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Volume 26, Issue 5
Mass- and Energy-Conserved Numerical Schemes for Nonlinear Schrödinger Equations

Xiaobing Feng, Hailiang Liu & Shu Ma

DOI: 10.4208/cicp.2019.js60.05

Commun. Comput. Phys., 26 (2019), pp. 1365-1396.

Published online: 2019-08

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  • Abstract

n this paper, we propose a family of time-stepping schemes for approximating general nonlinear Schrödinger equations. The proposed schemes all satisfy both mass and energy conservation (in a modified form for the latter). Truncation and dispersion error analyses are provided for four proposed schemes. Efficient fixed-point iterative solvers are also constructed to solve the resulting nonlinear discrete problems. As a byproduct, an efficient one-step implementation of the BDF schemes is obtained as well. Extensive numerical experiments are presented to demonstrate the convergence and the capability of capturing the blow-up time of the proposed schemes.

  • Keywords

Nonlinear Schrödinger equations, mass conservation and energy conservation, BDF schemes, finite element methods, finite time blow-ups.

  • AMS Subject Headings

65M06, 65M12

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

xfeng@math.utk.edu (Xiaobing Feng)

hliu@iastate.edu (Hailiang Liu)

mashu@mail.nwpu.edu.cn (Shu Ma)

  • BibTex
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  • TXT
@Article{CiCP-26-1365, author = {Feng , XiaobingLiu , Hailiang and Ma , Shu}, title = {Mass- and Energy-Conserved Numerical Schemes for Nonlinear Schrödinger Equations}, journal = {Communications in Computational Physics}, year = {2019}, volume = {26}, number = {5}, pages = {1365--1396}, abstract = {

n this paper, we propose a family of time-stepping schemes for approximating general nonlinear Schrödinger equations. The proposed schemes all satisfy both mass and energy conservation (in a modified form for the latter). Truncation and dispersion error analyses are provided for four proposed schemes. Efficient fixed-point iterative solvers are also constructed to solve the resulting nonlinear discrete problems. As a byproduct, an efficient one-step implementation of the BDF schemes is obtained as well. Extensive numerical experiments are presented to demonstrate the convergence and the capability of capturing the blow-up time of the proposed schemes.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.2019.js60.05}, url = {http://global-sci.org/intro/article_detail/cicp/13268.html} }
TY - JOUR T1 - Mass- and Energy-Conserved Numerical Schemes for Nonlinear Schrödinger Equations AU - Feng , Xiaobing AU - Liu , Hailiang AU - Ma , Shu JO - Communications in Computational Physics VL - 5 SP - 1365 EP - 1396 PY - 2019 DA - 2019/08 SN - 26 DO - http://doi.org/10.4208/cicp.2019.js60.05 UR - https://global-sci.org/intro/article_detail/cicp/13268.html KW - Nonlinear Schrödinger equations, mass conservation and energy conservation, BDF schemes, finite element methods, finite time blow-ups. AB -

n this paper, we propose a family of time-stepping schemes for approximating general nonlinear Schrödinger equations. The proposed schemes all satisfy both mass and energy conservation (in a modified form for the latter). Truncation and dispersion error analyses are provided for four proposed schemes. Efficient fixed-point iterative solvers are also constructed to solve the resulting nonlinear discrete problems. As a byproduct, an efficient one-step implementation of the BDF schemes is obtained as well. Extensive numerical experiments are presented to demonstrate the convergence and the capability of capturing the blow-up time of the proposed schemes.

Feng , XiaobingLiu , Hailiang and Ma , Shu. (2019). Mass- and Energy-Conserved Numerical Schemes for Nonlinear Schrödinger Equations. Communications in Computational Physics. 26 (5). 1365-1396. doi:10.4208/cicp.2019.js60.05
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