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

Balac, Stéphane; Mahé, Fabrice

doi:10.1016/j.cpc.2012.12.020

Cited by 46 publications

(42 citation statements)

References 19 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…[32,27,21]. Recently an efficient embedded RK method based on Dormand and Prince RK4(3)-T formula [12] and specifically designed for the IP method has been proposed in [3]. The name "Interaction Picture" and the change of unknown at the heart of the method originate from quantum mechanics [30,17] where it is usual to chose an appropriate "picture" in which the physical properties of the studied system can be easily revealed and the calculation made simpler.…”

Section: Presentation Of the Numerical Approachmentioning

confidence: 99%

“…As well, any other value z in the set ]z k , z k+1 [ could have been chosen rather than the particular value z k+ 1 2 in the change of unknown (2.2); however the benefit of the cancellation of 4 exponential operator terms in (2.10) would have been lost. An embedded Runge-Kutta scheme for the IP method preserving these nice features has been proposed in [3].…”

Section: Computing the Exp(mentioning

confidence: 99%

“…For convenience we assume a constant grid spacing h = L/K but this assumption is not a limitation of the method and an adaptive step-size version of the RK4-IP method is propounded in [3]. For k ∈ {0, .…”

Section: Presentation Of the Numerical Approachmentioning

confidence: 99%

“…We present in this section numerical results on two selected applications in optics: the propagation of optical solitons and the propagation of a picosecond pulse into a singlemode fiber where fiber losses, nonlinear Raman and Kerr effects and high order chromatic dispersion are taken into account. In both cases, we use the RK4-IP and S3F-RK4 methods with a constant step-size and with an adaptive step-size strategy based on the embedded RK4(3) scheme presented in [3].…”

Section: Numerical Comparisonmentioning

confidence: 99%

See 3 more Smart Citations

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

et al. 2016

Self Cite

View full text Add to dashboard Cite

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.

show abstract

Section: Presentation Of the Numerical Approachmentioning

confidence: 99%

Section: Computing the Exp(mentioning

confidence: 99%

Section: Presentation Of the Numerical Approachmentioning

confidence: 99%

Section: Numerical Comparisonmentioning

confidence: 99%

See 2 more Smart Citations

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

et al. 2016

Self Cite

View full text Add to dashboard Cite

show abstract

“…Numerical methods involved finite-difference approach [19,21,22], bi-k-Lagrange elements [23], spectral collocation methods with Chebyshev polynomials of the first and second kind [24] as well as basis set expansion technique [25]. Time-dependent equations were solved with implicit and semi-implicit Crank-Nicolson methods [26,27,18,28,29], Euler scheme [22], third and fourth-order adaptive Runge-Kutta methods [30], splitstep finite difference method [22] and time-splitting sine and Fourier pseudospectral methods [31,32]. In the latter case, space was discretized with second-and fourth-order finite differences, exponential splines [29] or with Chebyshev-Tau spectral discretization method [26].…”

Section: Introductionmentioning

confidence: 99%

Numerical modeling of exciton–polariton Bose–Einstein condensate in a microcavity

Voronych

Buraczewski

Matuszewski

et al. 2017

Computer Physics Communications

View full text Add to dashboard Cite

A novel, optimized numerical method of modeling of an exciton-polariton superfluid in a semiconductor microcavity was proposed. Exciton-polaritons are spin-carrying quasiparticles formed from photons strongly coupled to excitons. They possess unique properties, interesting from the point of view of fundamental research as well as numerous potential applications. However, their numerical modeling is challenging due to the structure of nonlinear differential equations describing their evolution. In this paper, we propose to solve the equations with a modified Runge-Kutta method of 4th order, further optimized for efficient computations. The algorithms were implemented in form of C++ programs fitted for parallel environments and utilizing vector instructions. The programs form the EPCGP suite which have been used for theoretical investigation of exciton-polaritons.Keywords: exciton-polariton superfluid; Bose-Einstein condensate; microcavity; Gross-Pitaevskii equation; Runge-Kutta method Nature of problem: An exciton-polariton superfluid is a novel, interesting physical system allowing investigation of high temperature Bose-Einstein condensation of exciton-polaritons-quasiparticles carrying spin. They have brought a lot of attention due to their unique properties and potential applications in polariton-based optoelectronic integrated circuits. This is an out-of-equilibrium quantum system confined within a semiconductor microcavity. It is described by a set of nonlinear differential equations similar in spirit to the Gross-Pitaevskii (GP) equation, but their unique properties do not allow standard GP solving frameworks to be utilized. Finding an accurate and efficient numerical algorithm as well as development of optimized numerical software is necessary for effective theoretical investigation of exciton-polaritons. Program summarySolution method: A Runge-Kutta method of 4th order was employed to solve the set of differential equations describing exciton-polariton superfluids. The method was fitted for the exciton-polariton equations and further optimized. The C++ programs utilize OpenMP extensions and vector operations in order to fully utilize the computer hardware.

show abstract

Stabilization of uni-directional water wave trains over an uneven bottom

et al. 2020

View full text Add to dashboard Cite

We study the evolution of nonlinear surface gravity water wave packets developing from modulational instability over an uneven bottom. A nonlinear Schrödinger equation (NLSE) with coefficients varying in space along propagation is used as a reference model. Based on a low-dimensional approximation obtained by considering only three complex harmonic modes, we discuss how to stabilize a one-dimensional pattern in the form of train of large peaks sitting on a background and propagating over a significant distance. Our approach is based on a gradual depth variation, while its conceptual framework is the theory of autoresonance in nonlinear systems and leads to a quasi-frozen state. Three main stages are identified: amplification from small sideband amplitudes, separatrix crossing and adiabatic conversion to orbits oscillating around an elliptic fixed point. Analytical estimates on the three stages are obtained from the low-dimensional approxi

show abstract

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

Cited by 46 publications

References 19 publications

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

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

Numerical modeling of exciton–polariton Bose–Einstein condensate in a microcavity

Stabilization of uni-directional water wave trains over an uneven bottom

Contact Info

Product

Resources

About