A high-order method for the numerical approximation of fractional nonlinear Schr\"odinger equations

TL;DR

This paper develops a high-order numerical scheme combining Fourier spectral Galerkin and implicit Runge-Kutta methods for solving fractional nonlinear Schrödinger equations, with proven convergence and demonstrated effectiveness through numerical experiments.

Contribution

It introduces a novel high-order discretization approach for fractional nonlinear Schrödinger equations, including rigorous error analysis and practical numerical validation.

Findings

The scheme achieves high accuracy in approximating solutions.

Error estimates confirm the method's convergence.

Numerical experiments demonstrate the scheme's efficiency.

Abstract

In this paper, the periodic initial-value problem for the fractional nonlinear Schr\"odinger (fNLS) equation is discretized in space by a Fourier spectral Galerkin method and in time by diagonally implicit, high-order Runge-Kutta schemes, based on the composition with the implicit midpoint rule (IMR). Some properties and error estimates for the semidiscretization in space and for the full discretization are proved. The convergence results and the general performance of the scheme are illustrated with several numerical experiments.