Sine-transform-based fast solvers for Riesz fractional nonlinear Schrödinger equations with attractive nonlinearities

TL;DR

The paper addresses the computational challenge of solving large, complex indefinite linear systems arising from discretizations of the RFNSE with Riesz derivatives. It develops a Toeplitz-based anti-symmetric and normal (TBAN) splitting and couples it with a sine-transform-based preconditioner, yielding a parameter-free preconditioned GMRES with eigenvalues clustered near $1$ and convergence independent of spatial mesh size and fractional order. The TBAN framework provides unconditional convergence for the iteration, and the sine-transform preconditioner achieves efficient, scalable performance, particularly in 1D and 2D RFNSE tests. Numerical experiments demonstrate substantial improvements over Circulant-based or unpreconditioned GMRES, with preserved discrete mass and energy and robust behavior across a range of fractional orders and grid refinements.

Abstract

This paper presents fast solvers for linear systems arising from the discretization of fractional nonlinear Schrödinger equations with Riesz derivatives and attractive nonlinearities. These systems are characterized by complex symmetry, indefiniteness, and a $d$-level Toeplitz-plus-diagonal structure. We propose a Toeplitz-based anti-symmetric and normal splitting iteration method for the equivalent real block linear systems, ensuring unconditional convergence. The derived optimal parameter is approximately equal to 1. By combining this iteration method with sine-transform-based preconditioning, we introduce a novel preconditioner that enhances the convergence rate of Krylov subspace methods. Both theoretical and numerical analyses demonstrate that the new preconditioner exhibits a parameter-free property (allowing the iteration parameter to be fixed at 1). The eigenvalues of the preconditioned system matrix are nearly clustered in a small neighborhood around 1, and the convergence rate of the corresponding preconditioned GMRES method is independent of the spatial mesh size and the fractional order of the Riesz derivatives.

Figures (11)

• Figure 1: The curves of IT versus the iteration parameter $\omega\in (0,4]$ of $\tau$-GMRES when $\alpha=1.1:0.2:1.9$, $\rho=2$, $M=6400$ and $N=200$.
• Figure 2: The eigenvalue distribution of $\mathcal{R}$, ${\mathcal{F}_{\tau}^{-1}\mathcal{R}}$, ${\mathcal{F}_{C}^{-1}\mathcal{R}}$ when $\alpha=1.1$, $\rho=2$, $N=200$, $M=1600$ (left) and $M=3200$ (right).
• Figure 3: The eigenvalue distribution of $\mathcal{R}$, ${\mathcal{F}_{\tau}^{-1}\mathcal{R}}$, ${\mathcal{F}_{C}^{-1}\mathcal{R}}$ when $\alpha=1.5$, $\rho=2$, $N=200$, $M=1600$ (left) and $M=3200$ (right).
• Figure 4: The eigenvalue distribution of $\mathcal{R}$, ${\mathcal{F}_{\tau}^{-1}\mathcal{R}}$, ${\mathcal{F}_{C}^{-1}\mathcal{R}}$ when $\alpha=1.9$, $\rho=2$, $N=200$, $M=1600$ (left) and $M=3200$ (right).
• Figure 5: The curves of IT of $\tau$-GMRES, C-GMRES and GMRES versus the nonlinear term parameter $\rho$ when $\alpha=1.3:0.2:1.7$, $M=6400$, $N=200$.

Theorems & Definitions (13)

• Theorem 3.1
• Remark 3.1
• Theorem 4.1
• Remark 4.1
• Lemma 4.1
• Lemma 4.2
• Lemma 4.3
• Lemma 4.4
• Lemma 4.5
• Lemma 4.6