Construction and Analysis of Structure-Preserving Numerical Algorithm for Two-Dimensional Damped Nonlinear Space Fractional Schrödinger equation

In this paper, we present a novel high-order structure-preserving numerical scheme for solving the damped nonlinear space fractional Schrödinger equation (DNSFSE) in two spatial dimensions. The main idea of constructing new algorithm consists of two parts. Firstly, we introduce an auxiliary exponent...

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Published in:	Journal of scientific computing Vol. 99; no. 3; p. 60
Main Authors:	Ding, Hengfei, Qu, Haidong, Yi, Qian
Format:	Journal Article
Language:	English
Published:	New York Springer US 01.06.2024 Springer Nature B.V
Subjects:	Algorithms Boundary conditions Computational Mathematics and Numerical Analysis Convergence Fourier transforms Iterative algorithms Mathematical and Computational Engineering Mathematical and Computational Physics Mathematics Mathematics and Statistics Numerical analysis Partial differential equations Schrodinger equation Theoretical Iterative algorithm The damped nonlinear space fractional Schrödinger equations Structure-preserving algorithm Numerical differential formula 65M12 65M06
ISSN:	0885-7474, 1573-7691
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Abstract	In this paper, we present a novel high-order structure-preserving numerical scheme for solving the damped nonlinear space fractional Schrödinger equation (DNSFSE) in two spatial dimensions. The main idea of constructing new algorithm consists of two parts. Firstly, we introduce an auxiliary exponential variable to transform the original DNSFSE into a modified one. The modified DNSFSE subjects to the conservation of mass and energy, which is crucial to develop structure-preserving numerical schemes. Secondly, we construct a high-order numerical differential formula to approximate the Riesz derivative in space, which contributes to a semi-discrete difference scheme for the modified DNSFSE. Combining the semi-discrete scheme with the variant Crank–Nicolson method in time, we can obtain the fully-discrete difference scheme for solving the modified DNSFSE. The advantage of the proposed scheme is that a fourth-order convergent accuracy can be achieved in space while maintaining the conservation of mass and energy. Subsequently, we conduct a detailed study on the boundedness, uniqueness, and convergence of solution for fully-discrete scheme. Furthermore, an improved efficient iterative algorithm is proposed for the fully-discrete scheme, which has the advantage of maintaining the same convergence order as the original difference scheme. Finally, extensive numerical results are reported to further verify the correctness of theoretical analysis and the effectiveness of the proposed numerical algorithm.
AbstractList	In this paper, we present a novel high-order structure-preserving numerical scheme for solving the damped nonlinear space fractional Schrödinger equation (DNSFSE) in two spatial dimensions. The main idea of constructing new algorithm consists of two parts. Firstly, we introduce an auxiliary exponential variable to transform the original DNSFSE into a modified one. The modified DNSFSE subjects to the conservation of mass and energy, which is crucial to develop structure-preserving numerical schemes. Secondly, we construct a high-order numerical differential formula to approximate the Riesz derivative in space, which contributes to a semi-discrete difference scheme for the modified DNSFSE. Combining the semi-discrete scheme with the variant Crank–Nicolson method in time, we can obtain the fully-discrete difference scheme for solving the modified DNSFSE. The advantage of the proposed scheme is that a fourth-order convergent accuracy can be achieved in space while maintaining the conservation of mass and energy. Subsequently, we conduct a detailed study on the boundedness, uniqueness, and convergence of solution for fully-discrete scheme. Furthermore, an improved efficient iterative algorithm is proposed for the fully-discrete scheme, which has the advantage of maintaining the same convergence order as the original difference scheme. Finally, extensive numerical results are reported to further verify the correctness of theoretical analysis and the effectiveness of the proposed numerical algorithm.
ArticleNumber	60
Author	Yi, Qian Ding, Hengfei Qu, Haidong
Author_xml	– sequence: 1 givenname: Hengfei surname: Ding fullname: Ding, Hengfei organization: School of Mathematics and Statistics, The Center for Applied Mathematics of Guangxi, Guangxi Normal University, Education Department of Guangxi Zhuang Autonomous Region, Key Laboratory of Mathematical Model and Application (Guangxi Normal University) – sequence: 2 givenname: Haidong surname: Qu fullname: Qu, Haidong organization: Department of Mathematics, Hanshan Normal University – sequence: 3 givenname: Qian orcidid: 0000-0003-3664-6519 surname: Yi fullname: Yi, Qian email: yiqianqianyi@163.com organization: School of Mathematics and Statistics, Guangxi Normal University
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SubjectTerms	Algorithms Boundary conditions Computational Mathematics and Numerical Analysis Convergence Fourier transforms Iterative algorithms Mathematical and Computational Engineering Mathematical and Computational Physics Mathematics Mathematics and Statistics Numerical analysis Partial differential equations Schrodinger equation Theoretical
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Title	Construction and Analysis of Structure-Preserving Numerical Algorithm for Two-Dimensional Damped Nonlinear Space Fractional Schrödinger equation
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