Exponential Runge–Kutta methods for the Schrödinger equation

Dujardin, Guillaume

doi:10.1016/j.apnum.2009.02.002

Cited by 36 publications

(46 citation statements)

References 11 publications

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“…x is a linear isometry in H r (see (6)) the error in u and v is the same, i.e., u(t n ) − u n r = v(t n ) − v n r . Thanks to the representations (34), (35) and the form of the remainder R 2γ (τ 1+2γ ) given in (11) we obtain with the help of Lemma 3.1 that…”

Section: Convergence Analysis In One Dimension (D = 1)mentioning

confidence: 99%

A Fourier Integrator for the Cubic Nonlinear Schrödinger Equation with Rough Initial Data

Knöller¹,

Ostermann²,

Schratz³

2019

SIAM J. Numer. Anal.

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Standard numerical integrators suffer from an order reduction when applied to nonlinear Schrödinger equations with low-regularity initial data. For example, standard Strang splitting requires the boundedness of the solution in H r+4 in order to be second-order convergent in H r , i.e., it requires the boundedness of four additional derivatives of the solution. We present a new type of integrator that is based on the variation-of-constants formula and makes use of certain resonance based approximations in Fourier space. The latter can be efficiently evaluated by fast Fourier methods. For second-order convergence, the new integrator requires two additional derivatives of the solution in one space dimension, and three derivatives in higher space dimensions. Numerical examples illustrating our convergence results are included. These examples demonstrate the clear advantage of the Fourier integrator over standard Strang splitting for initial data with low regularity.

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Section: Convergence Analysis In One Dimension (D = 1)mentioning

confidence: 99%

A Fourier Integrator for the Cubic Nonlinear Schrödinger Equation with Rough Initial Data

Knöller¹,

Ostermann²,

Schratz³

2019

SIAM J. Numer. Anal.

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“…Our goal in this paper is to introduce and analyse two classes of exponential methods for the time integration of (1.1) on a d-dimensional space. The first class is that of exponential Runge-Kutta methods [37,38,27]. The second class relies on the Lawson techniques [43,45].…”

Section: Introductionmentioning

confidence: 99%

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

Besse¹,

Dujardin²,

Lacroix–Violet³

2017

SIAM J. Numer. Anal.

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This article deals with the numerical integration in time of nonlinear Schrödinger equations. The main application is the numerical simulation of rotating Bose-Einstein condensates. The authors perform a change of unknown so that the rotation term disappears and they obtain as a result a nonautonomous nonlinear Schrödinger equation. They consider exponential integrators such as exponential Runge-Kutta methods and Lawson methods. They provide an analysis of the order of convergence and some preservation properties of these methods in a simplified setting and they supplement their results with numerical experiments with realistic physical parameters. Moreover, they compare these methods with the classical split-step methods applied to the same problem.

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“…The RK4-IP method can be interpreted as an exponential Runge-Kutta method according to the general form presented in [24] for the semi-linear parabolic problems, but this method is not of collocation type as studied in [23] for parabolic problems and in [13] for the Schrödinger equation.…”

Section: Presentation Of the Numerical Approachmentioning

confidence: 99%

The Interaction Picture method for solving the generalized nonlinear Schrödinger equation in optics

et al. 2016

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Abstract. The "interaction picture" (IP) method is a very promising alternative to Split-Step methods for solving certain type of partial differential equations such as the nonlinear Schrödinger equation involved in the simulation of wave propagation in optical fibers. The method exhibits interesting convergence properties and is likely to provide more accurate numerical results than cost comparable Split-Step methods such as the Symmetric Split-Step method. In this work we investigate in detail the numerical properties of the IP method and carry out a precise comparison between the IP method and the Symmetric Split-Step method.

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Exponential Runge–Kutta methods for the Schrödinger equation

Cited by 36 publications

References 11 publications

A Fourier Integrator for the Cubic Nonlinear Schrödinger Equation with Rough Initial Data

A Fourier Integrator for the Cubic Nonlinear Schrödinger Equation with Rough Initial Data

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

The Interaction Picture method for solving the generalized nonlinear Schrödinger equation in optics

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