Computational methods for the dynamics of the nonlinear Schrödinger/Gross–Pitaevskii equations

Antoine, Xavier; Bao, Weizhu; Besse, Christophe

doi:10.1016/j.cpc.2013.07.012

Cited by 339 publications

(310 citation statements)

References 196 publications

(557 reference statements)

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“…If ν = 0, then we will speak about the time-dependent Linear Schrödinger Equation (LSE). In the Physics literature, the first equation of system (1) is also called the Gross-Pitaevskii Equation (GPE) [2,11,16], when considering Bose-Einstein Condensates (BEC) (see e.g. [38,39]).…”

Section: Introductionmentioning

confidence: 99%

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Asymptotic estimates of the convergence of classical Schwarz waveform relaxation domain decomposition methods for two-dimensional stationary quantum waves

2018

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This paper is devoted to the analysis of convergence of Schwarz Waveform Relaxation (SWR) domain decomposition methods (DDM) for solving the stationary linear and nonlinear Schrödinger equations by the imaginary-time method. Although SWR are extensively used for numerically solving high-dimensional quantum and classical wave equations, the analysis of convergence and of the rate of convergence is still largely open for variable coefficients linear and nonlinear equations. The aim of this paper is to tackle this problem for both the linear and nonlinear Schrödinger equations in the two-dimensional setting. By extending ideas and concepts presented earlier [12] and by using pseudodifferential calculus, we prove the convergence and determine some approximate rates of convergence of the two-dimensional Classical SWR method for two subdomains with smooth boundary. Some numerical experiments are also proposed to validate the analysis.

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Section: Introductionmentioning

confidence: 99%

“…More general versions of the GPE include rotational terms, complex nonlinear (nonlocal) functions and coupled species of cold atomic gases [2,11,15,16,40].…”

Section: Introductionmentioning

confidence: 99%

Asymptotic estimates of the convergence of classical Schwarz waveform relaxation domain decomposition methods for two-dimensional stationary quantum waves

2018

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“…f (|ψ| 2 ) = β |ψ| 2 ). However, it is not invariant under gauge transformation and the dispersive relation does not hold [10,35]. The computational cost is O(M log M) since we again use FFTs.…”

Section: The Relaxation Scheme For the Rotating Gpementioning

confidence: 99%

“…The major disadvantages of this scheme are that it is conditionally stable and it does not satisfy most of the properties from Section 4.1 (see [10]). In Section 1.3 (page 11), we have seen that a change of variables is used for modeling a rotating condensate.…”

Section: Other Schemesmentioning

confidence: 99%

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Modeling and Computation of Bose-Einstein Condensates: Stationary States, Nucleation, Dynamics, Stochasticity

Antoine

Duboscq

2015

Lecture Notes in Mathematics

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An explicit spectral collocation method using nonpolynomial basis functions for the time‐dependent Schrödinger equation

Fang

Liu

2018

Math Methods in App Sciences

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We propose a spectral collocation method for the numerical solution of the time-dependent Schrödinger equation, where the newly developed nonpolynomial functions in a previous study are used as basis functions. Equipped with the new basis functions, various boundary conditions can be imposed exactly. The preferable semi-implicit time marching schemes are employed for temporal discretization. Moreover, the new basis functions build in a free parameter intrinsically, which can be chosen properly so that the semi-implicit scheme collapses to an explicit scheme. The method is further applied to linear Schrödinger equation set in unbounded domain. The transparent boundary conditions are constructed for time semidiscrete scheme of the linear Schrödinger equation. We employ spectral collocation method using the new basis functions for the spatial discretization, which allows for the exact imposition of the transparent boundary conditions. Comprehensive numerical tests both in bounded and unbounded domain are performed to demonstrate the attractive features of the proposed method. KEYWORDS collocation method, nonpolynomial basis functions, Schrödinger equation, transparent boundary conditions 186

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Computational methods for the dynamics of the nonlinear Schrödinger/Gross–Pitaevskii equations

Cited by 339 publications

References 196 publications

Asymptotic estimates of the convergence of classical Schwarz waveform relaxation domain decomposition methods for two-dimensional stationary quantum waves

Asymptotic estimates of the convergence of classical Schwarz waveform relaxation domain decomposition methods for two-dimensional stationary quantum waves

Modeling and Computation of Bose-Einstein Condensates: Stationary States, Nucleation, Dynamics, Stochasticity

An explicit spectral collocation method using nonpolynomial basis functions for the time‐dependent Schrödinger equation

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