A Conservative Crank-Nicolson Fourier Spectral Method for the Space Fractional Schrödinger Equation with Wave Operators

Lei Zhang,Rui Yang,Lisha Wang,Li Zhang,Xinguang Zhang

doi:10.1155/2021/5137845

Abstract

In this paper, the Crank-Nicolson Fourier spectral method is proposed for solving the space fractional Schrödinger equation with wave operators. The equation is treated with the conserved Crank-Nicolson Fourier Galerkin method and the conserved Crank-Nicolson Fourier collocation method, respectively. In addition, the ability of the constructed numerical method to maintain the conservation of mass and energy is studied in detail. Meanwhile, the convergence with spectral accuracy in space and second-order accuracy in time is verified for both Galerkin and collocation approximations. Finally, the numerical experiments verify the properties of the conservative difference scheme and demonstrate the correctness of theoretical results.

Highlights

The Schrödinger equation is one of the most basic equations in quantum mechanics, which was proposed by Austrian physicist Schrödinger in 1926
Few studies have been investigated on conservative methods for the fractional Schrödinger equation
Numerical examples are introduced to verify the correctness of the theoretical results and the validity of our numerical methods

Summary

Introduction

The Schrödinger equation is one of the most basic equations in quantum mechanics, which was proposed by Austrian physicist Schrödinger in 1926. Few studies have been investigated on conservative methods for the fractional Schrödinger equation. From the viewpoint of mathematics, the solution of the Schrödinger system has important geometric structures such as energy conservation and multisymplectic structure These properties should be maintained as much as possible in the construction of numerical methods. Bridges and Reich [12] first put forward the Hamiltonian system using the Fourier spectrum discrete method in 2001 Based on their theoretical ideas, Chen and Qin [13] in the same year proposed the Fourier pseudo-spectral method for the Hamiltonian partial differential equation and used it to integrate the nonlinear Schrödinger equation with periodic boundary conditions.

Crank-Nicolson Fourier Spectral Method and Conservation Laws

Theory Analysis of Conservation

Theory Analysis of Convergence

Numerical Example

Conclusion