Volume 10, Issue 5
A Galerkin Splitting Symplectic Method for the Two Dimensional Nonlinear Schrödinger Equation

Zhenguo Mu, Haochen Li, Yushun Wang & Wenjun Cai

Adv. Appl. Math. Mech., 10 (2018), pp. 1069-1089.

Published online: 2018-07

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  • Abstract

In this paper, we propose a Galerkin splitting symplectic (GSS) method for solving the 2D nonlinear Schrödinger equation based on the weak formulation of the equation. First, the model equation is discretized by the Galerkin method in spatial direction by a selected finite element method and the semi-discrete system is rewritten as a finite-dimensional canonical Hamiltonian system. Then the resulted Hamiltonian system is split into a linear Hamiltonian subsystem and a nonlinear subsystem. The linear Hamiltonian subsystem is solved by the implicit midpoint method and the nonlinear subsystem is integrated exactly. By the Strang splitting method, we obtain a fully implicit scheme for the 2D nonlinear Schrödinger equation (NLS), which is symmetric and of order 2 in time. Furthermore, we apply the FFT technique to improve computation efficiency of the new scheme. It is proven that our scheme preserves the mass conservation and the symplectic conservation. Comprehensive numerical experiments are carried out to illustrate the accuracy of the scheme as well as its conservative properties.

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@Article{AAMM-10-1069, author = {}, title = {A Galerkin Splitting Symplectic Method for the Two Dimensional Nonlinear Schrödinger Equation}, journal = {Advances in Applied Mathematics and Mechanics}, year = {2018}, volume = {10}, number = {5}, pages = {1069--1089}, abstract = {

In this paper, we propose a Galerkin splitting symplectic (GSS) method for solving the 2D nonlinear Schrödinger equation based on the weak formulation of the equation. First, the model equation is discretized by the Galerkin method in spatial direction by a selected finite element method and the semi-discrete system is rewritten as a finite-dimensional canonical Hamiltonian system. Then the resulted Hamiltonian system is split into a linear Hamiltonian subsystem and a nonlinear subsystem. The linear Hamiltonian subsystem is solved by the implicit midpoint method and the nonlinear subsystem is integrated exactly. By the Strang splitting method, we obtain a fully implicit scheme for the 2D nonlinear Schrödinger equation (NLS), which is symmetric and of order 2 in time. Furthermore, we apply the FFT technique to improve computation efficiency of the new scheme. It is proven that our scheme preserves the mass conservation and the symplectic conservation. Comprehensive numerical experiments are carried out to illustrate the accuracy of the scheme as well as its conservative properties.

}, issn = {2075-1354}, doi = {https://doi.org/10.4208/aamm.OA-2017-0222}, url = {http://global-sci.org/intro/article_detail/aamm/12589.html} }
TY - JOUR T1 - A Galerkin Splitting Symplectic Method for the Two Dimensional Nonlinear Schrödinger Equation JO - Advances in Applied Mathematics and Mechanics VL - 5 SP - 1069 EP - 1089 PY - 2018 DA - 2018/07 SN - 10 DO - http://dor.org/10.4208/aamm.OA-2017-0222 UR - https://global-sci.org/intro/article_detail/aamm/12589.html KW - AB -

In this paper, we propose a Galerkin splitting symplectic (GSS) method for solving the 2D nonlinear Schrödinger equation based on the weak formulation of the equation. First, the model equation is discretized by the Galerkin method in spatial direction by a selected finite element method and the semi-discrete system is rewritten as a finite-dimensional canonical Hamiltonian system. Then the resulted Hamiltonian system is split into a linear Hamiltonian subsystem and a nonlinear subsystem. The linear Hamiltonian subsystem is solved by the implicit midpoint method and the nonlinear subsystem is integrated exactly. By the Strang splitting method, we obtain a fully implicit scheme for the 2D nonlinear Schrödinger equation (NLS), which is symmetric and of order 2 in time. Furthermore, we apply the FFT technique to improve computation efficiency of the new scheme. It is proven that our scheme preserves the mass conservation and the symplectic conservation. Comprehensive numerical experiments are carried out to illustrate the accuracy of the scheme as well as its conservative properties.

Zhenguo Mu, Haochen Li, Yushun Wang & Wenjun Cai. (1970). A Galerkin Splitting Symplectic Method for the Two Dimensional Nonlinear Schrödinger Equation. Advances in Applied Mathematics and Mechanics. 10 (5). 1069-1089. doi:10.4208/aamm.OA-2017-0222
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