Diagonal and normal with Toeplitz-block splitting iteration method for space fractional coupled nonlinear Schrödinger equations with repulsive nonlinearities

TL;DR

Numerical experiments show that the new preconditioner can significantly improve the computational efficiency of the Krylov subspace iteration methods and shows space mesh size independent and almost fractional order parameter insensitive convergence behaviors.

Abstract

By applying the linearly implicit conservative difference scheme proposed in [D.-L. Wang, A.-G. Xiao, W. Yang. J. Comput. Phys. 2014;272:670-681], the system of repulsive space fractional coupled nonlinear Schrödinger equations leads to a sequence of linear systems with complex symmetric and Toeplitz-plus-diagonal structure. In this paper, we propose the diagonal and normal with Toeplitz-block splitting iteration method to solve the above linear systems. The new iteration method is proved to converge unconditionally, and the optimal iteration parameter is deducted. Naturally, this new iteration method leads to a diagonal and normal with circulant-block preconditioner which can be executed efficiently by fast algorithms. In theory, we provide sharp bounds for the eigenvalues of the discrete fractional Laplacian and its circulant approximation, and further analysis indicates that the spectral distribution of the preconditioned system matrix is tight. Numerical experiments show that the new preconditioner can significantly improve the computational efficiency of the Krylov subspace iteration methods. Moreover, the behavior of the corresponding preconditioned GMRES method exhibits a linear dependence on the space mesh size, which weakens as the fractional order parameter decreases.

Figures (14)

• Figure 1: The curves of IT of DNCB-GMRES versus the parameter $\omega\in (0,3]$ of $\mathcal{F}_{\hbox{\tiny DNCB}}$ in the DNLS case when $\alpha=1.1:0.2:1.9$ and $M=6400$: blue solid line with circle mark for $\alpha=1.1$, red solid line with diamond mark for $\alpha=1.3$, orange solid line with triangle mark for $\alpha=1.5$, purple solid line with square mark for $\alpha=1.7$, green solid line with pentagram mark for $\alpha=1.9$.
• Figure 2: The curves of IT of DNCB-GMRES versus the number of the inner spatial discrete points $M$ of the LICD scheme in the DNLS case when $\alpha=1.1:0.2:1.9$ and $\alpha=2.0$: blue solid line with circle mark for $\alpha=1.1$, red solid line with diamond mark for $\alpha=1.3$, orange solid line with triangle mark for $\alpha=1.5$, purple solid line with square mark for $\alpha=1.7$, green solid line with pentagram mark for $\alpha=1.9$, cyan solid line with cross mark for $\alpha=2.0$.
• Figure 3: The eigenvalue distribution of $\mathcal{R}$, $\mathcal{F}_{\hbox{\tiny DNTB}}^{-1}\mathcal{R}$, $\mathcal{F}_{\hbox{\tiny DNCB}}^{-1}\mathcal{R}$, $\mathcal{F}_{\hbox{\tiny PMHSS}}^{-1}\mathcal{R}$ and $\mathcal{F}_{\hbox{\tiny CPMHSS}}^{-1}\mathcal{R}$ in the case of $\alpha=1.3$ for $M=1600$ (left), $3200$ (right): blue circle mark for $\mathcal{R}$, red square mark for $\mathcal{F}_{\hbox{\tiny DNTB}}^{-1}\mathcal{R}$, orange solid hexagram mark for $\mathcal{F}_{\hbox{\tiny DNCB}}^{-1}\mathcal{R}$, purple diamond mark for $\mathcal{F}_{\hbox{\tiny PMHSS}}^{-1}\mathcal{R}$, green solid triangle mark for $\mathcal{F}_{\hbox{\tiny CPMHSS}}^{-1}\mathcal{R}$.
• Figure 4: The eigenvalue distribution of $\mathcal{R}$, $\mathcal{F}_{\hbox{\tiny DNTB}}^{-1}\mathcal{R}$, $\mathcal{F}_{\hbox{\tiny DNCB}}^{-1}\mathcal{R}$, $\mathcal{F}_{\hbox{\tiny PMHSS}}^{-1}\mathcal{R}$ and $\mathcal{F}_{\hbox{\tiny CPMHSS}}^{-1}\mathcal{R}$ in the case of $\alpha=1.7$ for $M=1600$ (left), $3200$ (right): blue circle mark for $\mathcal{R}$, red square mark for $\mathcal{F}_{\hbox{\tiny DNTB}}^{-1}\mathcal{R}$, orange solid hexagram mark for $\mathcal{F}_{\hbox{\tiny DNCB}}^{-1}\mathcal{R}$, purple diamond mark for $\mathcal{F}_{\hbox{\tiny PMHSS}}^{-1}\mathcal{R}$, green solid triangle mark for $\mathcal{F}_{\hbox{\tiny CPMHSS}}^{-1}\mathcal{R}$.
• Figure 5: The numerical solution (left) and its error (right) with the exact solution of the LICD scheme in the DNLS case when $\alpha=1.1$ and $M=800$.

Theorems & Definitions (12)

• Theorem 3.1
• Remark 3.1
• Theorem 3.2
• Remark 3.2
• Lemma 4.1
• Lemma 4.2
• Remark 4.1
• Theorem 4.1
• Theorem 4.2
• Remark 4.2