A conservative difference scheme with optimal pointwise error estimates for two-dimensional space fractional nonlinear Schr\"{o}dinger equations

TL;DR

This paper introduces a stable, mass- and energy-conserving finite difference scheme for 2D space fractional nonlinear Schrödinger equations, providing the first rigorous optimal pointwise error estimates and validated by numerical experiments.

Contribution

It presents the first rigorous proof of optimal pointwise error estimates for a 2D SFNSE scheme, with a novel approach to handle nonlinear terms.

Findings

Scheme is unconditionally stable with second order accuracy.

Numerical results align with theoretical error estimates.

Mass and energy are conserved at the discrete level.

Abstract

In this paper, a linearized semi-implicit finite difference scheme is proposed for solving the two-dimensional (2D) space fractional nonlinear Schr\"{o}dinger equation (SFNSE).The scheme has the property of mass and energy conservation on the discrete level, with an unconditional stability and a second order accuracy for both time and spatial variables. The main contribution of this paper is an optimal pointwise error estimate for the 2D SFNSE, which is rigorously established and proved for the first time. Moreover, a novel technique is proposed for dealing with the nonlinear term in the equation, which plays an essential role in the error estimation. Finally, the numerical results confirm well with the theoretical findings.