A linearized and conservative Fourier pseudo-spectral method for the damped nonlinear Schr\"{o}dinger equation in three dimensions

TL;DR

This paper introduces a linearized Fourier pseudo-spectral method for the 3D damped nonlinear Schrödinger equation that conserves mass and energy, providing optimal error estimates and validated by numerical experiments.

Contribution

It develops a novel linearized Fourier pseudo-spectral scheme that preserves key physical invariants and achieves optimal error bounds without grid restrictions.

Findings

The method conserves total mass and energy.

Optimal $L^2$-error estimates are established.

Numerical results confirm theoretical predictions.

Abstract

In this paper, we propose a linearized Fourier pseudo-spectral method, which preserves the total mass and energy conservation laws, for the damped nonlinear Schr\"{o}dinger equation in three dimensions. With the aid of the semi-norm equivalence between the Fourier pseudo-spectral method and the finite difference method, an optimal $L^2$-error estimate for the proposed method without any restriction on the grid ratio is established by analyzing the real and imaginary parts of the error function. Numerical results are addressed to confirm our theoretical analysis.