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

Yuto Miyatake; Tai Nakagawa; Tomohiro Sogabe; Shao-Liang Zhang

doi:10.3934/jcd.2019018

Article Contents

2019, Volume 6, Issue 2: 361-383. Doi: 10.3934/jcd.2019018

This issue Previous Article The Lie algebra of classical mechanics Next Article Discrete gradients for computational Bayesian inference

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

Received: February 2019

Revised: September 2019

Published: November 2019

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Abstract

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.

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

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

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

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

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

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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 $

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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 $

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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 $

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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 $

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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 $

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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

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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 $

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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 $

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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 $

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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 $

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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 $

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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 $

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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 $

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