Volume 54, Issue 4
A Fast Conservative Scheme for the Space Fractional Nonlinear Schrödinger Equation with Wave Operator

J. Math. Study, 54 (2021), pp. 407-426.

Published online: 2021-06

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• Abstract

A new efficient compact difference scheme is proposed for solving a space fractional nonlinear Schrödinger equation with wave operator. The scheme is proved to conserve the total mass and total energy in a discrete sense. Using the energy method, the proposed scheme is proved to be unconditionally stable and its convergence order is shown to be of $\mathcal{O}( h^6 + \tau^2)$ in the discrete $L_2$ norm with mesh size $h$ and the time step $\tau$. Moreover, a fast difference solver is developed to speed up the numerical computation of the scheme. Numerical experiments are given to support the theoretical analysis and to verify the efficiency, accuracy, and discrete conservation laws.

• Keywords

Space-fractional nonlinear Schrödinger equations, fast difference solver, convergence, conservation laws.

65M06, 65M12, 35Q41

mstf1985@mail.hust.edu.cn (Mustafa Almushaira)

liufei@hust.edu.cn (Fei Liu)

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@Article{JMS-54-407, author = {Mustafa and Almushaira and mstf1985@mail.hust.edu.cn and 16785 and School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, China and Mustafa Almushaira and Fei and Liu and liufei@hust.edu.cn and 16786 and School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, China and Fei Liu}, title = {A Fast Conservative Scheme for the Space Fractional Nonlinear Schrödinger Equation with Wave Operator}, journal = {Journal of Mathematical Study}, year = {2021}, volume = {54}, number = {4}, pages = {407--426}, abstract = {

A new efficient compact difference scheme is proposed for solving a space fractional nonlinear Schrödinger equation with wave operator. The scheme is proved to conserve the total mass and total energy in a discrete sense. Using the energy method, the proposed scheme is proved to be unconditionally stable and its convergence order is shown to be of $\mathcal{O}( h^6 + \tau^2)$ in the discrete $L_2$ norm with mesh size $h$ and the time step $\tau$. Moreover, a fast difference solver is developed to speed up the numerical computation of the scheme. Numerical experiments are given to support the theoretical analysis and to verify the efficiency, accuracy, and discrete conservation laws.

}, issn = {2617-8702}, doi = {https://doi.org/10.4208/jms.v54n4.21.06}, url = {http://global-sci.org/intro/article_detail/jms/19295.html} }
TY - JOUR T1 - A Fast Conservative Scheme for the Space Fractional Nonlinear Schrödinger Equation with Wave Operator AU - Almushaira , Mustafa AU - Liu , Fei JO - Journal of Mathematical Study VL - 4 SP - 407 EP - 426 PY - 2021 DA - 2021/06 SN - 54 DO - http://doi.org/10.4208/jms.v54n4.21.06 UR - https://global-sci.org/intro/article_detail/jms/19295.html KW - Space-fractional nonlinear Schrödinger equations, fast difference solver, convergence, conservation laws. AB -

A new efficient compact difference scheme is proposed for solving a space fractional nonlinear Schrödinger equation with wave operator. The scheme is proved to conserve the total mass and total energy in a discrete sense. Using the energy method, the proposed scheme is proved to be unconditionally stable and its convergence order is shown to be of $\mathcal{O}( h^6 + \tau^2)$ in the discrete $L_2$ norm with mesh size $h$ and the time step $\tau$. Moreover, a fast difference solver is developed to speed up the numerical computation of the scheme. Numerical experiments are given to support the theoretical analysis and to verify the efficiency, accuracy, and discrete conservation laws.

Mustafa Almushaira & Fei Liu. (2021). A Fast Conservative Scheme for the Space Fractional Nonlinear Schrödinger Equation with Wave Operator. Journal of Mathematical Study. 54 (4). 407-426. doi:10.4208/jms.v54n4.21.06
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