Analysis of finite element method for one-dimensional time-dependent Schrödinger equation on unbounded domain

Jicheng Jin; Xiaonan Wu

doi:10.1016/j.cam.2007.08.006

Analysis of finite element method for one-dimensional time-dependent Schrödinger equation on unbounded domain

Jicheng Jin^*, Xiaonan Wu

^*Corresponding author for this work

Department of Mathematics

Research output: Contribution to journal › Journal article › peer-review

23 Citations (Scopus)

Abstract

This paper addresses the theoretical analysis of a fully discrete scheme for the one-dimensional time-dependent Schrödinger equation on unbounded domain. We first reduce the original problem into an initial-boundary value problem in a bounded domain by introducing a transparent boundary condition, then fully discretize this reduced problem by applying Crank-Nicolson scheme in time and linear or quadratic finite element approximation in space. By a rigorous analysis, this scheme has been proved to be unconditionally stable and convergent, its convergence order has also be obtained. Finally, two numerical examples are performed to show the accuracy of the scheme.

Original language	English
Pages (from-to)	240-256
Number of pages	17
Journal	Journal of Computational and Applied Mathematics
Volume	220
Issue number	1-2
DOIs	https://doi.org/10.1016/j.cam.2007.08.006
Publication status	Published - 15 Oct 2008

Scopus Subject Areas

Computational Mathematics
Applied Mathematics

User-Defined Keywords

Artificial boundary
Finite element method
Schrödinger equation

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10.1016/j.cam.2007.08.006

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title = "Analysis of finite element method for one-dimensional time-dependent Schr{\"o}dinger equation on unbounded domain",

abstract = "This paper addresses the theoretical analysis of a fully discrete scheme for the one-dimensional time-dependent Schr{\"o}dinger equation on unbounded domain. We first reduce the original problem into an initial-boundary value problem in a bounded domain by introducing a transparent boundary condition, then fully discretize this reduced problem by applying Crank-Nicolson scheme in time and linear or quadratic finite element approximation in space. By a rigorous analysis, this scheme has been proved to be unconditionally stable and convergent, its convergence order has also be obtained. Finally, two numerical examples are performed to show the accuracy of the scheme.",

keywords = "Artificial boundary, Finite element method, Schr{\"o}dinger equation",

author = "Jicheng Jin and Xiaonan Wu",

note = "The research is supported by NSFC (10771178), Hunan Provincial Natural Science Foundation of China (05JJ40014) and Scientific Research Fund of Hunan Provincial Education Department (05C102, 07A068).",

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T1 - Analysis of finite element method for one-dimensional time-dependent Schrödinger equation on unbounded domain

AU - Jin, Jicheng

AU - Wu, Xiaonan

N1 - The research is supported by NSFC (10771178), Hunan Provincial Natural Science Foundation of China (05JJ40014) and Scientific Research Fund of Hunan Provincial Education Department (05C102, 07A068).

PY - 2008/10/15

Y1 - 2008/10/15

N2 - This paper addresses the theoretical analysis of a fully discrete scheme for the one-dimensional time-dependent Schrödinger equation on unbounded domain. We first reduce the original problem into an initial-boundary value problem in a bounded domain by introducing a transparent boundary condition, then fully discretize this reduced problem by applying Crank-Nicolson scheme in time and linear or quadratic finite element approximation in space. By a rigorous analysis, this scheme has been proved to be unconditionally stable and convergent, its convergence order has also be obtained. Finally, two numerical examples are performed to show the accuracy of the scheme.

AB - This paper addresses the theoretical analysis of a fully discrete scheme for the one-dimensional time-dependent Schrödinger equation on unbounded domain. We first reduce the original problem into an initial-boundary value problem in a bounded domain by introducing a transparent boundary condition, then fully discretize this reduced problem by applying Crank-Nicolson scheme in time and linear or quadratic finite element approximation in space. By a rigorous analysis, this scheme has been proved to be unconditionally stable and convergent, its convergence order has also be obtained. Finally, two numerical examples are performed to show the accuracy of the scheme.

KW - Artificial boundary

KW - Finite element method

KW - Schrödinger equation

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DO - 10.1016/j.cam.2007.08.006

M3 - Journal article

AN - SCOPUS:47849094600

SN - 0377-0427

VL - 220

SP - 240

EP - 256

JO - Journal of Computational and Applied Mathematics

JF - Journal of Computational and Applied Mathematics

IS - 1-2

ER -

Analysis of finite element method for one-dimensional time-dependent Schrödinger equation on unbounded domain

Abstract

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