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The radiation condition at infinity for the high-frequency Helmholtz equation with source term: a wave packet approach

François Castella — 2004

Journées Équations aux dérivées partielles

We consider the high-frequency Helmholtz equation with a given source term, and a small absorption parameter α > 0 . The high-frequency (or: semi-classical) parameter is ε > 0 . We let ε and α go to zero simultaneously. We assume that the zero energy is non-trapping for the underlying classical flow. We also assume that the classical trajectories starting from the origin satisfy a transversality condition, a generic assumption. Under these assumptions, we prove that the solution u ε radiates in...

On the derivation of a quantum Boltzmann equation from the periodic Von-Neumann equation

François Castella — 2010

ESAIM: Mathematical Modelling and Numerical Analysis

We present the semi-conductor Boltzmann equation, which is time-reversible, and indicate that it can be formally derived by considering the large time and small perturbing potential limit in the Von-Neumann equation (time-reversible). We then rigorously compute the corresponding asymptotics in the case of the Von-Neumann equation on the Torus. We show that the limiting equation we obtain does not coincide with the physically realistic model. The former is indeed an equation of Boltzmann type, yet...

Propagation of Gevrey regularity over long times for the fully discrete Lie Trotter splitting scheme applied to the linear Schrödinger equation

François CastellaGuillaume Dujardin — 2009

ESAIM: Mathematical Modelling and Numerical Analysis

In this paper, we study the linear Schrödinger equation over the -dimensional torus, with small values of the perturbing potential. We consider numerical approximations of the associated solutions obtained by a symplectic splitting method (to discretize the time variable) in combination with the Fast Fourier Transform algorithm (to discretize the space variable). In this fully discrete setting, we prove that the regularity of the initial datum is preserved over long times,  times that are exponentially...

High frequency limit of the Helmholtz equations.

Jean-David BenamouFrançois CastellaTheodoros KatsaounisBenoit Perthame — 2002

Revista Matemática Iberoamericana

We derive the high frequency limit of the Helmholtz equations in terms of quadratic observables. We prove that it can be written as a stationary Liouville equation with source terms. Our method is based on the Wigner Transform, which is a classical tool for evolution dispersive equations. We extend its use to the stationary case after an appropriate scaling of the Helmholtz equation. Several specific difficulties arise here; first, the identification of the source term ( which does not share the...

High Frequency limit of the Helmholtz Equations

Jean-David BenamouFrançois CastellaThodoros KatsaounisBenoît Perthame

Séminaire Équations aux dérivées partielles

We derive the high frequency limit of the Helmholtz equations in terms of quadratic observables. We prove that it can be written as a stationary Liouville equation with source terms. Our method is based on the Wigner Transform, which is a classical tool for evolution dispersive equations. We extend its use to the stationary case after an appropriate scaling of the Helmholtz equation. Several specific difficulties arise here; first, the identification of the source term (which does not share the...

Raman laser : mathematical and numerical analysis of a model

François CastellaPhilippe ChartierErwan FaouDominique BayartFlorence LeplingardCatherine Martinelli — 2004

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

In this paper we study a discrete Raman laser amplification model given as a Lotka-Volterra system. We show that in an ideal situation, the equations can be written as a Poisson system with boundary conditions using a global change of coordinates. We address the questions of existence and uniqueness of a solution. We deduce numerical schemes for the approximation of the solution that have good stability.

Raman laser: mathematical and numerical analysis of a model

François CastellaPhilippe ChartierErwan FaouDominique BayartFlorence LeplingardCatherine Martinelli — 2010

ESAIM: Mathematical Modelling and Numerical Analysis

In this paper we study a discrete Raman laser amplification model given as a Lotka-Volterra system. We show that in an ideal situation, the equations can be written as a Poisson system with boundary conditions using a global change of coordinates. We address the questions of existence and uniqueness of a solution. We deduce numerical schemes for the approximation of the solution that have good stability.

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