Title

Jacobi-Gauss-Lobatto collocation method for the numerical solution of 1+1 nonlinear Schrodinger equations

Authors

Authors

E. H. Doha; A. H. Bhrawy; M. A. Abdelkawy;R. A. Van Gorder

Comments

Authors: contact us about adding a copy of your work at STARS@ucf.edu

Abbreviated Journal Title

J. Comput. Phys.

Keywords

Nonlinear complex Schrodinger equations; Gross-Pitaevskii equation; Collocation method; Jacobi-Gauss-Lobatto quadrature; Implicit; Runge-Kutta method; VOLTERRA INTEGRAL-EQUATIONS; DIFFERENTIAL-EQUATIONS; PSEUDOSPECTRAL; METHOD; GALERKIN METHOD; SCHEME; APPROXIMATION; POLYNOMIALS; ALGORITHMS; SOLITON; VORTEX; Computer Science, Interdisciplinary Applications; Physics, Mathematical

Abstract

A Jacobi-Gauss-Lobatto collocation (J-GL-C) method, used in combination with the implicit Runge-Kutta method of fourth order, is proposed as a numerical algorithm for the approximation of solutions to nonlinear Schrodinger equations (NLSE) with initial-boundary data in 1 + 1 dimensions. Our procedure is implemented in two successive steps. In the first one, the J-GL-C is employed for approximating the functional dependence on the spatial variable, using (N - 1) nodes of the Jacobi-Gauss-Lobatto interpolation which depends upon two general Jacobi parameters. The resulting equations together with the two-point boundary conditions induce a system of 2(N - 1) first-order ordinary differential equations (ODEs) in time. In the second step, the implicit Runge-Kutta method of fourth order is applied to solve this temporal system. The proposed J-GL-C method, used in combination with the implicit Runge-Kutta method of fourth order, is employed to obtain highly accurate numerical approximations to four types of NLSE, including the attractive and repulsive NLSE and a Gross-Pitaevskii equation with space-periodic potential. The numerical results obtained by this algorithm have been compared with various exact solutions in order to demonstrate the accuracy and efficiency of the proposed method. Indeed, for relatively few nodes used, the absolute error in our numerical solutions is sufficiently small. (C) 2014 Elsevier Inc. All rights reserved.

Journal Title

Journal of Computational Physics

Volume

261

Publication Date

1-1-2014

Document Type

Article

Language

English

First Page

244

Last Page

255

WOS Identifier

WOS:000330577100013

ISSN

0021-9991

Share

COinS