An explicit spectral collocation method using nonpolynomial basis functions for the time‐dependent Schrödinger equation. (22nd October 2018)

• Record Type: Journal Article
• Authors: Fang, Jinwei; Wu, Boying; Liu, Wenjie
• Is Part Of: Mathematical methods in the applied sciences. Volume 42:Number 1(2019)
• Page Start: 186
• Page End: 203
• Publication Date: 2018-10-22
• Subjects: collocation method -- nonpolynomial basis functions -- Schrödinger equation -- transparent boundary conditions; Mathematics -- Periodicals; Technology -- Mathematics -- Periodicals; 519
• DOI: 10.1002/mma.5332
• Languages: English
• ISSNs: 0170-4214

Abstract

Abstract : We propose a spectral collocation method for the numerical solution of the time‐dependent Schrödinger equation, where the newly developed nonpolynomial functions in a previous study are used as basis functions. Equipped with the new basis functions, various boundary conditions can be imposed exactly. The preferable semi‐implicit time marching schemes are employed for temporal discretization. Moreover, the new basis functions build in a free parameter λ intrinsically, which can be chosen properly so that the semi‐implicit scheme collapses to an explicit scheme. The method is further applied to linear Schrödinger equation set in unbounded domain. The transparent boundary conditions are constructed for time semidiscrete scheme of the linear Schrödinger equation. We employ spectral collocation method using the new basis functions for the spatial discretization, which allows for the exact imposition of the transparent boundary conditions. Comprehensive numerical tests both in bounded and unbounded domain are performed to demonstrate the attractive features of the proposed method.