Florian Méhats

The Nonlinear Schrödinger Equation with a Strongly Anisotropic Harmonic Potential

The nonlinear Schrodinger equation with general nonlinearity of polynomial growth and harmonic confining potential is considered. More precisely, the confining potential is strongly anisotropic; i.e., the trap frequencies in different directions are of different orders of magnitude. The limit as the ratio of trap frequencies tends to zero is carried out. A concentration of mass on the ground state of the dominating harmonic oscillator is shown to be propagated, and the lower-dimensional modulation wave function again satisfies a nonlinear Schrodinger equation. The main tools of the analysis are energy and Strichartz estimates, as well as two anisotropic Sobolev inequalities. As an application, the dimension reduction of the three-dimensional Gross-Pitaevskii equation is discussed, which models the dynamics of Bose-Einstein condensates.

Uniformly accurate numerical schemes for highly oscillatory Klein---Gordon and nonlinear Schrödinger equations

This work is devoted to the numerical simulation of nonlinear Schrödinger and Klein–Gordon equations. We present a general strategy to construct numerical schemes which are uniformly accurate with respect to the oscillation frequency. This is a stronger feature than the usual so called “Asymptotic preserving” property, the last being also satisfied by our scheme in the highly oscillatory limit. Our strategy enables to simulate the oscillatory problem without using any mesh or time step refinement, and the orders of our schemes are preserved uniformly in all regimes. In other words, since our numerical method is not based on the derivation and the simulation of asymptotic models, it works in the regime where the solution does not oscillate rapidly, in the highly oscillatory limit regime, and in the intermediate regime with the same order of accuracy. In the same spirit as in Crouseilles et al. (J Comput Phys 248, 287–308, 2013), the method is based on two main ingredients. First, we embed our problem in a suitable “two-scale” reformulation with the introduction of an additional variable. Then a link is made with classical strategies based on Chapman–Enskog expansions in kinetic theory despite the dispersive context of the targeted equations, allowing to separate the fast time scale from the slow one. Uniformly accurate schemes are eventually derived from this new formulation and their properties and performances are assessed both theoretically and numerically.

An entropic quantum drift-diffusion model for electron transport in resonant tunneling diodes

We present an entropic quantum drift-diffusion model (eQDD) and show how it can be derived on a bounded domain as the diffusive approximation of the Quantum Liouville equation with a quantum BGK operator. Some links between this model and other existing models are exhibited, especially with the density gradient (DG) model and the Schrodinger-Poisson drift-diffusion model (SPDD). Then a finite difference scheme is proposed to discretize the eQDD model coupled to the Poisson equation and we show how this scheme can be slightly modified to discretize the other models. Numerical results show that the properties listed for the eQDD model are checked, as well as the model captures important features concerning the modeling of a resonant tunneling diode. To finish, some comparisons between the models stated above are realized.

Entropic Discretization of a Quantum Drift-Diffusion Model

This paper is devoted to the discretization and numerical simulation of a new quantum drift-diffusion model that was recently derived. In a first step, we introduce an implicit semi-discretization in time which possesses some interesting properties: this system is well-posed, it preserves the positivity of the density, the total charge is conserved, and it is entropic (a free energy is dissipated). Then, after a discretization of the space variable, we define a numerical scheme which has the same properties and is equivalent to a convex minimization problem. These results are illustrated by some numerical simulations.

Asymptotic Preserving schemes for highly oscillatory Vlasov-Poisson equations

This work is devoted to the numerical simulation of a Vlasov–Poisson model describing a charged particle beam under the action of a rapidly oscillating external field. We construct an Asymptotic Preserving numerical scheme for this kinetic equation in the highly oscillatory limit. This scheme enables to simulate the problem without using any time step refinement technique. Moreover, since our numerical method is not based on the derivation of the simulation of asymptotic models, it works in the regime where the solution does not oscillate rapidly, and in the highly oscillatory regime as well. Our method is based on a “two scale” reformulation of the initial equation, with the introduction of an additional periodic variable.

Micro-Macro Schemes for Kinetic Equations Including Boundary Layers

We introduce a new micro-macro decomposition of collisional kinetic equations in the specific case of the diffusion limit, which naturally incorporates the incoming boundary conditions. The idea is to write the distribution function

Quantum drift-diffusion modeling of spin transport in nanostructures

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Isothermal Quantum Hydrodynamics: Derivation, Asymptotic Analysis, and Simulation

in all its domain as the sum of an equilibrium adapted to the boundary (which is not the usual equilibrium associated with

Diffusive transport of partially quantized particles: Existence, uniqueness and long time behaviour

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On a Vlasov–Schrödinger–Poisson Model

) and a remaining kinetic part. This equilibrium is defined such that its incoming velocity moments coincide with the incoming velocity moments of the distribution function. A consequence of this strategy is that no artificial boundary condition is needed in the micro-macro models and the exact boundary condition on