Communications in Mathematical Sciences

Volume 20 (2022)

Number 2

A Fourier collocation method for Schrödinger–Poisson system with perfectly matched layer

Pages: 523 – 542

DOI: https://dx.doi.org/10.4310/CMS.2022.v20.n2.a10

Authors

Ronghua Cheng (School of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing, Jiangsu, China; and School of Statistics and Mathematics, Yunnan University of Finance and Economics, Kunming, Yunnan, China)

Liping Wu (Faculty of Information Engineering and Automation, Kunming University of Science and Technology, Kunming, Yunnan, China)

Chunping Pang (School of Statistics and Mathematics, Yunnan University of Finance and Economics, Kunming, Yunnan, China)

Hanquan Wang (School of Statistics and Mathematics, Yunnan University of Finance and Economics, Kunming, Yunnan, China)

Abstract

Fourier spectral method has been widely used to solve Schrödinger equation with constant coefficients. It meets difficulties and loses its efficiency when solving Schrödinger equation with variable coefficients. We show that Fourier collocation method can be applied to efficiently solve Schrödinger equation with variable coefficients. The method is characterized by the expansion of the solution in terms of Fourier series-based functions, while the expansion coefficients are computed so that the equation is satisfied exactly at a set of collocation points. We implement the method to solve the Schrödinger–Poisson (SP) system with perfectly matched layer (PML), which is a Schrödinger-type equation with variable coefficients. We carry out numerical simulation for the SP system by employing splitting method in time and Fourier collocation method in space, respectively. Numerical results show that the Fourier-collocation method coupled with PML technique can absorb well the outgoing waves governed by the Schrödinger equation when the wave goes out of the computational boundary.

Keywords

Schrödinger–Poisson system, perfectly matched layer, Fourier collocation method, time-splitting method

2010 Mathematics Subject Classification

65N12, 65N35, 65Zxx

Full Text (PDF format)

The research of H.Wang is supported in part by the Natural Science Foundation of China (NSFC) under grant Nos. 11871418 and 11971120, by Yunnan Fundamental Research Projects under grant No. 202101AS070044, and by Program for Innovative Reseach Team on Science and Technology in Universities of Yunnan Province.

Received 21 February 2020

Received revised 4 August 2021

Accepted 5 August 2021

Published 28 January 2022