Symplectic Euler Method for Nonlinear High Order Schrödinger Equation with a Trapped Term

Symplectic Euler Method for Nonlinear High Order Schrödinger Equation with a Trapped Term

Year:    2009

Author:    Fangfang Fu, Linghua Kong, Lan Wang

Advances in Applied Mathematics and Mechanics, Vol. 1 (2009), Iss. 5 : pp. 699–710

Abstract

In this paper, we establish a family of symplectic integrators for a class of high order Schrödinger equations with trapped terms. First, we find its symplectic structure and reduce it to a finite dimensional Hamilton system via spatial discretization. Then we apply the symplectic Euler method to the Hamiltonian system. It is demonstrated that the scheme not only preserves symplectic geometry structure of the original system, but also does not require to resolve coupled nonlinear algebraic equations which is different from the general implicit symplectic schemes. The linear stability of the symplectic Euler scheme and the errors of the numerical solutions are investigated. It shows that the semi-explicit scheme is conditionally stable, first order accurate in time and $2l^{th}$ order accuracy in space. Numerical tests suggest that the symplectic integrators are more effective than non-symplectic ones, such as backward Euler integrators.

You do not have full access to this article.

Already a Subscriber? Sign in as an individual or via your institution

Journal Article Details

Publisher Name:    Global Science Press

Language:    English

DOI:    https://doi.org/10.4208/aamm.09-m0929

Advances in Applied Mathematics and Mechanics, Vol. 1 (2009), Iss. 5 : pp. 699–710

Published online:    2009-01

AMS Subject Headings:    Global Science Press

Copyright:    COPYRIGHT: © Global Science Press

Pages:    12

Keywords:    Symplectic Euler integrator high order Schrödinger equation stability trapped term.

Author Details

Fangfang Fu

Linghua Kong

Lan Wang

  1. Conformal structure-preserving method for damped nonlinear Schrödinger equation

    Fu, Hao | Zhou, Wei-En | Qian, Xu | Song, Song-He | Zhang, Li-Ying

    Chinese Physics B, Vol. 25 (2016), Iss. 11 P.110201

    https://doi.org/10.1088/1674-1056/25/11/110201 [Citations: 13]
  2. Symplectic structure-preserving integrators for the two-dimensional Gross–Pitaevskii equation for BEC

    Kong, Linghua | Hong, Jialin | Fu, Fangfang | Chen, Jing

    Journal of Computational and Applied Mathematics, Vol. 235 (2011), Iss. 17 P.4937

    https://doi.org/10.1016/j.cam.2011.04.019 [Citations: 11]
  3. Classification of positive solutions for an elliptic system with a higher-order fractional Laplacian

    Dou, Jingbo | Qu, Changzheng

    Pacific Journal of Mathematics, Vol. 261 (2013), Iss. 2 P.311

    https://doi.org/10.2140/pjm.2013.261.311 [Citations: 7]
  4. A review of dynamic analysis on space solar power station

    Hu, Weipeng | Deng, Zichen

    Astrodynamics, Vol. 7 (2023), Iss. 2 P.115

    https://doi.org/10.1007/s42064-022-0144-2 [Citations: 13]