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The Lie algebra of classical mechanics
A structure-preserving Fourier pseudo-spectral linearly implicit scheme for the space-fractional nonlinear Schrödinger equation
1. | Cybermedia Center, Osaka University, 1-32 Machikaneyama, Toyonaka, Osaka 560-0043, Japan |
2. | Department of Computational Science and Engineering, Graduate School of Engineering, Nagoya University, Furo-cho, Chikusa-ku, Nagoya 464-8603, Japan |
3. | Department of Applied Physics, Graduate School of Engineering, Nagoya University, Furo-cho, Chikusa-ku, Nagoya 464-8603, Japan |
We propose a Fourier pseudo-spectral scheme for the space-fractional nonlinear Schrödinger equation. The proposed scheme has the following features: it is linearly implicit, it preserves two invariants of the equation, its unique solvability is guaranteed without any restrictions on space and time step sizes. The scheme requires solving a complex symmetric linear system per time step. To solve the system efficiently, we also present a certain variable transformation and preconditioner.
References:
[1] |
C. Besse,
A relaxation scheme for the nonlinear Schrödinger equation, SIAM J. Numer. Anal., 42 (2004), 934-952.
doi: 10.1137/S0036142901396521. |
[2] |
C. Besse, S. Descombes, G. Dujardin and I. Lacroix-Violet, Energy preserving methods for nonlinear Schrödinger equations, (2018). |
[3] |
E. Celledoni, V. Grimm, R. I. McLachlan, D. I. McLaren, D. O'Neale, B. Owren and G. R. W. Quispel,
Preserving energy resp. dissipation in numerical PDEs using the "average vector field" method, J. Comput. Phys., 231 (2012), 6770-6789.
doi: 10.1016/j.jcp.2012.06.022. |
[4] |
Y. Cho, G. Hwang, S. Kwon and S. Lee,
Well-posedness and ill-posedness for the cubic fractional Schrödinger equations, Discrete Contin. Dyn. Syst., 35 (2015), 2863-2880.
doi: 10.3934/dcds.2015.35.2863. |
[5] |
M. Dahlby and B. Owren,
A general framework for deriving integral preserving numerical methods for PDEs, SIAM J. Sci. Comput., 33 (2011), 2318-2340.
doi: 10.1137/100810174. |
[6] |
M. Delfour, M. Fortin and G. Payre,
Finite-difference solutions of a non-linear Schrödinger equation, J. Comput. Phys., 44 (1981), 277-288.
doi: 10.1016/0021-9991(81)90052-8. |
[7] |
S. W. Duo and Y. Z. Zhang,
Mass-conservative Fourier spectral methods for solving the fractional nonlinear Schrödinger equation, Comput. Math. Appl., 71 (2016), 2257-2271.
doi: 10.1016/j.camwa.2015.12.042. |
[8] |
E. Faou, Geometric Numerical Integration and Schrödinger Equations, Zurich Lectures in Advanced Mathematics, European Mathematical Society (EMS), Zürich, 2012.
doi: 10.4171/100. |
[9] |
R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals, Emended edition, Emended and with a preface by Daniel F. Styer. Dover Publications, Inc., Mineola, NY, 2010. |
[10] |
D. Furihata and T. Matsuo, Discrete Variational Derivative Method: A Structure-Preserving Numerical Method for Partial Differential Equations, Chapman & Hall/CRC Numerical Analysis and Scientific Computing, CRC Press, Boca Raton, FL, 2011.
![]() ![]() |
[11] |
B. L. Guo, Y. Q. Han and J. Xin,
Existence of the global smooth solution to the period boundary value problem of fractional nonlinear Schrödinger equation, Appl. Math. Comput., 204 (2008), 468-477.
doi: 10.1016/j.amc.2008.07.003. |
[12] |
B. L. Guo and Z. H. Huo,
Global well-posedness for the fractional nonlinear Schrödinger equation, Comm. Partial Differential Equations, 36 (2011), 247-255.
doi: 10.1080/03605302.2010.503769. |
[13] |
X. Y. Guo and M. Y. Xu, Some physical applications of fractional Schrödinger equation, J. Math. Phys., 47 (2006), 082104, 9 pp.
doi: 10.1063/1.2235026. |
[14] |
K. Kirkpatrick, E. Lenzmann and G. Staffilani,
On the continuum limit for discrete NLS with long-range lattice interactions, Comm. Math. Phys., 317 (2013), 563-591.
doi: 10.1007/s00220-012-1621-x. |
[15] |
N. Laskin, Fractional Quantum Mechanics, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2018.
doi: 10.1142/10541. |
[16] |
N. Laskin, Fractional Schrödinger equation, Phys. Rev. E (3), 66 (2002), 056108, 7 pp.
doi: 10.1103/PhysRevE.66.056108. |
[17] |
N. Laskin,
Fractional quantum mechanics and Lévy path integrals, Phys. Lett. A, 268 (2000), 298-305.
doi: 10.1016/S0375-9601(00)00201-2. |
[18] |
M. Li, X.-M. Gu, C. M. Huang, M. F. Fei and G. Y. Zhang,
A fast linearized conservative finite element method for the strongly coupled nonlinear fractional Schrödinger equations, J. Comput. Phys., 358 (2018), 256-282.
doi: 10.1016/j.jcp.2017.12.044. |
[19] |
M. Li, C. Huang and W. Ming, A relaxation-type Galerkin FEM for nonlinear fractional Schrödinger equations, Numer. Algorithms, (2019), 1–26.
doi: 10.1007/s11075-019-00672-3. |
[20] |
M. Li, C. M. Huang and P. D. Wang,
Galerkin finite element method for nonlinear fractional Schrödinger equations, Numer. Algorithms, 74 (2017), 499-525.
doi: 10.1007/s11075-016-0160-5. |
[21] |
S. Longhi, Fractional Schrödinger equation in optics, Opt. Lett., 40 (2015), 1117.
doi: 10.1364/OL.40.001117. |
[22] |
T. Matsuo and D. Furihata,
Dissipative or conservative finite-difference schemes for complex-valued nonlinear partial differential equations, J. Comput. Phys., 171 (2001), 425-447.
doi: 10.1006/jcph.2001.6775. |
[23] |
Y. Miyatake and T. Matsuo,
Conservative finite difference schemes for the Degasperis-Procesi equation, J. Comput. Appl. Math., 236 (2012), 3728-3740.
doi: 10.1016/j.cam.2011.09.004. |
[24] |
Y. Miyatake, T. Matsuo and D. Furihata,
Invariants-preserving integration of the modified Camassa-Holm equation, Jpn. J. Ind. Appl. Math., 28 (2011), 351-381.
doi: 10.1007/s13160-011-0043-z. |
[25] |
G. R. W. Quispel and D. I. McLaren, A new class of energy-preserving numerical integration methods, J. Phys. A, 41 (2008), 045206, 7 pp.
doi: 10.1088/1751-8113/41/4/045206. |
[26] |
T. Sogabe and S.-L. Zhang,
A COCR method for solving complex symmetric linear systems, J. Comput. Appl. Math., 199 (2007), 297-303.
doi: 10.1016/j.cam.2005.07.032. |
[27] |
H. A. van der Vorst,
Bi-CGSTAB: A fast and smoothly converging variant of Bi-CG for the solution of nonsymmetric linear systems, SIAM J. Sci. Statist. Comput., 13 (1992), 631-644.
doi: 10.1137/0913035. |
[28] |
H. A. van der Vorst and J. B. Melissen,
A Petrov-Galerkin type method for solving ${A}x = b$, where ${A}$ is symmetric complex, IEEE Trans. Mag., 26 (1990), 706-708.
|
[29] |
D. L. Wang, A. G. Xiao and W. Yang,
A linearly implicit conservative difference scheme for the space fractional coupled nonlinear Schrödinger equations, J. Comput. Phys., 272 (2014), 644-655.
doi: 10.1016/j.jcp.2014.04.047. |
[30] |
P. D. Wang and C. M. Huang,
A conservative linearized difference scheme for the nonlinear fractional Schrödinger equation, Numer. Algorithms, 69 (2015), 625-641.
doi: 10.1007/s11075-014-9917-x. |
[31] |
P. D. Wang and C. M. Huang,
An energy conservative difference scheme for the nonlinear fractional Schrödinger equations, J. Comput. Phys., 293 (2015), 238-251.
doi: 10.1016/j.jcp.2014.03.037. |
[32] |
P. D. Wang and C. M. Huang,
Split-step alternating direction implicit difference scheme for the fractional Schrödinger equation in two dimensions, Comput. Math. Appl., 71 (2016), 1114-1128.
doi: 10.1016/j.camwa.2016.01.022. |
[33] |
P. D. Wang and C. M. Huang,
Structure-preserving numerical methods for the fractional Schrödinger equation, Appl. Numer. Math., 129 (2018), 137-158.
doi: 10.1016/j.apnum.2018.03.008. |
[34] |
P. D. Wang, C. M. Huang and L. B. Zhao,
Point-wise error estimate of a conservative difference scheme for the fractional Schrödinger equation, J. Comput. Appl. Math., 306 (2016), 231-247.
doi: 10.1016/j.cam.2016.04.017. |
[35] |
Y. Q. Zhang, X. Liu, M. R. Belić, W. P. Zhong, Y. P. Zhang and M. Xiao, Propagation dynamics of a light beam in a fractional Schrödinger equation, Phys. Rev. Lett., 115 (2015), 180403.
doi: 10.1103/PhysRevLett.115.180403. |
show all references
References:
[1] |
C. Besse,
A relaxation scheme for the nonlinear Schrödinger equation, SIAM J. Numer. Anal., 42 (2004), 934-952.
doi: 10.1137/S0036142901396521. |
[2] |
C. Besse, S. Descombes, G. Dujardin and I. Lacroix-Violet, Energy preserving methods for nonlinear Schrödinger equations, (2018). |
[3] |
E. Celledoni, V. Grimm, R. I. McLachlan, D. I. McLaren, D. O'Neale, B. Owren and G. R. W. Quispel,
Preserving energy resp. dissipation in numerical PDEs using the "average vector field" method, J. Comput. Phys., 231 (2012), 6770-6789.
doi: 10.1016/j.jcp.2012.06.022. |
[4] |
Y. Cho, G. Hwang, S. Kwon and S. Lee,
Well-posedness and ill-posedness for the cubic fractional Schrödinger equations, Discrete Contin. Dyn. Syst., 35 (2015), 2863-2880.
doi: 10.3934/dcds.2015.35.2863. |
[5] |
M. Dahlby and B. Owren,
A general framework for deriving integral preserving numerical methods for PDEs, SIAM J. Sci. Comput., 33 (2011), 2318-2340.
doi: 10.1137/100810174. |
[6] |
M. Delfour, M. Fortin and G. Payre,
Finite-difference solutions of a non-linear Schrödinger equation, J. Comput. Phys., 44 (1981), 277-288.
doi: 10.1016/0021-9991(81)90052-8. |
[7] |
S. W. Duo and Y. Z. Zhang,
Mass-conservative Fourier spectral methods for solving the fractional nonlinear Schrödinger equation, Comput. Math. Appl., 71 (2016), 2257-2271.
doi: 10.1016/j.camwa.2015.12.042. |
[8] |
E. Faou, Geometric Numerical Integration and Schrödinger Equations, Zurich Lectures in Advanced Mathematics, European Mathematical Society (EMS), Zürich, 2012.
doi: 10.4171/100. |
[9] |
R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals, Emended edition, Emended and with a preface by Daniel F. Styer. Dover Publications, Inc., Mineola, NY, 2010. |
[10] |
D. Furihata and T. Matsuo, Discrete Variational Derivative Method: A Structure-Preserving Numerical Method for Partial Differential Equations, Chapman & Hall/CRC Numerical Analysis and Scientific Computing, CRC Press, Boca Raton, FL, 2011.
![]() ![]() |
[11] |
B. L. Guo, Y. Q. Han and J. Xin,
Existence of the global smooth solution to the period boundary value problem of fractional nonlinear Schrödinger equation, Appl. Math. Comput., 204 (2008), 468-477.
doi: 10.1016/j.amc.2008.07.003. |
[12] |
B. L. Guo and Z. H. Huo,
Global well-posedness for the fractional nonlinear Schrödinger equation, Comm. Partial Differential Equations, 36 (2011), 247-255.
doi: 10.1080/03605302.2010.503769. |
[13] |
X. Y. Guo and M. Y. Xu, Some physical applications of fractional Schrödinger equation, J. Math. Phys., 47 (2006), 082104, 9 pp.
doi: 10.1063/1.2235026. |
[14] |
K. Kirkpatrick, E. Lenzmann and G. Staffilani,
On the continuum limit for discrete NLS with long-range lattice interactions, Comm. Math. Phys., 317 (2013), 563-591.
doi: 10.1007/s00220-012-1621-x. |
[15] |
N. Laskin, Fractional Quantum Mechanics, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2018.
doi: 10.1142/10541. |
[16] |
N. Laskin, Fractional Schrödinger equation, Phys. Rev. E (3), 66 (2002), 056108, 7 pp.
doi: 10.1103/PhysRevE.66.056108. |
[17] |
N. Laskin,
Fractional quantum mechanics and Lévy path integrals, Phys. Lett. A, 268 (2000), 298-305.
doi: 10.1016/S0375-9601(00)00201-2. |
[18] |
M. Li, X.-M. Gu, C. M. Huang, M. F. Fei and G. Y. Zhang,
A fast linearized conservative finite element method for the strongly coupled nonlinear fractional Schrödinger equations, J. Comput. Phys., 358 (2018), 256-282.
doi: 10.1016/j.jcp.2017.12.044. |
[19] |
M. Li, C. Huang and W. Ming, A relaxation-type Galerkin FEM for nonlinear fractional Schrödinger equations, Numer. Algorithms, (2019), 1–26.
doi: 10.1007/s11075-019-00672-3. |
[20] |
M. Li, C. M. Huang and P. D. Wang,
Galerkin finite element method for nonlinear fractional Schrödinger equations, Numer. Algorithms, 74 (2017), 499-525.
doi: 10.1007/s11075-016-0160-5. |
[21] |
S. Longhi, Fractional Schrödinger equation in optics, Opt. Lett., 40 (2015), 1117.
doi: 10.1364/OL.40.001117. |
[22] |
T. Matsuo and D. Furihata,
Dissipative or conservative finite-difference schemes for complex-valued nonlinear partial differential equations, J. Comput. Phys., 171 (2001), 425-447.
doi: 10.1006/jcph.2001.6775. |
[23] |
Y. Miyatake and T. Matsuo,
Conservative finite difference schemes for the Degasperis-Procesi equation, J. Comput. Appl. Math., 236 (2012), 3728-3740.
doi: 10.1016/j.cam.2011.09.004. |
[24] |
Y. Miyatake, T. Matsuo and D. Furihata,
Invariants-preserving integration of the modified Camassa-Holm equation, Jpn. J. Ind. Appl. Math., 28 (2011), 351-381.
doi: 10.1007/s13160-011-0043-z. |
[25] |
G. R. W. Quispel and D. I. McLaren, A new class of energy-preserving numerical integration methods, J. Phys. A, 41 (2008), 045206, 7 pp.
doi: 10.1088/1751-8113/41/4/045206. |
[26] |
T. Sogabe and S.-L. Zhang,
A COCR method for solving complex symmetric linear systems, J. Comput. Appl. Math., 199 (2007), 297-303.
doi: 10.1016/j.cam.2005.07.032. |
[27] |
H. A. van der Vorst,
Bi-CGSTAB: A fast and smoothly converging variant of Bi-CG for the solution of nonsymmetric linear systems, SIAM J. Sci. Statist. Comput., 13 (1992), 631-644.
doi: 10.1137/0913035. |
[28] |
H. A. van der Vorst and J. B. Melissen,
A Petrov-Galerkin type method for solving ${A}x = b$, where ${A}$ is symmetric complex, IEEE Trans. Mag., 26 (1990), 706-708.
|
[29] |
D. L. Wang, A. G. Xiao and W. Yang,
A linearly implicit conservative difference scheme for the space fractional coupled nonlinear Schrödinger equations, J. Comput. Phys., 272 (2014), 644-655.
doi: 10.1016/j.jcp.2014.04.047. |
[30] |
P. D. Wang and C. M. Huang,
A conservative linearized difference scheme for the nonlinear fractional Schrödinger equation, Numer. Algorithms, 69 (2015), 625-641.
doi: 10.1007/s11075-014-9917-x. |
[31] |
P. D. Wang and C. M. Huang,
An energy conservative difference scheme for the nonlinear fractional Schrödinger equations, J. Comput. Phys., 293 (2015), 238-251.
doi: 10.1016/j.jcp.2014.03.037. |
[32] |
P. D. Wang and C. M. Huang,
Split-step alternating direction implicit difference scheme for the fractional Schrödinger equation in two dimensions, Comput. Math. Appl., 71 (2016), 1114-1128.
doi: 10.1016/j.camwa.2016.01.022. |
[33] |
P. D. Wang and C. M. Huang,
Structure-preserving numerical methods for the fractional Schrödinger equation, Appl. Numer. Math., 129 (2018), 137-158.
doi: 10.1016/j.apnum.2018.03.008. |
[34] |
P. D. Wang, C. M. Huang and L. B. Zhao,
Point-wise error estimate of a conservative difference scheme for the fractional Schrödinger equation, J. Comput. Appl. Math., 306 (2016), 231-247.
doi: 10.1016/j.cam.2016.04.017. |
[35] |
Y. Q. Zhang, X. Liu, M. R. Belić, W. P. Zhong, Y. P. Zhang and M. Xiao, Propagation dynamics of a light beam in a fractional Schrödinger equation, Phys. Rev. Lett., 115 (2015), 180403.
doi: 10.1103/PhysRevLett.115.180403. |













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