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Volume 41, Issue 5
A Linearly-Implicit Energy-Preserving Algorithm for the Two-Dimensional Space-Fractional Nonlinear Schrödinger Equation Based on the SAV Approach

Yayun Fu, Wenjun Cai & Yushun Wang

DOI: 10.4208/jcm.2111-m2020-0177

J. Comp. Math., 41 (2023), pp. 797-816.

Published online: 2023-05

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  • Abstract

The main objective of this paper is to present an efficient structure-preserving scheme, which is based on the idea of the scalar auxiliary variable approach, for solving the two-dimensional space-fractional nonlinear Schrödinger equation. First, we reformulate the equation as an canonical Hamiltonian system, and obtain a new equivalent system via introducing a scalar variable. Then, we construct a semi-discrete energy-preserving scheme by using the Fourier pseudo-spectral method to discretize the equivalent system in space direction. After that, applying the Crank-Nicolson method on the temporal direction gives a linearly-implicit scheme in the fully-discrete version. As expected, the proposed scheme can preserve the energy exactly and more efficient in the sense that only decoupled equations with constant coefficients need to be solved at each time step. Finally, numerical experiments are provided to demonstrate the efficiency and conservation of the scheme.

  • Keywords

Fractional nonlinear Schrödinger equation, Hamiltonian system, Scalar auxiliary variable approach, Structure-preserving algorithm.

  • AMS Subject Headings

35R11, 65M70

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

fyyly@xcu.edu.cn (Yayun Fu)

caiwenjun@njnu.edu.cn (Wenjun Cai)

wangyushun@njnu.edu.cn (Yushun Wang)

  • BibTex
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  • TXT
@Article{JCM-41-797, author = {Fu , YayunCai , Wenjun and Wang , Yushun}, title = {A Linearly-Implicit Energy-Preserving Algorithm for the Two-Dimensional Space-Fractional Nonlinear Schrödinger Equation Based on the SAV Approach}, journal = {Journal of Computational Mathematics}, year = {2023}, volume = {41}, number = {5}, pages = {797--816}, abstract = {

The main objective of this paper is to present an efficient structure-preserving scheme, which is based on the idea of the scalar auxiliary variable approach, for solving the two-dimensional space-fractional nonlinear Schrödinger equation. First, we reformulate the equation as an canonical Hamiltonian system, and obtain a new equivalent system via introducing a scalar variable. Then, we construct a semi-discrete energy-preserving scheme by using the Fourier pseudo-spectral method to discretize the equivalent system in space direction. After that, applying the Crank-Nicolson method on the temporal direction gives a linearly-implicit scheme in the fully-discrete version. As expected, the proposed scheme can preserve the energy exactly and more efficient in the sense that only decoupled equations with constant coefficients need to be solved at each time step. Finally, numerical experiments are provided to demonstrate the efficiency and conservation of the scheme.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.2111-m2020-0177}, url = {http://global-sci.org/intro/article_detail/jcm/21674.html} }
TY - JOUR T1 - A Linearly-Implicit Energy-Preserving Algorithm for the Two-Dimensional Space-Fractional Nonlinear Schrödinger Equation Based on the SAV Approach AU - Fu , Yayun AU - Cai , Wenjun AU - Wang , Yushun JO - Journal of Computational Mathematics VL - 5 SP - 797 EP - 816 PY - 2023 DA - 2023/05 SN - 41 DO - http://doi.org/10.4208/jcm.2111-m2020-0177 UR - https://global-sci.org/intro/article_detail/jcm/21674.html KW - Fractional nonlinear Schrödinger equation, Hamiltonian system, Scalar auxiliary variable approach, Structure-preserving algorithm. AB -

The main objective of this paper is to present an efficient structure-preserving scheme, which is based on the idea of the scalar auxiliary variable approach, for solving the two-dimensional space-fractional nonlinear Schrödinger equation. First, we reformulate the equation as an canonical Hamiltonian system, and obtain a new equivalent system via introducing a scalar variable. Then, we construct a semi-discrete energy-preserving scheme by using the Fourier pseudo-spectral method to discretize the equivalent system in space direction. After that, applying the Crank-Nicolson method on the temporal direction gives a linearly-implicit scheme in the fully-discrete version. As expected, the proposed scheme can preserve the energy exactly and more efficient in the sense that only decoupled equations with constant coefficients need to be solved at each time step. Finally, numerical experiments are provided to demonstrate the efficiency and conservation of the scheme.

Fu , YayunCai , Wenjun and Wang , Yushun. (2023). A Linearly-Implicit Energy-Preserving Algorithm for the Two-Dimensional Space-Fractional Nonlinear Schrödinger Equation Based on the SAV Approach. Journal of Computational Mathematics. 41 (5). 797-816. doi:10.4208/jcm.2111-m2020-0177
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