High Order Exponential Integrators for Nonlinear Schrödinger Equations with Application to Rotating Bose--Einstein Condensates

Besse, Christophe; Dujardin, Guillaume; Lacroix–Violet, Ingrid

doi:10.1137/15m1029047

Cited by 31 publications

(26 citation statements)

References 48 publications

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“…Besse et al [11] studies exponential time propagation methods for the rotational Gross-Pitaevskii equation by comparing exponential Runge-Kutta methods with Lawson and splitting methods, with no clear advantage for either approach. An error analysis given there only considers the transformed problem, without taking into account the stiffness hidden in the Lawson transformation.…”

Section: Lawson Methodsmentioning

confidence: 99%

Convergence of exponential Lawson-multistep methods for the MCTDHF equations

Koch

2019

ESAIM: M2AN

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We consider exponential Lawson multistep methods for the time integration of the equations of motion associated with the multi-configuration time-dependent Hartree-Fock (MCTDHF) approximation for high-dimensional quantum dynamics. These provide high-order approximations at a minimum of evaluations of the computationally expensive nonlocal potential terms, and have been found to enable stable long-time integration. In this work, we prove convergence of the numerical approximation on finite time intervals under minimal regularity assumptions on the exact solution. A numerical illustration shows adaptive time propagation based on our methods.Mathematics Subject Classification. 65L05, 65L06, 65M12, 65M15, 65M20, 81-08. on the Hilbert space L 2 . Here, A : D ⊆ L 2 → L 2 is a self-adjoint differential operator and B a generally unbounded nonlinear operator. We will denote the flow of −iĤ by E −iĤ , that is, ψ(t) = E −iĤ (t, ψ 0 ) is the solution of ∂ t ψ = −iĤ(ψ), ψ(0) = ψ 0 , and similarly for E A and E B .Equations of this type commonly arise from model reductions of high-dimensional quantum dynamical systems serving to make many-particle quantum simulations computationally tractable. Examples of such model reductions are the Gross-Pitaevskii equation (GPE) for Bose-Einstein condensates (BEC) [41], which constitutes a meanfield theory for ultracold dilute bosonic gases, the multiconfigurational time-dependent Hartree for bosons (MCTDHB) method for ultracold bosonic atoms [1], the multiconfigurational time-dependent Hartree-Fock (MCTDHF) method [27,32,48] and its most current extension, the time-dependent complete-active-space self-consistent-field (TD-CASSCF) method for electron dynamics in atoms and small molecules [44], and timedependent density functional theory (TDDFT) for extended systems such as large molecules, nanostructures

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Section: Lawson Methodsmentioning

confidence: 99%

Convergence of exponential Lawson-multistep methods for the MCTDHF equations

Koch

2019

ESAIM: M2AN

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“…when the system is non-autonomous. Recent high-order schemes (exponential integrators [23,75,77], IMEXSP [6]...) with time-stepping techniques have been designed to get new efficient solvers that could be combined with pseudospectral approximation schemes. We also note that ReFD can be extended to include the pseudospectral approximation, resulting in the ReSP scheme given in [10].…”

Section: Overview Of Popular Numerical Schemesmentioning

confidence: 99%

On the numerical solution and dynamical laws of nonlinear fractional Schrödinger/Gross–Pitaevskii equations

Antoine

Tang

Zhang

2018

International Journal of Computer Mathematics

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The purpose of this paper is to discuss some recent developments concerning the numerical simulation of space and time fractional Schrödinger and Gross-Pitaevskii equations. In particular, we address some questions related to the discretization of the models (order of accuracy and fast implementation) and clarify some of their dynamical properties. Some numerical simulations illustrate these points.

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“…Exponential integrators for the time integration of deterministic semi-linear problems of the formẏ = Ly + N (y), are nowadays widely used and studied, as witnessed by the recent review [22]. Applications of such numerical schemes to the deterministic (nonlinear) Schrödinger equation can be found in, for example, [4][5][6][7][8][9][10]17,21] and references therein. Furthermore, these numerical methods were investigated for stochastic parabolic partial differential equations in, for example, [23][24][25], more recently for the stochastic wave equations in [2,11,12,27], where they are termed stochastic trigonometric methods, and lately to stochastic Schrödinger equations driven by Ito noise in [1].…”

Section: Introductionmentioning

confidence: 99%

Exponential integrators for nonlinear Schrödinger equations with white noise dispersion

Cohen

Dujardin

2017

Stoch PDE: Anal Comp

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This article deals with the numerical integration in time of the nonlinear Schrödinger equation with power law nonlinearity and random dispersion. We introduce a new explicit exponential integrator for this purpose that integrates the noisy part of the equation exactly. We prove that this scheme is of mean-square order 1 and we draw consequences of this fact. We compare our exponential integrator with several other numerical methods from the literature. We finally propose a second exponential integrator, which is implicit and symmetric and, in contrast to the first one, preserves the L 2 -norm of the solution.

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High Order Exponential Integrators for Nonlinear Schrödinger Equations with Application to Rotating Bose--Einstein Condensates

Cited by 31 publications

References 48 publications

Convergence of exponential Lawson-multistep methods for the MCTDHF equations

Convergence of exponential Lawson-multistep methods for the MCTDHF equations

On the numerical solution and dynamical laws of nonlinear fractional Schrödinger/Gross–Pitaevskii equations

Exponential integrators for nonlinear Schrödinger equations with white noise dispersion

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