Lie-Trotter operator splitting spectral method for linear semiclassical fractional Schrödinger equation. (English) Zbl 1504.65227
Summary: In this paper the error estimates are derived for Lie-Trotter operator splitting spectral method for semiclassical linear fractional Schrödinger equation. We first establish a priori estimates for the solution in fractional Sobolev space. On the basis of these a priori estimates, we obtain local error bound for the well-known Lie-Trotter splitting operator associated with the linear fractional Schrödinger equation in the semiclassical regime by using a formula for the fractional Laplacian of the product of two functions. The convergence orders of the fully discrete scheme based on Fourier spectral methods for the space approximation are then analyzed and provided with respect to the time step-size \(\Delta t\) and the small (scaled) Planck constant \(\varepsilon\). As far as we know, this is the first rigorous proof of local error estimate for an semi-discrete operator splitting method for semiclassical fractional Schrödinger equation. Computational experiments confirm the theoretical results.
MSC:
| 65M70 | Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs |
| 35R11 | Fractional partial differential equations |
| 65J08 | Numerical solutions to abstract evolution equations |
| 65M15 | Error bounds for initial value and initial-boundary value problems involving PDEs |
Keywords:
fractional Schrödinger equation; a priori estimates; Lie-Trotter operator splitting; semiclassical regime; error estimates; spectral methodsReferences:
| [1] | Ainsworth, M.; Mao, Z. P., Analysis and approximation of a fractional Cahn-Hilliard equation, SIAM J. Numer. Anal., 55, 1689-1718 (2017) · Zbl 1369.65124 |
| [2] | Antoine, X.; Tang, Q.; Zhang, J., On the numerical solution and dynamical laws of nonlinear fractional Schrödinger/Gross-Pitaevskii equations, Int. J. Comput. Math., 95, 1423-1443 (2018) · Zbl 1499.35522 |
| [3] | Antoine, X.; Tang, Q.; Zhang, Y., On the ground states and dynamics of space fractional nonlinear Schrödinger/Gross-Pitaevskii equations with rotation term and nonlocal nonlinear interactions, J. Comput. Phys., 325, 74-97 (2016) · Zbl 1380.65296 |
| [4] | Bader, P.; Iserles, A.; Kropielnicka, K.; Singh, P., Efficient approximation for the semiclassical Schrödinger equation, Found. Comput. Math., 14, 689-720 (2014) · Zbl 1302.65230 |
| [5] | Bao, W.; Jin, S.; Markowich, P. A., On time-splitting spectral approximations for the Schrödinger equation in the semiclassical regime, J. Comput. Phys., 175, 487-524 (2002) · Zbl 1006.65112 |
| [6] | Bao, W.; Jin, S.; Markowich, P. A., Numerical study of time-splitting spectral discretizations of nonlinear Schrodinger equations in the semiclassical regimes, SIAM J. Sci. Comput., 25, 27-64 (2003) · Zbl 1038.65099 |
| [7] | U. Biccari, A.B. Aceves, WKB expansion for a fractional Schrödinger equation with applications to controllability, 2018, hal-01758576. |
| [8] | Biccari, U.; Warma, M.; Zuazua, E., Local elliptic regularity for the Dirichlet fractional Laplacian, Adv. Nonlinear Stud., 17, 387-409 (2017) · Zbl 1360.35033 |
| [9] | Chen, S. S.; Tian, B.; Chai, J.; Wu, X. Y.; Du, Z., Lax pair, binary Darboux transformations and dark-soliton interaction of a fifth-order defocusing nonlinear Schrödinger equation for the attosecond pulses in the optical fiber communication, Waves Random Complex Media, 30, 389-402 (2020) · Zbl 1498.78032 |
| [10] | Du, X. X.; Tian, B.; Qu, Q. X.; Yuan, Y. Q.; Zhao, X-H., Lie group analysis, solitons, self-adjointness and conservation laws of the modified Zakharov-Kuznetsov equation in an electron-positron-ion magnetoplasma, Chaos Solitons Fractals, 134, Article 109709 pp. (2020) · Zbl 1483.35177 |
| [11] | Descombes, S.; Thalhammer, M., An exact local error representation of exponential operator splitting methods for evolutionary problems and applications to linear Schrödinger equations in the semi-classical regime, BIT Numer. Math., 50, 729-749 (2010) · Zbl 1205.65250 |
| [12] | Descombes, S.; Thalhammer, M., The Lie-Trotter splitting method for nonlinear evolutionary problems involving critical parameters. An exact local error representation and applications to nonlinear Schrödinger equations in the semi-classical regime, IMA J. Numer. Anal., 33, 722-745 (2013) · Zbl 1266.65153 |
| [13] | Duo, S.; Zhang, Y., Mass-conservative Fourier spectral methods for solving the fraction nonlinear Schrödinger equation, Comput. Math. Appl., 71, 2257-2271 (2016) · Zbl 1443.65242 |
| [14] | Faou, E., Geometric Numerical Integration and Schrödinger Equations, Zurich Lectures in Advanced Mathematics (2012), Europ. Math. Soc.: Europ. Math. Soc. Zürich · Zbl 1239.65078 |
| [15] | Ford, N. J.; Rodrigues, M. M.; Vieira, N., A numerical method for the fractional Schrödinger type equation of spatial dimension two, Fract. Calc. Appl. Anal., 16, 454-468 (2013) · Zbl 1312.65132 |
| [16] | Fröhlich, J.; Jonsson, B. L.G.; Lenzmann, E., Boson stars as solitary waves, Commun. Math. Phys., 274, 1-30 (2007) · Zbl 1126.35064 |
| [17] | Guo, B.; Pu, X. K.; Huang, F. H., Fractional Partial Differential Equations and Their Numerical Solutions (2008), Science Press: Science Press Beijing |
| [18] | Guo, B.; Huang, D., Existence and stability of standing waves for nonlinear fractional Schrödinger equations, J. Math. Phys., 53, Article 083702 pp. (2012) · Zbl 1278.35229 |
| [19] | Gao, X. Y.; Guo, Y. J.; Shan, W. R., Hetero-Bäcklund transformation and similarity reduction of an extended (2+1)-dimensional coupled Burgers system in fluid mechanics, Phys. Lett. A, 384, Article 126788 pp. (2020) · Zbl 1448.37084 |
| [20] | Gao, X. Y.; Guo, Y. J.; Shan, W. R., Shallow water in an open sea or a wide channel: auto- and non-auto-Bäcklund transformations with solitons for a generalized (2+1)-dimensional dispersive long-wave system, Chaos Solitons Fractals, 138, Article 109950 pp. (2020) · Zbl 1490.35314 |
| [21] | Gao, X. Y.; Guo, Y. J.; Shan, W. R., Water-wave symbolic computation for the Earth, Enceladus and Titan: the higher-order Boussinesq-Burgers system, auto- and non-auto-Bäcklund transformations, Appl. Math. Lett., 104, Article 106170 pp. (2020) · Zbl 1437.86001 |
| [22] | Gao, X. Y.; Guo, Y. J.; Shan, W. R., Cosmic dusty plasmas via a (3+1)-dimensional generalized variable-coefficient Kadomtsev-Petviashvili-Burgers-type equation: auto-Bäcklund transformations, solitons and similarity reductions plus observational/experimental supports, Waves Random Complex Media, 20, 1-21 (July 20 2021), Published online |
| [23] | Holden, H.; Karlsen, K. H.; Risebro, N. H.; Tao, T., Operator splitting for the KDV equation, Math. Comput., 80, 821-846 (2011) · Zbl 1219.35235 |
| [24] | Huang, Y.; Li, X.; Xiao, A., Fourier pseudospectral method on generalized sparse grids for the space-fractional Schrödinger equation, Comput. Math. Appl., 75, 4241-4255 (2018) · Zbl 1419.65079 |
| [25] | Hundsdorfer, W.; Verwer, J. G., Numerical Solution of Time-Dependent Advection-Diffusion-Reaction Equations (2003), Springer: Springer Berlin · Zbl 1030.65100 |
| [26] | Jin, S.; Markowich, P. A.; Sparber, C., Mathematical and computational methods for semicalssical Schrödinger equations, Acta Numer., 20, 121-209 (2011) · Zbl 1233.65071 |
| [27] | Jumarie, G., Modified Riemann-Liouville derivative and fractional Taylor series of nondifferentiable functions further result, Comput. Math. Appl., 51, 1367-1376 (2006) · Zbl 1137.65001 |
| [28] | Khaliq, A. Q.M.; Liang, X.; Furati, K. M., A fourth-order implicit-explicit scheme for the space fractional nonlinear Schrödinger equations, Numer. Algorithms, 75, 145-172 (2017) · Zbl 1365.65195 |
| [29] | Kirkpatrick, K.; Lenzmann, E.; Staffilani, G., On the continuum limit for discrete NLS with long-range lattice interanctions, Commun. Math. Phys., 317, 563-591 (2013) · Zbl 1258.35182 |
| [30] | Kirkpatrick, K.; Zhang, Y., Fractional Schrödinger dynamics and decoherence, Physica D, 332, 41-54 (2016) · Zbl 1415.65235 |
| [31] | Klein, C.; Sparber, C.; Markowich, P., Numerical study of fractional nonlinear Schrödinger equations, Proc. R. Soc. A, 470, Article 20140364 pp. (2014) · Zbl 1372.65284 |
| [32] | Laskin, N., Fractional quantum mechanics and Lévy path integrals, Phys. Lett. A, 268, 298-305 (2000) · Zbl 0948.81595 |
| [33] | Laskin, N., Fractional Schrödinger equation, Phys. Rev. E, 66, Article 056108 pp. (2002) |
| [34] | Lenzmann, E., Well-posedness for semi-relativistic Hartree equations of critical type, Math. Phys. Anal. Geom., 10, 43-64 (2007) · Zbl 1171.35474 |
| [35] | Li, M.; Huang, C.; Wang, P., Galerkin finite element method for nonlinear fractional Schrödinger equations, Numer. Algorithms, 74, 499-525 (2017) · Zbl 1359.65208 |
| [36] | Liang, X.; Khaliq, A. Q.M.; Bhatt, H.; Furati, K. M., The locally extrapolated exponential splitting scheme for multi-dimensional nonlinear space-fractional Schrödinger equations, Numer. Algorithms, 76, 939-958 (2017) · Zbl 1380.65162 |
| [37] | Lischke, A.; Pang, G.; Gulian, M., What is the fractional Laplacian? A comparative review with new results, J. Comput. Phys., 404, Article 109009 pp. (2020) · Zbl 1453.35179 |
| [38] | Lubich, Ch., From Quantum to Classical Molecular Dynamics: Reduced Models and Numerical Analysis, Zurich Lectures in Advanced Mathematics (2008), Europ. Math. Soc.: Europ. Math. Soc. Zürich · Zbl 1160.81001 |
| [39] | Markowich, P. A.; Pietra, P.; Pohl, C., Numerical approximation of quadratic observables of Schrödinger-type equations in the semi-classical limit, Numer. Math., 81, 595-630 (1999) · Zbl 0928.65109 |
| [40] | McLachlan, R. I.; Quispel, G. R.W., Splitting methods, Acta Numer., 11, 341-434 (2002) · Zbl 1105.65341 |
| [41] | Di Nezza, E.; Palatucci, G.; Valdinoci, E., Hitchhiker’s guide to the fractional Sobolev spaces, Bull. Sci. Math., 136, 521-573 (2012) · Zbl 1252.46023 |
| [42] | Odibat, Z. M.; Shawagfeh, N. T., Generalized Taylor’s formula, Appl. Math. Comput., 186, 286-293 (2007) · Zbl 1122.26006 |
| [43] | Ran, M.; Zhang, C., A conservative difference scheme for solving the strongly coupled nonlinear fractional Schrödinger equations, Commun. Nonlinear Sci. Numer. Simul., 41, 64-83 (2016) · Zbl 1458.65112 |
| [44] | Ros-Oton, X.; Serra, J., The Dirichlet problem for the fractional Laplacian: regularity up to the boundary, J. Math. Pures Appl., 101, 275-302 (2014) · Zbl 1285.35020 |
| [45] | Ros-Oton, X.; Serra, J., The Pohozaev identity for the fractional Laplacian, Arch. Ration. Mech. Anal., 213, 587-628 (2014) · Zbl 1361.35199 |
| [46] | Samko, S. G.; Kilbas, A. A.; Marichev, O. I., Fractional Integrals and Derivatives: Theory and Applications (1993), Gordon and Breach Science Publ.: Gordon and Breach Science Publ. Switzerland, India, Japan · Zbl 0818.26003 |
| [47] | Shen, J.; Tang, T.; Wang, L. L., Spectral Methods: Algorithms, Analyses and Applications (2011), Springer: Springer Berlin · Zbl 1227.65117 |
| [48] | Secchi, S.; Squassina, M., Soliton dynamics for fractional Schrödinger equations, Appl. Anal., 93, 1702-1729 (2014) · Zbl 1298.35166 |
| [49] | Trujillo, J.; Rivero, M.; Bonilla, B., On a Riemann-Liouville generalized Taylor’s formula, J. Math. Anal. Appl., 1, 255-265 (1999) · Zbl 0931.26004 |
| [50] | Uchaikin, V. V., Fractional Derivatives for Physicists and Engineers: Volume I. Backgroud and Theory (2013), Higher Education Press: Higher Education Press Beijing · Zbl 1312.26002 |
| [51] | 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 |
| [52] | Wang, H.; Du, N., Fast alternating-direction finite difference methods for three-dimensional space-fractional diffusion equations, J. Comput. Phys., 258, 305-318 (2014) · Zbl 1349.65342 |
| [53] | 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 |
| [54] | Wang, W.; Tang, J., Efficient exponential splitting spectral methods for linear Schrödinger equation in the semiclassical regime, Appl. Numer. Math., 153, 132-146 (2020) · Zbl 1437.65153 |
| [55] | Zhai, S.; Wang, D.; Weng, Z.; Zhao, X., Error analysis and numerical simulations of Strang splitting method for space fractional nonlinear Schrödinger equation, J. Sci. Comput., 81, 965-989 (2019) · Zbl 1459.65204 |
| [56] | Wang, M.; Tian, B.; Sun, Y.; Zhang, Z., Lump, mixed lump-stripe and rogue wave-stripe solutions of a (3+1)-dimensional nonlinear wave equation for a liquid with gas bubbles, Comput. Math. Appl., 79, 576-587 (2020) · Zbl 1443.76233 |
| [57] | Xiang, X. M., Numerical Analysis of Spectral Methods (2000), Science Press: Science Press Beijing |
| [58] | Zhao, X.; Sun, Z.; Hao, Z., A fourth-order compact ADI scheme for two-dimensional nonlinear space fractional Schrödinger equations, SIAM J. Sci. Comput., 36, A2865-A2886 (2014) · Zbl 1328.65187 |
| [59] | Zhang, C. R.; Tian, B.; Qu, Q. X.; Liu, L.; Tian, H. Y., Vector bright solitons and their interactions of the couple Fokas-Lenells system in a birefringent optical fiber, Z. Angew. Math. Phys., 71, 18 (2020) · Zbl 1508.35164 |
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.
