# Christophe Besse

Professeur à l'Université Toulouse 3 Paul Sabatier

Directeur du LabEx CIMI

Institut
de Mathématiques de Toulouse U.M.R CNRS 5219

Université Paul Sabatier Toulouse 3

118 Route de Narbonne

31062 Toulouse Cedex 9

Tél. : +33 (0)5.61.55.75.87

Courriel : Christophe.Besse_at_math.univ-toulouse.fr

(remplacer _at_ par @)

## Dernières nouvelles ...

### Livre

- C. Besse, J.-C. Garreau
*Editors*,, At the Interface of PHysics and Mathematics, Lecture Notes in Mathematics 2146, Springer, Link*Nonlinear Optical and Atomic Systems*

### Nouvelles prépublications

- C. Besse, G. Dujardin, I. Lacroix-Violet,
, submitted, 2015, Link*High order exponential integrators for nonlinear Schrödinger equations with application to rotating Bose-Einstein condensates* - C. Besse, F. Xing,
, submitted, 2015, Link*Domain decomposition algorithms for two dimensional linear Schrödinger equation* - X. Antoine, C. Besse, V. Rispoli,
, submitted, 2016, Link*High-order IMEX-spectral schemes for computing the dynamics of systems of nonlinear Schrödinger /Gross-Pitaevskii equations* - C. Besse, F. Xing,
, submitted, 2016, Link*Domain decomposition algorithms for the two dimensional nonlinear Schrödinger equation and simulation of Bose-Einstein condensates* - C. Besse, B. Mesognon, P. Noble,
, submitted, 2016, Link*Artificial boundary conditions for the linearized Benjamin-Bona-Mahony equation*

**Abstract:**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.

**Abstract:**This paper deals with two domain decomposition methods for two dimensional linear Schrödinger equation, the Schwarz waveform relaxation method and the domain decomposition in space method. After presenting the classical algorithms, we propose a new algorithm for the free Schrödinger equation and a preconditioned algorithm for the general Schrödinger equation. These algorithms are studied numerically, which shows that the two new algorithms could accelerate the convergence and reduce the computation time. Besides the traditional Robin transmission condition, we also propose to use a newly constructed absorbing condition as the transmission condition.

**Abstract:**The aim of this paper is to build and validate some explicit high-order schemes, both in space and time, for simulating the dynamics of systems of nonlinear Schrödinger/Gross-Pitaevskii equations. The method is based on the combination of high-order IMplicit-EXplicit (IMEX) schemes in time and Fourier pseudo-spectral approximations in space. The resulting IMEXSP schemes are highly accurate, efficient and easy to implement. They are also robust when used in conjunction with an adaptive time stepping strategy and appear as an interesting alternative to time-splitting pseudo-spectral (TSSP) schemes. Finally, a complete numerical study is developed to investigate the properties of the IMEXSP schemes, in comparison with TSSP schemes, for one- and two-components systems of Gross-Pitaevskii equations.

**Abstract:**In this paper, we apply the optimized Schwarz method to the two dimensional nonlinear Schrödinger equation and extend this method to the simulation of Bose-Einstein condensates (Gross-Pitaevskii equation). We propose an extended version of the Schwartz method by introducing a preconditioned algorithm. The two algorithms are studied numerically. The experiments show that the preconditioned algorithm improves the convergence rate and reduces the computation time. In addition, the classical Robin condition and a newly constructed absorbing condition are used as transmission conditions.

**Abstract:**We consider various approximations of artificial boundary conditions for linearized Benjamin-Bona-Mahoney equation. Continuous (respectively discrete) artificial boundary conditions involve non local operators in time which in turn requires to compute time convolutions and invert the Laplace transform of an analytic function (respectively the Z-transform of an holomorphic function). In this paper, we derive explicit transparent boundary conditions both continuous and discrete for the linearized BBM equation. The equation is discretized with the Crank Nicolson time discretization scheme and we focus on the difference between the upwind and the centered discretization of the convection term. We use these boundary conditions to compute solutions with compact support in the computational domain and also in the case of an incoming plane wave which is an exact solution of the linearized BBM equation. We prove consistency, stability and convergence of the numerical scheme and provide many numerical experiments to show the efficiency of our tranparent boundary conditions.

### ANR

- Coordinateur local de l'ANR BECASIM (2013-2016), programme Méthodes Numérique
- Membre de l'ANR Moonrise , (2015-2019)
- Membre de l'ANR Bond , (2013-2017)
- Membre de l'ANR LODIQUAS, programme Blanc International, (2012-2014)
- Coordinateur de l'ANR IODISSEE (2009-2014), programme Cosinus, Conception et Simulation
- Coordinateur local de l'ANR MicroWave (2009-2013), programme Blanc

### Mini-cours et présentations

- Invitation au workshop "Phénomènes non linéaires en optique : théorie et expériences"> , 4-5 Novembre 2015, Besançon, France