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# 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 $3.286$ $3.291$ $3.323$
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