Communications in Mathematical Sciences
Volume 20 (2022)
A Fourier collocation method for Schrödinger–Poisson system with perfectly matched layer
Pages: 523 – 542
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.
Schrödinger–Poisson system, perfectly matched layer, Fourier collocation method, time-splitting method
2010 Mathematics Subject Classification
65N12, 65N35, 65Zxx
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