A high-order accurate wavelet method for solving Schrödinger equations with general nonlinearity

doi:10.1007/s10483-018-2299-6

Applied Mathematics and Mechanics (English Edition) ›› 2018, Vol. 39 ›› Issue (2): 275-290.doi: https://doi.org/10.1007/s10483-018-2299-6

A high-order accurate wavelet method for solving Schrödinger equations with general nonlinearity

Jiaqun WANG¹, Xiaojing LIU^1,2, Youhe ZHOU¹

1. Key Laboratory of Mechanics on Disaster and Environment in Western China, the Ministry of Education, College of Civil Engineering and Mechanics, Lanzhou University, Lanzhou 730000, China;
2. State Key Laboratory of Structural Analysis for Industrial Equipment, Dalian University of Technology, Dalian 116024, China

收稿日期:2016-12-08 修回日期:2017-08-01 出版日期:2018-02-01 发布日期:2018-02-01
通讯作者: Youhe ZHOU E-mail:zhouyh@lzu.edu.cn
基金资助:
Project supported by the National Natural Science Foundation of China (Nos. 11502103 and 11421062) and the Open Fund of State Key Laboratory of Structural Analysis for Industrial Equipment of China (No. GZ15115)

A high-order accurate wavelet method for solving Schrödinger equations with general nonlinearity

Jiaqun WANG¹, Xiaojing LIU^1,2, Youhe ZHOU¹

1. Key Laboratory of Mechanics on Disaster and Environment in Western China, the Ministry of Education, College of Civil Engineering and Mechanics, Lanzhou University, Lanzhou 730000, China;
2. State Key Laboratory of Structural Analysis for Industrial Equipment, Dalian University of Technology, Dalian 116024, China

Received:2016-12-08 Revised:2017-08-01 Online:2018-02-01 Published:2018-02-01
Contact: Youhe ZHOU E-mail:zhouyh@lzu.edu.cn
Supported by:
Project supported by the National Natural Science Foundation of China (Nos. 11502103 and 11421062) and the Open Fund of State Key Laboratory of Structural Analysis for Industrial Equipment of China (No. GZ15115)

摘要/Abstract

摘要： A sampling approximation for a function defined on a bounded interval is proposed by combining the Coiflet-type wavelet expansion and the boundary extension technique. Based on such a wavelet approximation scheme, a Galerkin procedure is developed for the spatial discretization of the generalized nonlinear Schrödinger (NLS) equations, and a system of ordinary differential equations for the time dependent unknowns is obtained. Then, the classical fourth-order explicit Runge-Kutta method is used to solve this semi-discretization system. To justify the present method, several widely considered problems are solved as the test examples, and the results demonstrate that the proposed wavelet algorithm has much better accuracy and a faster convergence rate in space than many existing numerical methods.

关键词: vector bundle dynamics, nonuniform hyperbolicity, ergodicity, admissible perturbation, pointwise boundedness, Galerkin method, wavelet, high-order convergence, generalized nonlinear Schrödinger (NLS) equation

Abstract: A sampling approximation for a function defined on a bounded interval is proposed by combining the Coiflet-type wavelet expansion and the boundary extension technique. Based on such a wavelet approximation scheme, a Galerkin procedure is developed for the spatial discretization of the generalized nonlinear Schrödinger (NLS) equations, and a system of ordinary differential equations for the time dependent unknowns is obtained. Then, the classical fourth-order explicit Runge-Kutta method is used to solve this semi-discretization system. To justify the present method, several widely considered problems are solved as the test examples, and the results demonstrate that the proposed wavelet algorithm has much better accuracy and a faster convergence rate in space than many existing numerical methods.

Key words: vector bundle dynamics, nonuniform hyperbolicity, ergodicity, admissible perturbation, pointwise boundedness, high-order convergence, Galerkin method, generalized nonlinear Schrödinger (NLS) equation, wavelet

中图分类号:

Jiaqun WANG, Xiaojing LIU, Youhe ZHOU. A high-order accurate wavelet method for solving Schrödinger equations with general nonlinearity[J]. Applied Mathematics and Mechanics (English Edition), 2018, 39(2): 275-290.

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A high-order accurate wavelet method for solving Schrödinger equations with general nonlinearity

A high-order accurate wavelet method for solving Schrödinger equations with general nonlinearity

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