Error Analysis of the Strang Time-Splitting Laguerre–Hermite/Hermite Collocation Methods for the Gross–Pitaevskii Equation

Shen, Jie; Wang, Zhong-Qing

doi:10.1007/s10208-012-9124-x

Cited by 25 publications

(8 citation statements)

References 20 publications

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“…It conserves the mass (2.4) in the discretized level as (2.9) [29,36,39,40,53], and it is therefore unconditionally stable. In addition, it can be rigorously proven that the TSSP method is second-order accurate in time and spectral order accurate in space for any fixed ε = ε 0 = O(1) [29,62,113,139,151,168,175]. In addition, it is time transverse invariant, i.e.…”

Section: )mentioning

confidence: 96%

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

Antoine

Bao

Besse

2013

Computer Physics Communications

332

297

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In this paper, we begin with the nonlinear Schrödinger/Gross-Pitaevskii equation (NLSE/GPE) for modeling Bose-Einstein condensation (BEC) and nonlinear optics as well as other applications, and discuss their dynamical properties ranging from time reversible, time transverse invariant, mass and energy conservation, dispersion relation to soliton solutions. Then, we review and compare different numerical methods for solving the NLSE/GPE including finite difference time domain methods and time-splitting spectral method, and discuss different absorbing boundary conditions. In addition, these numerical methods are extended to the NLSE/GPE with damping terms and/or an angular momentum rotation term as well as coupled NLSEs/GPEs. Finally, applications to simulate a quantized vortex lattice dynamics in a rotating BEC are reported.

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Section: )mentioning

confidence: 96%

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

Antoine

Bao

Besse

2013

Computer Physics Communications

332

297

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“…The present works contributes to a rigorous error analysis of full discretizations for low-dimensional nonlinear Schrödinger equations by higherorder time-splitting pseudospectral methods. For this purpose, unifying and extending techniques exploited in [15,20,22,23,28,30], we utilize an analytical framework of nonlinear evolutionary Schrödinger equations, fractional power spaces defined by the principal linear part, the formal calculus of Lie-derivatives, bounds for iterated Liecommutators, and results on the approximation error of pseudospectral methods; see also [9,11,14,17,19,27].…”

Section: Introductionmentioning

confidence: 99%

Convergence Analysis of High-Order Time-Splitting Pseudospectral Methods for Nonlinear Schrödinger Equations

Thalhammer¹

2012

SIAM J. Numer. Anal.

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In this work, the issue of favorable numerical methods for the space and time discretization of low-dimensional nonlinear Schrödinger equations is addressed. The objective is to provide a stability and error analysis of high-accuracy discretizations that rely on spectral and splitting methods. As a model problem, the time-dependent Gross-Pitaevskii equation arising in the description of Bose-Einstein condensates is considered. For the space discretization pseudospectral methods collocated at the associated quadrature nodes are analyzed. For the time integration highorder exponential operator splitting methods are studied, where the decomposition of the function defining the partial differential equation is chosen in accordance with the underlying spectral method. The convergence analysis relies on a general framework of abstract nonlinear evolution equations and fractional power spaces defined by the principal linear part. Essential tools in the derivation of a temporal global error estimate are further the formal calculus of Lie-derivatives and bounds for iterated Lie-commutators. Numerical examples for higher-order time-splitting pseudospectral methods applied to time-dependent Gross-Pitaevskii equations illustrate the theoretical result.

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“…The convergence of splitting schemes for semilinear Schrödinger equations was analyzed in [17,19,24,36,45,49]. In the present work, we extend the previous works to the quasilinear Schrödinger equation.…”

Section: Introductionmentioning

confidence: 75%

Strang splitting methods for a quasilinear Schrödinger equation: convergence, instability, and dynamics

Marzuola

2015

Communications in Mathematical Sciences

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We study the Strang splitting scheme for quasilinear Schrödinger equations. We establish the convergence of the scheme for solutions with small initial data. We analyze the linear instability of the numerical scheme, which explains the numerical blow-up of large data solutions and connects to the analytical breakdown of regularity of solutions to quasilinear Schrödinger equations. Numerical tests are performed for a modified version of the superfluid thin film equation. and wishes to especially thank his collaborators Jason Metcalfe and Daniel Tataru for introducing him to quasilinear Schrödinger theory. We also wish to thank the anonymous reviewers for many helpful comments that help improve the exposition in the manuscript. We thank Ludwig Gauckler also for pointing out an error in the convergence proof in an earlier version of the draft. 1 arXiv:1305.7205v5 [math.NA]

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Error Analysis of the Strang Time-Splitting Laguerre–Hermite/Hermite Collocation Methods for the Gross–Pitaevskii Equation

Cited by 25 publications

References 20 publications

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

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

Convergence Analysis of High-Order Time-Splitting Pseudospectral Methods for Nonlinear Schrödinger Equations

Strang splitting methods for a quasilinear Schrödinger equation: convergence, instability, and dynamics

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