Fourth order exponential time differencing method with local discontinuous Galerkin approximation for coupled nonlinear Schrodinger equations

Liang, Xiao; Khaliq, Abdul Q. M.; Xing, Yulong

doi:10.4208/cicp.060414.190914a

Title: Fourth order exponential time differencing method with local discontinuous Galerkin approximation for coupled nonlinear Schrodinger equations

Abstract

In this paper, we study a local discontinuous Galerkin method combined with fourth order exponential time differencing Runge-Kutta time discretization and a fourth order conservative method for solving the nonlinear Schrödinger equations. Based on different choices of numerical fluxes, we propose both energy-conserving and energy-dissipative local discontinuous Galerkin methods, and have proven the error estimates for the semi-discrete methods applied to linear Schrödinger equation. The numerical methods are proven to be highly efficient and stable for long-range soliton computations. Finally, extensive numerical examples are provided to illustrate the accuracy, efficiency and reliability of the proposed methods.

Authors:

Liang, Xiao ^[1]; Khaliq, Abdul Q. M. ^[1]; Xing, Yulong ^[2]

Middle Tennessee State Univ., Murfreesboro, TN (United States). Dept. of Mathematical Sciences. Center for Computational Science
Oak Ridge National Lab. (ORNL), Oak Ridge, TN (United States). Computer Science and Mathematics Division; Univ. of Tennessee, Knoxville, TN (United States). Dept. of Mathematics

Publication Date:: Fri Jan 23 00:00:00 EST 2015

Research Org.:: Oak Ridge National Laboratory (ORNL), Oak Ridge, TN (United States)

Sponsoring Org.:: USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR)

Contributing Org.:: Middle Tennessee State Univ., Murfreesboro, TN (United States)

OSTI Identifier:: 1286709

Grant/Contract Number:: AC05-00OR22725

Resource Type:: Accepted Manuscript

Journal Name:: Communications in Computational Physics

Additional Journal Information:: Journal Volume: 17; Journal Issue: 02; Journal ID: ISSN 1815-2406

Publisher:: Global Science Press

Country of Publication:: United States

Language:: English

Subject:: 97 MATHEMATICS AND COMPUTING; exponential time differencing; local discontinuous Galerkin; nonlinear Schrödinger equation; energy conserving; error estimate

Citation Formats


                    Liang, Xiao, Khaliq, Abdul Q. M., and Xing, Yulong. Fourth order exponential time differencing method with local discontinuous Galerkin approximation for coupled nonlinear Schrodinger equations.  United States: N. p., 2015. 
Web.  doi:10.4208/cicp.060414.190914a.

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                    Liang, Xiao, Khaliq, Abdul Q. M., & Xing, Yulong. Fourth order exponential time differencing method with local discontinuous Galerkin approximation for coupled nonlinear Schrodinger equations.  United States.  https://doi.org/10.4208/cicp.060414.190914a

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                    Liang, Xiao, Khaliq, Abdul Q. M., and Xing, Yulong. Fri .  
"Fourth order exponential time differencing method with local discontinuous Galerkin approximation for coupled nonlinear Schrodinger equations".  United States.  https://doi.org/10.4208/cicp.060414.190914a.  https://www.osti.gov/servlets/purl/1286709.

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@article{osti_1286709,

  title        = {Fourth order exponential time differencing method with local discontinuous Galerkin approximation for coupled nonlinear Schrodinger equations},

  author       = {Liang, Xiao and Khaliq, Abdul Q. M. and Xing, Yulong},

  abstractNote = {In this paper, we study a local discontinuous Galerkin method combined with fourth order exponential time differencing Runge-Kutta time discretization and a fourth order conservative method for solving the nonlinear Schrödinger equations. Based on different choices of numerical fluxes, we propose both energy-conserving and energy-dissipative local discontinuous Galerkin methods, and have proven the error estimates for the semi-discrete methods applied to linear Schrödinger equation. The numerical methods are proven to be highly efficient and stable for long-range soliton computations. Finally, extensive numerical examples are provided to illustrate the accuracy, efficiency and reliability of the proposed methods.},

  doi          = {10.4208/cicp.060414.190914a},

  journal      = {Communications in Computational Physics},

  number       = 02,

  volume       = 17,

  place        = {United States},

  year         = {Fri Jan 23 00:00:00 EST 2015},

  month        = {Fri Jan 23 00:00:00 EST 2015}

}

Copy to clipboard

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Works referenced in this record:

On the L∞ convergence of a difference scheme for coupled nonlinear Schrödinger equations
journal, May 2010

Sun, Zhi-zhong; Zhao, Dan-dan
Computers & Mathematics with Applications, Vol. 59, Issue 10
DOI: 10.1016/j.camwa.2010.03.012

Numerical simulation of singular solutions to the two-dimensional cubic schrödinger equation
journal, November 1984

Sulem, P. L.; Sulem, C.; Patera, A.
Communications on Pure and Applied Mathematics, Vol. 37, Issue 6
DOI: 10.1002/cpa.3160370603

Exponential time differencing Crank–Nicolson method with a quartic spline approximation for nonlinear Schrödinger equations
journal, May 2014

Liang, Xiao; Khaliq, Abdul Q. M.; Sheng, Qin
Applied Mathematics and Computation, Vol. 235
DOI: 10.1016/j.amc.2014.02.063

A space-time finite element method for the nonlinear Schrödinger equation: the discontinuous Galerkin method
journal, April 1998

Karakashian, Ohannes; Makridakis, Charalambos
Mathematics of Computation, Vol. 67, Issue 222
DOI: 10.1090/S0025-5718-98-00946-6

Local discontinuous Galerkin methods for nonlinear Schrödinger equations
journal, May 2005

Xu, Yan; Shu, Chi-Wang
Journal of Computational Physics, Vol. 205, Issue 1
DOI: 10.1016/j.jcp.2004.11.001

Optimal Error Estimates of the Semidiscrete Local Discontinuous Galerkin Methods for High Order Wave Equations
journal, January 2012

Xu, Yan; Shu, Chi-Wang
SIAM Journal on Numerical Analysis, Vol. 50, Issue 1
DOI: 10.1137/11082258X

Analytical and numerical aspects of certain nonlinear evolution equations. II. Numerical, nonlinear Schrödinger equation
journal, August 1984

Taha, Thiab R.; Ablowitz, Mark I.
Journal of Computational Physics, Vol. 55, Issue 2
DOI: 10.1016/0021-9991(84)90003-2

The Propagation of Nonlinear Wave Envelopes
journal, April 1967

Benney, D. J.; Newell, A. C.
Journal of Mathematics and Physics, Vol. 46, Issue 1-4
DOI: 10.1002/sapm1967461133

Solving the Generalized Nonlinear Schrödinger Equation via Quartic Spline Approximation
journal, January 2001

Sheng, Q.; Khaliq, A. Q. M.; Al-Said, E. A.
Journal of Computational Physics, Vol. 166, Issue 2
DOI: 10.1006/jcph.2000.6668

A New Class of Time Discretization Schemes for the Solution of Nonlinear PDEs
journal, December 1998

Beylkin, Gregory; Keiser, James M.; Vozovoi, Lev
Journal of Computational Physics, Vol. 147, Issue 2
DOI: 10.1006/jcph.1998.6093

Exponential Time Differencing for Stiff Systems
journal, March 2002

Cox, S. M.; Matthews, P. C.
Journal of Computational Physics, Vol. 176, Issue 2
DOI: 10.1006/jcph.2002.6995

The Local Discontinuous Galerkin Method for Time-Dependent Convection-Diffusion Systems
journal, December 1998

Cockburn, Bernardo; Shu, Chi-Wang
SIAM Journal on Numerical Analysis, Vol. 35, Issue 6
DOI: 10.1137/S0036142997316712

Smoothing schemes for reaction-diffusion systems with nonsmooth data
journal, January 2009

Khaliq, A. Q. M.; Martín-Vaquero, J.; Wade, B. A.
Journal of Computational and Applied Mathematics, Vol. 223, Issue 1
DOI: 10.1016/j.cam.2008.01.017

Runge-Kutta Discontinuous Galerkin Methods for Convection-Dominated Problems
journal, September 2001

Cockburn, Bernardo; Shu, Chi-Wang
Journal of Scientific Computing, Vol. 16, Issue 3, p. 173-261
DOI: 10.1023/A:1012873910884

Numerical simulation of coupled nonlinear Schrödinger equation
journal, July 2001

Ismail, M. S.; Taha, Thiab R.
Mathematics and Computers in Simulation, Vol. 56, Issue 6
DOI: 10.1016/S0378-4754(01)00324-X

Highly accurate finite difference method for coupled nonlinear Schrödinger equation
journal, March 2004

Ismail, M. S.; Alamri, S. Z.
International Journal of Computer Mathematics, Vol. 81, Issue 3
DOI: 10.1080/00207160410001661339

Efficient time discretization for local discontinuous Galerkin methods
journal, July 2007

Shu, Chi-Wang; Xu, Yan; Xia, Yinhua
Discrete and Continuous Dynamical Systems - Series B, Vol. 8, Issue 3
DOI: 10.3934/dcdsb.2007.8.677

Energy conserving local discontinuous Galerkin methods for wave propagation problems
journal, August 2013

Xing, Yulong; Chou, Ching-Shan; Shu, Chi-Wang
Inverse Problems and Imaging, Vol. 7, Issue 3
DOI: 10.3934/ipi.2013.7.967

Optimal energy conserving local discontinuous Galerkin methods for second-order wave equation in heterogeneous media
journal, September 2014

Chou, Ching-Shan; Shu, Chi-Wang; Xing, Yulong
Journal of Computational Physics, Vol. 272
DOI: 10.1016/j.jcp.2014.04.009

Pseudo-spectral solution of nonlinear Schrödinger equations
journal, March 1990

Pathria, D.; Morris, J. Ll.
Journal of Computational Physics, Vol. 87, Issue 1
DOI: 10.1016/0021-9991(90)90228-S

A numerical study of the nonlinear Schrödinger equation
journal, September 1984

Griffiths, D. F.; Mitchell, A. R.; Morris, J. Li.
Computer Methods in Applied Mechanics and Engineering, Vol. 45, Issue 1-3
DOI: 10.1016/0045-7825(84)90156-7

Conservative, discontinuous Galerkin–methods for the generalized Korteweg–de Vries equation
journal, January 2013

Bona, J. L.; Chen, H.; Karakashian, O.
Mathematics of Computation, Vol. 82, Issue 283
DOI: 10.1090/S0025-5718-2013-02661-0

Conservation of phase space properties using exponential integrators on the cubic Schrödinger equation
journal, July 2007

Berland, Håvard; Islas, Alvaro L.; Schober, Constance M.
Journal of Computational Physics, Vol. 225, Issue 1
DOI: 10.1016/j.jcp.2006.11.030

Conerservative and Nonconservative Schemes for the Solution of the Nonlinear Schrödinger Equation
journal, January 1986

Sanz-Serna, J. M.; Verwer, J. G.
IMA Journal of Numerical Analysis, Vol. 6, Issue 1
DOI: 10.1093/imanum/6.1.25

An exponential time differencing method for the nonlinear Schrödinger equation
journal, October 2008

de la Hoz, F.; Vadillo, F.
Computer Physics Communications, Vol. 179, Issue 7
DOI: 10.1016/j.cpc.2008.04.013

On Dysthe's nonlinear Schrödinger equation for deep water waves
journal, April 2000

Lange, Horst
Transport Theory and Statistical Physics, Vol. 29, Issue 3-5
DOI: 10.1080/00411450008205888

Nonlinear pulse propagation in birefringent optical fibers
journal, February 1987

Menyuk, C.
IEEE Journal of Quantum Electronics, Vol. 23, Issue 2
DOI: 10.1109/JQE.1987.1073308

Variational approach to nonlinear pulse propagation in optical fibers
journal, June 1983

Anderson, D.
Physical Review A, Vol. 27, Issue 6
DOI: 10.1103/PhysRevA.27.3135

Exponential integrators
journal, May 2010

Hochbruck, Marlis; Ostermann, Alexander
Acta Numerica, Vol. 19
DOI: 10.1017/S0962492910000048

Exponential integrators
text, January 2010

Hochbruck, Marlis; Ostermann, Alexander
Karlsruhe
DOI: 10.5445/ir/1000024914

Works referencing / citing this record:

An energy-preserving Crank-Nicolson Galerkin method for Hamiltonian partial differential equations: Energy-Preserving Crank-Nicolson Galerkin Method
journal, April 2016

Li, Haochen; Wang, Yushun; Sheng, Qin
Numerical Methods for Partial Differential Equations, Vol. 32, Issue 5
DOI: 10.1002/num.22062

Superconvergence of Ultra-Weak Discontinuous Galerkin Methods for the Linear Schrödinger Equation in One Dimension
journal, January 2020

Chen, Anqi; Cheng, Yingda; Liu, Yong
Journal of Scientific Computing, Vol. 82, Issue 1
DOI: 10.1007/s10915-020-01124-0

Compact and efficient conservative schemes for coupled nonlinear Schrödinger equations: Compact and Efficient Conservative Schemes for CNLS
journal, February 2015

Kong, Linghua; Hong, Jialin; Ji, Lihai
Numerical Methods for Partial Differential Equations, Vol. 31, Issue 6
DOI: 10.1002/num.21969

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Title: Fourth order exponential time differencing method with local discontinuous Galerkin approximation for coupled nonlinear Schrodinger equations

Abstract

Citation Formats

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