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Article

Zhang, Chun-Hua ; Jin, Jun-Wei ; Sun, Hai-Wei ; Sheng, Qin
A spatially sixth-order hybrid $L1$-CCD method for solving time fractional Schrödinger equations. (English). Applications of Mathematics, vol. 66 (2021), issue 2, pp. 213-232
MSC: 65M06, 65M20, 65M60 | MR 4226457 | Zbl 07332696 | DOI: 10.21136/AM.2020.0339-19
Keywords:
nonlinear time fractional Schrödinger equations; $L1$ formula; hybrid compact difference method; linearization; unconditional stability
Summary:
We consider highly accurate schemes for nonlinear time fractional Schrödinger equations (NTFSEs). While an $L1$ strategy is employed for approximating the Caputo fractional derivative in the temporal direction, compact CCD finite difference approaches are incorporated in the space. A highly effective hybrid $L1$-CCD method is implemented successfully. The accuracy of this linearized scheme is order six in space, and order $2-\gamma $ in time, where $0<\gamma <1$ is the order of the Caputo fractional derivative involved. It is proved rigorously that the hybrid numerical method accomplished is unconditionally stable in the Fourier sense. Numerical experiments are carried out with typical testing problems to validate the effectiveness of the new algorithms.
References:
[1] Baleanu, D., Diethelm, K., Scalas, E., Trujillo, J. J.: Fractional Calculus: Models and Numerical Methods. Series on Complexity, Nonlinearity and Chaos 5. World Scientific, Hackensack (2012). DOI 10.1142/8180 | MR 2894576 | Zbl 1248.26011
[2] Benson, D. A., Schumer, R., Meerschaert, M. M., Wheatcraft, S. W.: Fractional dispersion, Lévy motion, and the MADE tracer tests. Transp. Porous Media 42 (2001), 211-240. DOI 10.1023/A:1006733002131 | MR 1948593
[3] Bhrawy, A. H., Doha, E. H., Ezz-Eldien, S. S., Gorder, R. A. Van: A new Jacobi spectral collocation method for solving 1+1 fractional Schrödinger equations and fractional coupled Schrödinger systems. Eur. Phys. J. Plus 129 (2014), Article ID 260, 21 pages. DOI 10.1140/epjp/i2014-14260-6 | MR 2930392
[4] Bratsos, A. G.: A linearized finite-difference scheme for the numerical solution of the nonlinear cubic Schrödinger equation. Korean J. Comput. Appl. Math. 8 (2001), 459-467. DOI 10.1007/BF02941979 | MR 1848910 | Zbl 1015.65042
[5] Chang, Q., Jia, E., Sun, W.: Difference schemes for solving the generalized nonlinear Schrödinger equation. J. Comput. Phys. 148 (1999), 397-415. DOI 10.1006/jcph.1998.6120 | MR 1669707 | Zbl 0923.65059
[6] Chen, X., Di, Y., Duan, J., Li, D.: Linearized compact ADI schemes for nonlinear timefractional Schrödinger equations. Appl. Math. Lett. 84 (2018), 160-167. DOI 10.1016/j.aml.2018.05.007 | MR 3808513 | Zbl 06892656
[7] Chen, B., He, D., Pan, K.: A linearized high-order combined compact difference scheme for multi-dimensional coupled Burgers' equations. Numer. Math., Theory Methods Appl. 11 (2018), 299-320. DOI 10.4208/nmtma.OA-2017-0090 | MR 3849456 | Zbl 1424.65124
[8] Chen, B., He, D., Pan, K.: A CCD-ADI method for two-dimensional linear and nonlinear hyperbolic telegraph equations with variable coefficients. Int. J. Comput. Math. 96 (2019), 992-1004. DOI 10.1080/00207160.2018.1478415 | MR 3911287
[9] Chu, P. C., Fan, C.: A three-point combined compact difference scheme. J. Comput. Phys. 140 (1998), 370-399. DOI 10.1006/jcph.1998.5899 | MR 1616146 | Zbl 0923.65071
[10] Cui, M.: Compact finite difference method for the fractional diffusion equation. J. Comput. Phys. 228 (2009), 7792-7804. DOI 10.1016/j.jcp.2009.07.021 | MR 2561843 | Zbl 1179.65107
[11] Dehghan, M., Taleei, A.: A compact split-step finite difference method for solving the nonlinear Schrödinger equations with constant and variable coefficients. Comput. Phys. Commun. 181 (2010), 43-51. DOI 10.1016/j.cpc.2009.08.015 | MR 2575184 | Zbl 1206.65207
[12] Feynman, R. P., Hibbs, A. R.: Quantum Mechanics and Path Integrals. Dover Publications, New York (2010). MR 2797644 | Zbl 1220.81156
[13] Gao, G.-H., Sun, H.-W.: Three-point combined compact alternating direction implicit difference schemes for two-dimensional time-fractional advection-diffusion equations. Commun. Comput. Phys. 17 (2015), 487-509. DOI 10.4208/cicp.180314.010914a | MR 3371511 | Zbl 1388.65052
[14] Gao, G.-H., Sun, H.-W.: Three-point combined compact difference schemes for time-fractional advection-diffusion equations with smooth solutions. J. Comput. Phys. 298 (2015), 520-538. DOI 10.1016/j.jcp.2015.05.052 | MR 3374563 | Zbl 1349.65294
[15] Gao, G.-H., Sun, H.-W., Sun, Z.-Z.: Stability and convergence of finite difference schemes for a class of time-fractional sub-diffusion equations based on certain superconvergence. J. Comput. Phys. 280 (2015), 510-528. DOI 10.1016/j.jcp.2014.09.033 | MR 3273149 | Zbl 1349.65295
[16] Gao, G.-H., Sun, Z.-Z.: A compact finite difference scheme for the fractional sub-diffusion equations. J. Comput. Phys. 230 (2011), 586-595. DOI 10.1016/j.jcp.2010.10.007 | MR 2745445 | Zbl 1211.65112
[17] Gao, G.-H., Sun, Z.-Z., Zhang, H.-W.: A new fractional numerical differentiation formula to approximate the Caputo fractional derivative and its applications. J. Comput. Phys. 259 (2014), 33-50. DOI 10.1016/j.jcp.2013.11.017 | MR 3148558 | Zbl 1349.65088
[18] Golmankhaneh, A. K., Baleanu, D.: Calculus on fractals. Fractional Dynamics De Gruyter, Berlin (2015), 307-332. DOI 10.1515/9783110472097-019 | MR 3468175 | Zbl 1333.00024
[19] He, D.: An unconditionally stable spatial sixth-order CCD-ADI method for the twodimensional linear telegraph equation. Numer. Algorithms 72 (2016), 1103-1117. DOI 10.1007/s11075-015-0082-7 | MR 3529835 | Zbl 1350.65088
[20] He, D., Pan, K.: A fifth-order combined compact difference scheme for the Stokes flow on polar geometries. East Asian J. Appl. Math. 7 (2017), 714-727. DOI 10.4208/eajam.200816.300517a | MR 3766403 | Zbl 1426.76471
[21] 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 (2017), 2360-2374. DOI 10.1016/j.camwa.2017.04.009 | MR 3648018 | Zbl 1373.65056
[22] Jones, T. N., Sheng, Q.: Asymptotic stability of a dual-scale compact method for approximating highly oscillatory Helmholtz solutions. J. Comput. Phys. 392 (2019), 403-418. DOI 10.1016/j.jcp.2019.04.046 | MR 3948727
[23] Khan, N. A., Jamil, M., Ara, A.: Approximate solutions to time-fractional Schrödinger equation via homotopy analysis method. Int. Sch. Res. Not. 2012 (2012), Article ID 197068, 11 pages. DOI 10.5402/2012/197068
[24] Klages, R., Radons, G., (eds.), I. M. Sokolov: Anomalous Transport: Foundations and Applications. Wiley, Weinheim (2008). DOI 10.1002/9783527622979
[25] Laskin, N.: Fractional quantum mechanics and Lévy path integrals. Phys. Lett., A 268 (2000), 298-305. DOI 10.1016/S0375-9601(00)00201-2 | MR 1755089 | Zbl 0948.81595
[26] Lee, S. T., Liu, J., Sun, H.-W.: Combined compact difference scheme for linear second-order partial differential equations with mixed derivative. J. Comput. Appl. Math. 264 (2014), 23-37. DOI 10.1016/j.cam.2014.01.004 | MR 3164100 | Zbl 1294.65098
[27] Li, L. Z., Sun, H.-W., Tam, S.-C.: A spatial sixth-order alternating direction implicit method for two-dimensional cubic nonlinear Schrödinger equations. Comput. Phys. Commun. 187 (2015), 38-48. DOI 10.1016/j.cpc.2014.10.008 | MR 3283134 | Zbl 1348.35238
[28] Li, D., Wang, J., Zhang, J.: Unconditionally convergent $L1$-Galerkin FEMs for nonlinear time-fractional Schrödinger equations. SIAM. J. Sci. Comput. 39 (2017), A3067--A3088. DOI 10.1137/16M1105700 | MR 3738320 | Zbl 1379.65079
[29] Liao, H.-L., Shi, H.-S., Zhao, Y.: Numerical study of fourth-order linearized compact schemes for generalized NLS equations. Comput. Phys. Commun. 185 (2014), 2240-2249. DOI 10.1016/j.cpc.2014.05.002 | MR 3211950 | Zbl 1344.35137
[30] Mohebbi, A., Abbaszadeh, M., Dehghan, M.: The use of a meshless technique based on collocation and radial basis functions for solving the time fractional nonlinear Schrödinger equation arising in quantum mechanics. Eng. Anal. Bound. Elem. 37 (2013), 475-485. DOI 10.1016/j.enganabound.2012.12.002 | MR 3018826 | Zbl 1352.65397
[31] Murio, D. A.: Implicit finite difference approximation for time fractional diffusion equations. Comput. Math. Appl. 56 (2008), 1138-1145. DOI 10.1016/j.camwa.2008.02.015 | MR 2437884 | Zbl 1155.65372
[32] Naber, M.: Time fractional Schrödinger equation. J. Math. Phys. 45 (2004), 3339-3352. DOI 10.1063/1.1769611 | MR 2077514 | Zbl 1071.81035
[33] Rida, S. Z., El-Sherbiny, H. M., Arafa, A. A. M.: On the solution of the fractional nonlinear Schrödinger equation. Phys. Lett., A 372 (2008), 553-558. DOI 10.1016/j.physleta.2007.06.071 | MR 2378723 | Zbl 1217.81068
[34] Scalas, E., Gorenflo, R., Mainardi, F.: Fractional calculus and continuous-time finance. Phys. A 284 (2000), 376-384. DOI 10.1016/S0378-4371(00)00255-7 | MR 1773804
[35] Shivanian, E., Jafarabadi, A.: Error and stability analysis of numerical solution for the time fractional nonlinear Schrödinger equation on scattered data of general-shaped domains. Numer. Methods Partial Differ. Equations 33 (2017), 1043-1069. DOI 10.1002/num.22126 | MR 3652177 | Zbl 1379.65082
[36] Sun, H.-W., Li, L. Z.: A CCD-ADI method for unsteady convection-diffusion equations. Comput. Phys. Commun. 185 (2014), 790-797. DOI 10.1016/j.cpc.2013.11.009 | MR 3161142 | Zbl 1360.35193
[37] Sun, Z.-Z., Wu, X.: A fully discrete difference scheme for a diffusion-wave system. Appl. Numer. Math. 56 (2006), 193-209. DOI 10.1016/j.apnum.2005.03.003 | MR 2200938 | Zbl 1094.65083
[38] Wang, Y.-M., Ren, L.: Efficient compact finite difference methods for a class of timefractional convection-reaction-diffusion equations with variable coefficients. Int. J. Comput. Math. 96 (2019), 264-297. DOI 10.1080/00207160.2018.1437262 | MR 3893479
[39] Wang, Z., Vong, S.: Compact difference schemes for the modified anomalous fractional sub-diffusion equation and the fractional diffusion-wave equation. J. Comput. Phys. 277 (2014), 1-15. DOI 10.1016/j.jcp.2014.08.012 | MR 3254221 | Zbl 1349.65348
[40] Wei, L., He, Y., Zhang, X., Wang, S.: Analysis of an implicit fully discrete local discontinuous Galerkin method for the time-fractional Schrödinger equation. Finite Elem. Anal. Des. 59 (2012), 28-34. DOI 10.1016/j.finel.2012.03.008 | MR 2949417
[41] Xu, Y., Zhang, L.: Alternating direction implicit method for solving two-dimensional cubic nonlinear Schrödinger equation. Comput. Phys. Commun. 183 (2012), 1082-1093. DOI 10.1016/j.cpc.2012.01.006 | MR 2888993 | Zbl 1277.65073
[42] Zhu, L., Sheng, Q.: A note on the adaptive numerical solution of a Riemann-Liouville space-fractional Kawarada problem. J. Comput. Appl. Math. 374 (2020), Article ID 112714, 14 pages. DOI 10.1016/j.cam.2020.112714 | MR 4066721 | Zbl 1435.65136
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