Article Contents
Article Contents

# A structure-preserving Fourier pseudo-spectral linearly implicit scheme for the space-fractional nonlinear Schrödinger equation

• 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.

Mathematics Subject Classification: Primary: 65M70, 65F08; Secondary: 65N22.

 Citation:

• Figure 1.  Numerical solutions for the case $\alpha = 2$ obtained by (LEFT) the linearly implicit scheme (9) and (RIGHT) the nonlinear scheme (14)

Figure 2.  Numerical solutions for the case $\alpha = 1.6$ obtained by (LEFT) the linearly implicit scheme (9) and (RIGHT) the nonlinear scheme (14)

Figure 3.  Numerical solutions for the case $\alpha = 1.2$ obtained by (LEFT) the linearly implicit scheme (9) and (RIGHT) the nonlinear scheme (14)

Figure 4.  Numerical solutions for the case $\alpha = 1.6$ obtained by the linearly implicit scheme (9) with $N = 61$

Figure 5.  Error behaviour obtained by the linearly implicit scheme (9) for the case $\alpha = 1.6$: (LEFT) global error, and (RIGHT) error at $t = 20$. Errors are measured by $\max_k | U_k^{(n)} - U_{\text{ref},k}^{(n)}|$, where the reference solution was generated by the nonlinear scheme (14) with $N = 303$ and $\Delta t = 0.001$

Figure 6.  Errors of the discrete mass $\mathcal{M}_ \mathrm{d} ( {\mathit{\boldsymbol{U}}}^{(n)}$ and energy $\mathcal{H}_ \mathrm{d} ( {\mathit{\boldsymbol{U}}}^{(n+1)}, {\mathit{\boldsymbol{U}}}^{(n)})$ obtained by the linearly implicit scheme (9) for the case $\alpha = 2.0$

Figure 7.  Errors of the discrete mass $\mathcal{M}_ \mathrm{d}( {\boldsymbol{U}}^{(n)}$ and energy $\mathcal{H}_ \mathrm{d} ( {\boldsymbol{U}}^{(n+1)}, {\boldsymbol{U}}^{(n)})$ obtained by the linearly implicit scheme (9) for the case $\alpha = 1.6$

Figure 8.  Errors of the discrete mass $\mathcal{M}_ \mathrm{d} ( {\boldsymbol{U}}^{(n)})$ and energy $\mathcal{H}_ \mathrm{d} ( {\boldsymbol{U}}^{(n+1)}, {\boldsymbol{U}}^{(n)})$ obtained by the linearly implicit scheme (9) for the case $\alpha = 1.2$

Figure 9.  Errors of the discrete energy $\tilde{ \mathcal{H}}_ \mathrm{d} ( {\mathit{\boldsymbol{U}}}^{(n)})$ obtained by the linearly implicit scheme (9) for the case $\alpha = 1.6$

Figure 10.  The number of iterations for the COCG method to the convergence at each time step. (LEFT) $N = 101$ and $\Delta t = 0.02$ are fixed, (RIGHT) $\alpha = 2$ and $\Delta t = 0.02$ are fixed, (BOTTOM) $\alpha = 2$ and $N = 401$ are fixed

Figure 11.  The number of iterations for the Bi-CGSTAB method applied to (10). The parameters are set to $\alpha = 2$, $N = 401$ and $\Delta t = 0.02$

Figure 12.  The number of iterations for the preconditioned COCG method applied to (19) with the matrix $M = I + \mathrm{i} \Delta t D_\alpha$. The initial condition is set to $u_0 (x) = 2\exp (0.5 \mathrm{i} x){\rm{sech}}(\sqrt{2}(x-10))$. The parameters are set to $\alpha = 2$ and $N = 401$. (LEFT) $\Delta t = 0.01,0.02$, (RIGHT) $\Delta t = 0.05$

Figure 13.  The number of iterations for the preconditioned COCG method applied to (19) with the matrix $M = I + \mathrm{i} \Delta t D_\alpha$. The initial condition is set to $u_0 (x) = 2\exp (0.5 \mathrm{i} x){\rm{sech}}(4(x-10))$. The parameters are set to $\alpha = 2$ and $N = 401$

Table 1.  The maximum, minimum and average number of iterations of the preconditioned Bi-CGSTAB method: the time step size is set to $\Delta t = 0.02$, and the initial value $u_0 (x) = 2\exp (0.5 \mathrm{i} x){\rm{sech}}(\sqrt{2}(x-10))$

 $N$ $401$ $1001$ $4001$ maximum $3$ $3$ $3$ minimum $3$ $3$ $3$ average $3$ $3$ $3$

Table 2.  Average CPU time of $10$ simulations at $T = 8$ (the cost for obtaining ${\mathit{\boldsymbol{U}}}^{(1)}$ by the nonlinear scheme (14) is excluded): the time step size is set to $\Delta t = 0.02$, and the initial value $u_0 (x) = 2\exp (0.5 \mathrm{i} x){\rm{sech}}(\sqrt{2}(x-10))$

 $N$ $1001$ $2001$ $4001$ $8001$ CPU time $0.794$ $1.524$ $5.615$ $8.223$

Table 3.  The maximum, minimum and average number of iterations of the preconditioned Bi-CGSTAB method: the time step size is set to $\Delta t = 0.2$, and the initial value $u_0 (x) = 2\exp (0.5 \mathrm{i} x){\rm{sech}}(\sqrt{2}(x-10))$

 $N$ $401$ $1001$ $4001$ maximum $6$ $6$ $6$ minimum $5$ $5$ $5$ average $5.020$ $5.020$ $5.020$

Table 4.  The maximum, minimum and average number of iterations of the preconditioned Bi-CGSTAB method: the time step size is set to $\Delta t = 0.02$, and the initial value $u_0 (x) = 2\exp (0.5 \mathrm{i} x){\rm{sech}}(x-10)$

 $N$ $401$ $1001$ $4001$ maximum $4$ $4$ $4$ minimum $3$ $3$ $3$ average $3.286$ $3.291$ $3.323$
•  [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.

Figures(13)

Tables(4)