Mass- And energy-conserved numerical schemes for nonlinear Schrödinger equations

Xiaobing Feng; Hailiang Liu; Shu Ma

doi:10.4208/cicp.2019.js60.05

Mass- And energy-conserved numerical schemes for nonlinear Schrödinger equations

Xiaobing Feng^*
, Hailiang Liu^*
, Shu Ma

^*Corresponding author for this work

Department of Mathematics

Research output: Contribution to journal › Journal article › peer-review

26 Citations (Scopus)

Abstract

In 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.

Original language	English
Pages (from-to)	1365-1396
Number of pages	32
Journal	Communications in Computational Physics
Volume	26
Issue number	5
DOIs	https://doi.org/10.4208/cicp.2019.js60.05
Publication status	Published - Nov 2019

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SDG 7 Affordable and Clean Energy

User-Defined Keywords

BDF schemes
Finite element methods
Finite time blow-ups
Mass conservation and energy conservation
Nonlinear Schrödinger equations

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10.4208/cicp.2019.js60.05

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@article{525ddaacf71f4325b1a6f736a38fb7a0,

title = "Mass- And energy-conserved numerical schemes for nonlinear Schr{\"o}dinger equations",

abstract = "In this paper, we propose a family of time-stepping schemes for approximating general nonlinear Schr{\"o}dinger 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 = "BDF schemes, Finite element methods, Finite time blow-ups, Mass conservation and energy conservation, Nonlinear Schr{\"o}dinger equations",

author = "Xiaobing Feng and Hailiang Liu and Shu Ma",

note = "The work of the first author was partially supported by the NSF Grant DMS-1620168, and the work of the second author was partially supported by NSF Grant DMS-1812666. Part of the third author{\textquoteright}s work was done during a recent visit to the University of Tennessee at Knoxville (UTK), the author would like to thank Department of Mathematics of UTK for the support and hospitality. The visit was financially supported by a scholarship from the author{\textquoteright}s home institution, Northwestern Polytechnical University of China. Publisher Copyright: {\textcopyright} 2019 Global-Science Press.",

year = "2019",

month = nov,

doi = "10.4208/cicp.2019.js60.05",

language = "English",

volume = "26",

pages = "1365--1396",

journal = "Communications in Computational Physics",

issn = "1815-2406",

publisher = "Global Science Press",

number = "5",

}

TY - JOUR

T1 - Mass- And energy-conserved numerical schemes for nonlinear Schrödinger equations

AU - Feng, Xiaobing

AU - Liu, Hailiang

AU - Ma, Shu

N1 - The work of the first author was partially supported by the NSF Grant DMS-1620168, and the work of the second author was partially supported by NSF Grant DMS-1812666. Part of the third author’s work was done during a recent visit to the University of Tennessee at Knoxville (UTK), the author would like to thank Department of Mathematics of UTK for the support and hospitality. The visit was financially supported by a scholarship from the author’s home institution, Northwestern Polytechnical University of China. Publisher Copyright: © 2019 Global-Science Press.

PY - 2019/11

Y1 - 2019/11

N2 - In 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.

AB - In 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.

KW - BDF schemes

KW - Finite element methods

KW - Finite time blow-ups

KW - Mass conservation and energy conservation

KW - Nonlinear Schrödinger equations

UR - https://www.scopus.com/pages/publications/85071834406

UR - https://global-sci.com/article/79862/mass-and-energy-conserved-numerical-schemes-for-nonlinear-schrodinger-equations

U2 - 10.4208/cicp.2019.js60.05

DO - 10.4208/cicp.2019.js60.05

M3 - Journal article

AN - SCOPUS:85071834406

SN - 1815-2406

VL - 26

SP - 1365

EP - 1396

JO - Communications in Computational Physics

JF - Communications in Computational Physics

IS - 5

ER -

Mass- And energy-conserved numerical schemes for nonlinear Schrödinger equations

Abstract

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