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Fourth Order Exponential Time Differencing Method with Local Discontinuous Galerkin Approximation for Coupled Nonlinear Schrödinger Equations

Published online by Cambridge University Press:  23 January 2015

X. Liang ,
A. Q. M. Khaliq  and
Y. Xing
X. Liang*
Affiliation:
Department of Mathematical Sciences and Center for Computational ScienceMiddle Tennessee State University, Murfreesboro, TN 37132-0001, USA
A. Q. M. Khaliq
Affiliation:
Department of Mathematical Sciences and Center for Computational ScienceMiddle Tennessee State University, Murfreesboro, TN 37132-0001, USA
Y. Xing
Affiliation:
Computer Science and Mathematics Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831 and Department of Mathematics, University of Tennessee, Knoxville, TN 37996, USA
*
*Email addresses: xl2h@mtmail.mtsu.edu (X. Liang), Abdul.Khaliq@mtsu.edu (A. Q. M. Khaliq), xingy@math.utk.edu (Y. Xing)
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Abstract

This paper studies 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. Extensive numerical examples are provided to illustrate the accuracy, efficiency and reliability of the proposed methods.

Keywords

65M2065M6065Z05Exponential time differencinglocal discontinuous Galerkinnonlinear Schrödinger equationenergy conservingerror estimate
Type
Research Article
Information
Communications in Computational Physics , Volume 17 , Issue 2 , February 2015 , pp. 510 - 541
Copyright
Copyright © Global Science Press Limited 2015 

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