arXiv subjects

Guillaume Dujardin

Publications and source records attributed to Guillaume Dujardin.

17 recordsLinked to original sources

arXiv2024

A numerical study of vortex nucleation in 2D rotating Bose-Einstein condensates

This article introduces a new numerical method for the minimization under constraints of a discrete energy modeling multicomponents rotating Bose-Einstein condensates in the regime of strong confinement and with rotation. Moreover, we consider both segregation and coexistence regimes between the components. The method includes a discretization of a continuous energy in space dimension 2 and a gradient algorithm with adaptive time step and projection for the minimization. It is well known that, depending on the regime, the minimizers may display different structures, sometimes with vorticity (from singly quantized vortices, to vortex sheets and giant holes). In order to study numerically the structures of the minimizers, we introduce in this paper a numerical algorithm for the computation of the indices of the vortices, as well as an algorithm for the computation of the indices of vortex sheets. Several computations are carried out, to illustrate the efficiency of the method, to cover different physical cases, to validate recent theoretical results as well as to support conjectures. Moreover, we compare this method with an alternative method from the literature.

math.AP↗

arXiv2024

Numerical study of the Gross-Pitaevskii equation on a two-dimensional ring and vortex nucleation

We consider the Gross-Pitaevskii equation with a confining ring potential with a Gaussian profile. By introducing a rotating sinusoidal perturbation, we numerically highlight the nucleation of quantum vortices in a particular regime throughout the dynamics. Numerical computations are made via a Strang splitting time integration and a two-point flux approximation Finite Volume scheme based on a particular admissible triangulation. We also develop numerical algorithms for vortex tracking adapted to our finite volume framework.

math.NA↗

arXiv2024

Uniform estimates for a fully discrete scheme integrating the linear heat equation on a bounded interval with pure Neumann boundary conditions

This manuscript deals with the analysis of numerical methods for the full discretization (in time and space) of the linear heat equation with Neumann boundary conditions, and it provides the reader with error estimates that are uniform in time. First, we consider the homogeneous equation with homogeneous Neumann boundary conditions over a finite interval. Using finite differences in space and the Euler method in time, we prove that our method is of order 1 in space, uniformly in time, under a classical CFL condition, and despite its lack of consistency at the boundaries. Second, we consider the nonhomogeneous equation with nonhomogeneous Neumann boundary conditions over a finite interval. Using a tailored similar scheme, we prove that our method is also of order 1 in space, uniformly in time, under a classical CFL condition. We indicate how this numerical method allows for a new way to compute steady states of such equations when they exist. We conclude by several numerical experiments to illustrate the sharpness and relevance of our theoretical results, as well as to examine situations that do not meet the hypotheses of our theoretical results, and to illustrate how our results extend to higher dimensions.

math.NA↗

arXiv2023

High order linearly implicit methods for semilinear evolution PDEs

This paper considers the numerical integration of semilinear evolution PDEs using the high order linearly implicit methods developped in a previous paper in the ODE setting. These methods use a collocation Runge--Kutta method as a basis, and additional variables that are updated explicitly and make the implicit part of the collocation Runge--Kutta method only linearly implicit. In this paper, we introduce several notions of stability for the underlying Runge--Kutta methods as well as for the explicit step on the additional variables necessary to fit the context of evolution PDE. We prove a main theorem about the high order of convergence of these linearly implicit methods in this PDE setting, using the stability hypotheses introduced before. We use nonlinear Schr\''odinger equations and heat equations as main examples but our results extend beyond these two classes of evolution PDEs. We illustrate our main result numerically in dimensions 1 and 2, and we compare the efficiency of the linearly implicit methods with other methods from the litterature. We also illustrate numerically the necessity of the stability conditions of our main result.

math.NA↗

arXiv2023

Numerical computation of dark solitons of a nonlocal nonlinear Schrödinger equation

The existence and decay properties of dark solitons for a large class of nonlinear nonlocal Gross-Pitaevskii equations with nonzero boundary conditions in dimension one has been established recently in [de Laire and S. López-Martínez, Comm. Partial Differential Equations, 2022]. Mathematically, these solitons correspond to minimizers of the energy at fixed momentum and are orbitally stable. This paper provides a numerical method to compute approximations of such solitons for these types of equations, and provides actual numerical experiments for several types of physically relevant nonlocal potentials. These simulations allow us to obtain a variety of dark solitons, and to comment on their shapes in terms of the parameters of the nonlocal potential. In particular, they suggest that, given the dispersion relation, the speed of sound and the Landau speed are important values to understand the properties of these dark solitons. They also allow us to test the necessity of some sufficient conditions in the theoretical result proving existence of the dark solitons.

math.NA↗

arXiv2023

Â-and Î-stability of collocation Runge-Kutta methods

This paper deals with stability of classical Runge-Kutta collocation methods. When such methods are embedded in linearly implicit methods as developed in [12] and used in [13] for the time integration of nonlinear evolution PDEs, the stability of these methods has to be adapted to this context. For this reason, we develop in this paper several notions of stability, that we analyze. We provide sufficient conditions that can be checked algorithmically to ensure that these stability notions are fulfilled by a given Runge-Kutta collocation method. We also introduce examples and counterexamples used in [13] to highlight the necessity of these stability conditions in this context.

math.NA↗

arXiv2022

Modulational instability in randomly dispersion-managed fiber links

We study modulational instability in a dispersion-managed system where the sign of the group-velocity dispersion is changed at uniformly distributed random distances around a reference length. An analytical technique is presented to estimate the instability gain from the linearized nonlinear Schr{ö}dinger equation, which is also solved numerically. The comparison of numerical and analytical results confirms the validity of our approach. Modulational instability of purely stochastic origin appears. A competition between instability bands of periodic and stochastic origin is also discussed. We find an instability gain comparable to the conventional values found in a homogeneous anomalous dispersion fiber.

physics.optics↗

arXiv2021

Stochastic modulational instability in the nonlinear Schrödinger equation with colored random dispersion

We study modulational instability (MI) in optical fibers with random group-velocity dispersion (GVD). We consider Gaussian and dichotomous colored stochastic processes. We resort to different analytical methods (namely, the cumulant expansion and the functional approach) and assess their reliability in estimating the MI gain of stochastic origin. If the power spectral density (PSD) of the GVD fluctuations is centered at null wavenumber, we obtain low-frequency MI sidelobes which converge to those given by a white noise perturbation when the correlation length tends to 0. If instead the stochastic processes are modulated in space, one or more MI sidelobe pairs corresponding to the well-known parametric resonance (PR) condition can be found. A transition from small and broad sidelobes to peaks nearly indistinguishable from PR-MI is predicted, in the limit of large perturbation amplitudes and correlation lengths of the random process. We find that the cumulant expansion provides good analytical estimates for small PSD values and small correlation lengths, when the MI gain is very small. The functional approach is rigorous only for the dichotomous processes, but allows us to model a wider range of parameters and to predict the existence of MI sidelobes comparable to those observed in homogeneous fibers of anomalous GVD

nlin.PS↗

arXiv2021

High order linearly implicit methods for evolution equations

This paper introduces a new class of numerical methods for the time integration of evolution equations set as Cauchy problems of ODEs or PDEs. The systematic design of these methods mixes the Runge-Kutta collocation formalism with collocation techniques, in such a way that the methods are linearly implicit and have high order. The fact that these methods are implicit allows to avoid CFL conditions when the large systems to integrate come from the space discretization of evolution PDEs. Moreover, these methods are expected to be efficient since they only require to solve one linear system of equations at each time step, and efficient techniques from the literature can be used to do so. After the introduction of the methods, we set suitable definitions of consistency and stability for these methods. This allows for a proof that arbitrarily high order linearly implicit methods exist and converge when applied to ODEs. Eventually, we perform numerical experiments on ODEs and PDEs that illustrate our theoretical results for ODEs, and compare our methods with standard methods for several evolution PDEs.

math.AP↗

arXiv2020

Lie-Trotter Splitting for the Nonlinear Stochastic Manakov System

This article analyses the convergence of the Lie-Trotter splitting scheme for the stochastic Manakov equation, a system arising in the study of pulse propagation in randomly birefringent optical fibers. First, we prove that the strong order of the numerical approximation is 1/2 if the nonlinear term in the system is globally Lipschitz. Then, we show that the splitting scheme has convergence order 1/2 in probability and almost sure order 1/2- in the case of a cubic nonlinearity. We provide several numerical experiments illustrating the aforementioned results and the efficiency of the Lie-Trotter splitting scheme. Finally, we numerically investigate the possible blowup of solutions for some power-law nonlinearities.

math.AP↗

arXiv2020

Exponential integrators for the stochastic Manakov equation

This article presents and analyses an exponential integrator for the stochastic Manakov equation, a system arising in the study of pulse propagation in randomly birefringent optical fibers. We first prove that the strong order of the numerical approximation is $1/2$ if the nonlinear term in the system is globally Lipschitz-continuous. Then, we use this fact to prove that the exponential integrator has convergence order $1/2$ in probability and almost sure order $1/2$, in the case of the cubic nonlinear coupling which is relevant in optical fibers. Finally, we present several numerical experiments in order to support our theoretical findings and to illustrate the efficiency of the exponential integrator as well as a modified version of it.

math.NA↗

arXiv2018

Energy preserving methods for nonlinear Schrödinger equations

This paper is concerned with the numerical integration in time of nonlinear Schrödinger equations using different methods preserving the energy or a discrete analog of it. The Crank-Nicolson method is a well known method of order 2 but is fully implicit and one may prefer a linearly implicit method like the relaxation method introduced in [10] for the cubic nonlinear Schr{ö}dinger equation. This method is also an energy preserving method and numerical simulations have shown that its order is 2. In this paper we give a rigorous proof of the order of this relaxation method and propose a generalized version that allows to deal with general power law nonlinearites. Numerical simulations for different physical models show the efficiency of these methods.

math.NA↗

arXiv2018

Coercivity, hypocoercivity, exponential time decay and simulations for discrete Fokker-Planck equations

In this article, we propose and study several discrete versions of homogeneous and inhomogeneous one-dimensional Fokker-Planck equations. In particular, for these discretizations of velocity and space, we prove the exponential convergence to the equilibrium of the solutions, for time-continuous equations as well as for time-discrete equations. Our method uses new types of discrete Poincaré inequalities for a "two-direction" discretization of the derivative in velocity. For the inhomogeneous problem, we adapt hypocoercive methods to the discrete cases.

math.NA↗

arXiv2017

High order exponential integrators for nonlinear Schrödinger equations with application to rotating Bose-Einstein condensates

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.

math.AP↗

arXiv2015

Asymptotic behavior of splitting schemes involving time-subcycling techniques

This paper deals with the numerical integration of well-posed multiscale systems of ODEs or evolutionary PDEs. As these systems appear naturally in engineering problems, time-subcycling techniques are widely used every day to improve computational efficiency. These methods rely on a decomposition of the vector field in a fast part and a slow part and take advantage of that decomposition. This way, if an unconditionnally stable (semi-)implicit scheme cannot be easily implemented, one can integrate the fast equations with a much smaller time step than that of the slow equations, instead of having to integrate the whole system with a very small time-step to ensure stability. Then, one can build a numerical integrator using a standard composition method, such as a Lie or a Strang formula for example. Such methods are primarily designed to be convergent in short-time to the solution of the original problems. However, their longtime behavior rises interesting questions, the answers to which are not very well known. In particular, when the solutions of the problems converge in time to an asymptotic equilibrium state, the question of the asymptotic accuracy of the numerical longtime limit of the schemes as well as that of the rate of convergence is certainly of interest. In this context, the asymptotic error is defined as the difference between the exact and numerical asymptotic states. The goal of this paper is to apply that kind of numerical methods based on splitting schemes with subcycling to some simple examples of evolutionary ODEs and PDEs that have attractive equilibrium states, to address the aforementioned questions of asymptotic accuracy, to perform a rigorous analysis, and to compare them with their counterparts without subcycling. Our analysis is developed on simple linear ODE and PDE toy-models and is illustrated with several numerical experiments on these toy-models as well as on more complex systems. Lie and

math.AP↗

arXiv2015

Peregrine comb: multiple compression points for Peregrine rogue waves in periodically modulated nonlinear Schr{ö}dinger equations

It is shown that sufficiently large periodic modulations in the coefficients of a nonlinear Schr{ö}dinger equation can drastically impact the spatial shape of the Peregrine soliton solutions: they can develop multiple compression points of the same amplitude, rather than only a single one, as in the spatially homogeneous focusing nonlinear Schr{ö}dinger equation. The additional compression points are generated in pairs forming a comb-like structure. The number of additional pairs depends on the amplitude of the modulation but not on its wavelength, which controls their separation distance. The dynamics and characteristics of these generalized Peregrine soliton are analytically described in the case of a completely integrable modulation. A numerical investigation shows that their main properties persist in nonintegrable situations, where no exact analytical expression of the generalized Peregrine soliton is available. Our predictions are in good agreement with numerical findings for an interesting specific case of an experimentally realizable periodically dispersion modulated photonic crystal fiber. Our results therefore pave the way for the experimental control and manipulation of the formation of generalized Peregrine rogue waves in the wide class of physical systems modeled by the nonlinear Schr{ö}dinger equation.

nlin.PS↗

arXiv2014

Excitation of radial collective modes in a quantum dot: Beyond linear response

We compare the response of five different models of two interacting electrons in a quantum dot to an external short lived radial excitation that is strong enough to excite the system well beyond the linear response regime. The models considered describe the Coulomb interaction between the electrons in different ways ranging from mean-field approaches to configuration interaction (CI) models, where the two-electron Hamiltonian is diagonalized in a large truncated Fock space. The radially symmetric excitation is selected in order to severely put to test the different approaches to describe the interaction and correlations of an electron system in a nonequilibrium state. As can be expected for the case of only two electrons none of the mean-field models can in full details reproduce the results obtained by the CI model. Nonetheless, some linear and nonlinear characteristics are reproduced reasonably well. All the models show activation of an increasing number of collective modes as the strength of the excitation is increased. By varying slightly the confinement potential of the dot we observe how sensitive the properties of the excitation spectrum are to the Coulomb interaction and its correlation effects. In order to approach closer the question of nonlinearity we solve one of the mean-field models directly in a nonlinear fashion without resorting to iterations.

cond-mat.mes-hall↗