B-spline finite element studies of the non-linear Schrödinger equation

Gardner, L. R. T.; Gardner, G. A.; Zaki, S.I.; Sahrawi, Z. El

doi:10.1016/0045-7825(93)90007-k

Cited by 69 publications

(34 citation statements)

References 12 publications

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“…where [31][32][33][34]. It is observed that two waves, traveling in opposite direction, collide and separate, but conserve initial shapes afterwards.…”

Section: Test 2 Soliton Interactionmentioning

confidence: 99%

A differential quadrature algorithm for nonlinear Schrödinger equation

Korkmaz¹,

Dağ

2008

Nonlinear Dyn

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Numerical solutions of a nonlinear Schrö-dinger equation is obtained using the differential quadrature method based on polynomials for space discretization and Runge-Kutta of order four for time discretization. Five well-known test problems are studied to test the efficiency of the method. For the first two test problems, namely motion of single soliton and interaction of two solitons, numerical results are compared with earlier works. It is shown that results of other test problems agrees the theoretical results. The lowest two conserved quantities and their relative changes are computed for all test examples. In all cases, the differential quadrature Runge-Kutta combination generates numerical results with high accuracy.

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“…where [31][32][33][34]. It is observed that two waves, traveling in opposite direction, collide and separate, but conserve initial shapes afterwards.…”

Section: Test 2 Soliton Interactionmentioning

confidence: 99%

A differential quadrature algorithm for nonlinear Schrödinger equation

Korkmaz¹,

Dağ

2008

Nonlinear Dyn

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“…(see also [19]). Numerical methods for solving the CNLS equation were found among others in [20][21][22][23][24][25] using finite-difference methods, in [26][27][28][29] with finite-elements, in [30] with pseudospectral methods, in [31] with Lanczos' Tau method, in [32,33] with the Adomian decomposition method, in [34] with the variation iteration method, etc. For more about the numerical methods for the CNLS equation, see the review article [35].…”

Section: Introductionmentioning

confidence: 99%

A modified numerical scheme for the cubic Schrödinger equation

Bratsos

2009

Numer. Methods Partial Differential Eq.

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A numerical scheme based on the use of rational approximants in a two-time level recurrence relation is applied to the cubic Schrödinger equation. The resulting nonlinear finite-difference scheme, which is analyzed for stability, is solved using an already known modified predictor-corrector (MPC) scheme. The efficiency of the proposed method is tested to the single and the double-soliton waves and the results arising from the experiments are compared with the relevant ones known in the available bibliography.

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“…This is a time-dependent example with complex-valued functions, which had been used to illustrate the efficiency of different numerical methods [4] [25]. We used it in this work to demonstrate that the expansion relying on Equation (2) is still applicable, except that all the coefficients are now complex-valued.…”

Section: D) Example Of the Nonlinear Schrödinger (Nls) Equation In Onmentioning

confidence: 99%

“…In general, the collocation approach does not require basis functions to be orthogonal. For example, splines [4], canonical polynomials [5], or even mesh-free formulations, based on specially constructed functions [6], had been proposed. So far, none of the non-orthogonal approaches could compete with the standard pseudospectral methods in popularity because there is no clear and decisive advantage in those alternatives.…”

Section: Introductionmentioning

confidence: 99%

The Lanczos-Chebyshev Pseudospectral Method for Solution of Differential Equations

Chen¹

2016

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In this paper, we propose to replace the Chebyshev series used in pseudospectral methods with the equivalent Chebyshev economized power series that can be evaluated more rapidly. We keep the rest of the implementation the same as the spectral method so that there is no new mathematical principle involved. We show by numerical examples that the new approach works well and there is indeed no significant loss of solution accuracy. The advantages of using power series also include simplicity in its formulation and implementation such that it could be used for complex systems. We investigate the important issue of collocation point selection. Our numerical results indicate that there is a clear accuracy advantage of using collocation points corresponding to roots of the Chebyshev polynomial.

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B-spline finite element studies of the non-linear Schrödinger equation

Cited by 69 publications

References 12 publications

A differential quadrature algorithm for nonlinear Schrödinger equation

A differential quadrature algorithm for nonlinear Schrödinger equation

A modified numerical scheme for the cubic Schrödinger equation

The Lanczos-Chebyshev Pseudospectral Method for Solution of Differential Equations

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