 27, 1340-1363, (2011) · Zbl 1233.65062 · doi:10.1002/num.20588 |
| [26] | Zhang, L., Long time behavior of difference approximations for the two-dimensional complex Ginzburg-Landau equation, Numer. Func. Anal. Opt., 31, 1190-1211, (2010) · Zbl 1410.65335 · doi:10.1080/01630563.2010.510974 |
| [27] | Zhang, Y.; Sun, Z.; Wang, T., Convergence analysis of a linearized Crank-Nicolson scheme for the two-dimensional complex Ginzburg-Landau equation, Numer. Meth. Part. D. E., 29, 1487-1503, (2013) · Zbl 1274.65262 · doi:10.1002/num.21763 |
| [28] | Hao, Z.; Sun, Z.; Cao, W., A three-level linearized compact difference scheme for the Ginzburg-Landau equation, Numer. Meth. Part. D. E., 31, 876-899, (2015) · Zbl 1320.65116 · doi:10.1002/num.21925 |
| [29] | 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 |
| [30] | Hao, Z.; Sun, Z., A linearized high-order difference scheme for the fractional Ginzburg-Landau equation, Numer. Meth. Part. D. E., 33, 105-124, (2017) · Zbl 1359.65150 · doi:10.1002/num.22076 |
| [31] | Meerschaert, MM; Tadjeran, C., Finite difference approximations for fractional advection-dispersion flow equations, J. Comput. Appl. Math., 172, 65-77, (2004) · Zbl 1126.76346 · doi:10.1016/j.cam.2004.01.033 |
| [32] | Meerschaert, MM; Scheffler, H-P; Tadjeran, C., Finite difference methods for two-dimensional fractional dispersion equation, J. Comput. Phys., 211, 249-261, (2006) · Zbl 1085.65080 · doi:10.1016/j.jcp.2005.05.017 |
| [33] | Meerschaert, MM; Tadjeran, C., Finite difference approximations for two-sided space-fractional partial differential equations, Appl. Numer. Math., 56, 80-90, (2006) · Zbl 1086.65087 · doi:10.1016/j.apnum.2005.02.008 |
| [34] | Ilić, M.; Liu, F.; Turner, I.; Anh, V., Numerical approximation of a fractional-in-space diffusion equation, Fract. Calc. Appl. Anal., 8, 323-341, (2005) · Zbl 1126.26009 |
| [35] | Yang, Q.; Turner, I.; Liu, F.; Ilis, M., Novel numerical methods for solving the time-space fractional diffusion equation in two dimensions, SIAM J. Sci. Comput., 33, 1159-1180, (2011) · Zbl 1229.35315 · doi:10.1137/100800634 |
| [36] | Tian, W.; Zhou, H.; Deng, W., A class of second order difference approximation for solving space fractional diffusion equations, Math. Comput., 84, 1703-1727, (2015) · Zbl 1318.65058 · doi:10.1090/S0025-5718-2015-02917-2 |
| [37] | Zhou, H.; Tian, W.; Deng, W., Quasi-compact finite difference schemes for space fractional diffusion equations, J. Sci. Comput., 56, 45-66, (2013) · Zbl 1278.65130 · doi:10.1007/s10915-012-9661-0 |
| [38] | Zhang, H.; Liu, F.; Anh, V., Galerkin finite element approximations of symmetric space-fractional partial differential equations, Appl. Math. Comput., 217, 2534-2545, (2010) · Zbl 1206.65234 |
| [39] | Ortigueira, Manuel Duarte, Riesz potential operators and inverses via fractional centred derivatives, International Journal of Mathematics and Mathematical Sciences, 2006, 1-12, (2006) · Zbl 1122.26007 · doi:10.1155/IJMMS/2006/48391 |
| [40] | 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 |
| [41] | Shen, S.; Liu, F.; Anh, V.; Turner, I.; Chen, J., A novel numerical approximation for the Riesz space fractional advection-dispersion equation, IMA J. Appl. Math., 79, 431-444, (2014) · Zbl 1297.65098 · doi:10.1093/imamat/hxs073 |
| [42] | 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 |
| [43] | Wang, D.; Xiao, A.; Yang, W., Maximum-norm error analysis of a difference scheme for the space fractional CNLS, Appl. Math. Comput., 257, 241-251, (2015) · Zbl 1339.65137 |
| [44] | Wang, P.; Huang, C., An energy conservative difference scheme for the nonlinear fractional Schrödinger equations, J. Comput. Phys., 293, 238-251, (2015) · Zbl 1349.65346 · doi:10.1016/j.jcp.2014.03.037 |
| [45] | Wang, P.; Huang, C., A conservative linearized difference scheme for the nonlinear fractional Schrödinger equation, Numer. Algorithms., 69, 625-641, (2015) · Zbl 1325.65127 · doi:10.1007/s11075-014-9917-x |
| [46] | Ran, M.; Zhang, C., A linearly implicit conservative scheme for the fractional nonlinear Schrödinger equation with wave operator, Int. J. Comput. Math., 93, 1103-1118, (2015) · Zbl 1390.65079 · doi:10.1080/00207160.2015.1016924 |
| [47] | 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 |
| [48] | Holte, JM, Discrete Gronwall lemma and applications, MAA-NCS meeting at the University of North Dakota, 24, 1-7, (2009) |
| [49] | He, D.; Pan, K., A linearly implicit conservative difference scheme for the generalized rosenau-kawahara-RLW equation, Appl. Math. Comput., 271, 323-336, (2015) · Zbl 1410.65312 |
| [50] | Akhmediev, NN; Afanasjev, VV; Soto-Crespo, JM, Singularities and special soliton solutions of the cubic-quintic complex Ginzburg-Landau equation, Phys. Rev. E, 53, 1190-1201, (1996) · doi:10.1103/PhysRevE.53.1190 |
| [51] | Pan, K.; He, D.; Hu, H.; Ren, Z., A new extrapolation cascadic multigrid method for three dimensional elliptic boundary value problems, J. Comput. Phys., 344, 499-515, (2017) · Zbl 1380.65407 · doi:10.1016/j.jcp.2017.04.069 |
| [52] | Pan, K.; He, D.; Hu, H., An extrapolation cascadic multigrid method combined with a fourth-order compact scheme for 3D Poisson equation, J. Sci. Comput., 70, 1180-1203, (2017) · Zbl 1366.65095 · doi:10.1007/s10915-016-0275-9 |
| [53] | Hu, H.; Ren, Z.; He, D.; Pan, K., On the convergence of an extrapolation cascadic multigrid method for elliptic problems, Comput. Math. Appl., 74, 759-771, (2017) · Zbl 1384.65087 · doi:10.1016/j.camwa.2017.05.023 |
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