Split-Step Methods for the Solution of the Nonlinear Schrödinger Equation

Weideman, J. A. C.; Herbst, B. M.

doi:10.1137/0723033

Cited by 368 publications

(311 citation statements)

References 21 publications

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“…Note that H is positive if κ ≤ 0. In the case κ > 0 the solution is known to possibly exhibit instabilities, see [37] and references therein, but these are not what we see in the examples below. Explicit soliton solutions are provided in [29] for the case κ > 0.…”

Section: The Nonlinear Schrödinger Equation In 1dmentioning

confidence: 59%

“…But the phenomenon discussed here concerns purely numerical, high frequency instabilities. For the case κ > 0 and using the spectral splitting method (6c), Weideman and Herbst derive in [37] the stability condition…”

Section: Examplementioning

confidence: 99%

“…See [37,19,2]. Since this method is symplectic, norm-preserving and 2nd order accurate in both t and x, so is the ensuing staggered composition with the exact solver of the ODE part of the splitting.…”

Section: The Nonlinear Schrödinger Equation In 1dmentioning

confidence: 99%

“…In [37] there is a detailed analysis of instabilites, based on linearization around a uniform wave train, for both methods (6a) and (6c) experimented with here. First, there are instabilities in the problem itself that are mirrored, generally speaking, by the numerical methods.…”

Section: Examplementioning

confidence: 99%

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Surprising computations

Ascher

2012

Applied Numerical Mathematics

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In the course of simulation of differential equations, especially of marginally stable differential problems using marginally stable numerical methods, one occasionally comes across a correct computation that yields surprising, or unexpected results. We examine several instances of such computations. These include (i) solving Hamiltonian ODE systems using almost conservative explicit Runge-Kutta methods, (ii) applying splitting methods for the nonlinear Schrödinger equation, and (iii) applying strong stability preserving Runge-Kutta methods in conjunction with weighted essentially non-oscillatory semi-discretizations for nonlinear conservation laws with discontinuous solutions.For each problem and method class we present a simple numerical example that yields results that in our experience many active researchers are finding unexpected and unintuitive. Each numerical example is then followed by an explanation and a resolution of the practical problem.

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Section: The Nonlinear Schrödinger Equation In 1dmentioning

confidence: 59%

Section: Examplementioning

confidence: 99%

Section: The Nonlinear Schrödinger Equation In 1dmentioning

confidence: 99%

Section: Examplementioning

confidence: 99%

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Surprising computations

Ascher

2012

Applied Numerical Mathematics

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“…From a computational point of view the Interaction Picture method is very close to the Symmetric Split-Step method based on the Strang splitting formula widely used to solve the nonlinear Schrödinger equation [12,13]. It has been for the first time applied for solving the GNLSE in optics by J. Hult in [4] where an experimental comparison of the RK4-IP method to other SS methods based on their numerical efficiency on benchmark problems in optics is presented.…”

Section: Introductionmentioning

confidence: 99%

Embedded Runge–Kutta scheme for step-size control in the interaction picture method

Balac¹,

Mahé²

2013

Computer Physics Communications

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To cite this version:Stéphane Balac, Fabrice Mahé. Embedded Runge-Kutta scheme for step-size control in the interaction picture method. Computer Physics Communications, Elsevier, 2013, 184 (4) AbstractWhen solving certain evolution type PDE such as the Schrödinger equation, the Interaction Picture method is a valuable alternative to Split-Step methods. The Interaction Picture method has good computational features when used together with the standard 4th order Runge-Kutta scheme (giving rise to the RK4-IP method). In this paper we present an embedded Runge-Kutta scheme with orders 3 and 4with the aim to deliver an estimation of the local error for adaptive step-size control purposes in the Interaction Picture method. The corresponding ERK4(3)-IP method preserves the features of the RK4-IP method and provide a local error estimate at no significant extra cost.

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A spectral expansion‐based Fourier split‐step method for uncertainty quantification of the propagation factor in a stochastic environment

Enstedt

Wellander

2016

Radio Science

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A chaos expanded Fourier split‐step method is derived and applied to a narrow‐angle parabolic equation. The parabolic equation has earlier been used to study deterministic settings. In this paper we develop a spectral‐based Fourier split‐step method that will take a limited degree of information about the environment into account. Our main focus is on proposing an efficient method for computational electromagnetics in stochastic settings. In this paper we study electromagnetic wave propagation in the troposphere in the case when the refraction index belongs to a uniform distribution.

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Split-Step Methods for the Solution of the Nonlinear Schrödinger Equation

Cited by 368 publications

References 21 publications

Surprising computations

Surprising computations

Embedded Runge–Kutta scheme for step-size control in the interaction picture method

A spectral expansion‐based Fourier split‐step method for uncertainty quantification of the propagation factor in a stochastic environment

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