Sine-transform-based fast solvers for Riesz fractional nonlinear Schr\"odinger equations with attractive nonlinearities

TL;DR

This paper develops fast, efficient solvers for fractional nonlinear Schrödinger equations with Riesz derivatives, employing sine-transform-based preconditioning to improve convergence and robustness of Krylov subspace methods.

Contribution

It introduces a novel preconditioner combining Toeplitz-based splitting and sine transforms, ensuring mesh-size and fractional-order independent convergence.

Findings

Preconditioner clusters eigenvalues near 1 for rapid convergence.

Convergence rate is independent of mesh size and fractional order.

The method is validated through theoretical analysis and numerical experiments.

Abstract

This paper presents fast solvers for linear systems arising from the discretization of fractional nonlinear Schr\"odinger 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.