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

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  • 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:

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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)

    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 $
     | Show Table
    DownLoad: CSV

    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 $
     | Show Table
    DownLoad: CSV

    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 $
     | Show Table
    DownLoad: CSV

    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 $
     | Show Table
    DownLoad: CSV
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