Numerical Computations of Nonlocal Schrödinger Equations on the Real Line

doi:10.1007/s42967-019-00052-7

Communications on Applied Mathematics and Computation ›› 2020, Vol. 2 ›› Issue (2): 241-260.doi: 10.1007/s42967-019-00052-7

• ORIGINAL PAPER • 上一篇下一篇

Numerical Computations of Nonlocal Schrödinger Equations on the Real Line

Yonggui Yan¹, Jiwei Zhang², Chunxiong Zheng^3,4

1 Beijing Computational Science Research Center, Beijing 100093, China;
2 School of Mathematics and Statistics, and Hubei Key Laboratory of Computational Science, Wuhan University, Wuhan 430072, China;
3 Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China;
4 College of Mathematics and Systems Science, Xinjiang University, Ürümqi 830046, China

收稿日期:2018-10-05 修回日期:2019-07-14 出版日期:2020-06-20 发布日期:2020-02-19
通讯作者: Jiwei Zhang, Yonggui Yan, Chunxiong Zheng E-mail:jiweizhang@whu.edu.cn;yan_yonggui@csrc.ac.cn;czheng@tsinghua.edu.cn

Numerical Computations of Nonlocal Schrödinger Equations on the Real Line

Yonggui Yan¹, Jiwei Zhang², Chunxiong Zheng^3,4

1 Beijing Computational Science Research Center, Beijing 100093, China;
2 School of Mathematics and Statistics, and Hubei Key Laboratory of Computational Science, Wuhan University, Wuhan 430072, China;
3 Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China;
4 College of Mathematics and Systems Science, Xinjiang University, Ürümqi 830046, China

Received:2018-10-05 Revised:2019-07-14 Online:2020-06-20 Published:2020-02-19
Contact: Jiwei Zhang, Yonggui Yan, Chunxiong Zheng E-mail:jiweizhang@whu.edu.cn;yan_yonggui@csrc.ac.cn;czheng@tsinghua.edu.cn

摘要/Abstract

摘要： The numerical computation of nonlocal Schrödinger equations (SEs) on the whole real axis is considered. Based on the artifcial boundary method, we frst derive the exact artifcial nonrefecting boundary conditions. For the numerical implementation, we employ the quadrature scheme proposed in Tian and Du (SIAM J Numer Anal 51:3458-3482, 2013) to discretize the nonlocal operator, and apply the z-transform to the discrete nonlocal system in an exterior domain, and derive an exact solution expression for the discrete system. This solution expression is referred to our exact nonrefecting boundary condition and leads us to reformulate the original infnite discrete system into an equivalent fnite discrete system. Meanwhile, the trapezoidal quadrature rule is introduced to discretize the contour integral involved in exact boundary conditions. Numerical examples are fnally provided to demonstrate the efectiveness of our approach.

关键词: Nonrefecting boundary conditions, Artifcial boundary method, Nonlocal Schrödinger equation, z-transform, Nonlocal models

Abstract: The numerical computation of nonlocal Schrödinger equations (SEs) on the whole real axis is considered. Based on the artifcial boundary method, we frst derive the exact artifcial nonrefecting boundary conditions. For the numerical implementation, we employ the quadrature scheme proposed in Tian and Du (SIAM J Numer Anal 51:3458-3482, 2013) to discretize the nonlocal operator, and apply the z-transform to the discrete nonlocal system in an exterior domain, and derive an exact solution expression for the discrete system. This solution expression is referred to our exact nonrefecting boundary condition and leads us to reformulate the original infnite discrete system into an equivalent fnite discrete system. Meanwhile, the trapezoidal quadrature rule is introduced to discretize the contour integral involved in exact boundary conditions. Numerical examples are fnally provided to demonstrate the efectiveness of our approach.

Key words: Nonrefecting boundary conditions, Artifcial boundary method, Nonlocal Schrödinger equation, z-transform, Nonlocal models

中图分类号:

Yonggui Yan, Jiwei Zhang, Chunxiong Zheng. Numerical Computations of Nonlocal Schrödinger Equations on the Real Line[J]. Communications on Applied Mathematics and Computation, 2020, 2(2): 241-260.

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Numerical Computations of Nonlocal Schrödinger Equations on the Real Line

Numerical Computations of Nonlocal Schrödinger Equations on the Real Line

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