# 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, R. Carles, S. Ervedoza,
, submitted, 2018, Link*A conservation law with spatially localized sublinear damping* - M. Maunoury, C. Besse, V. Mouysset, S. Pernet, P.-H. Haas,
, submitted, 2018, Link*Well-suited and adaptive post-processing for the visualization of hp simulation results* - M. Maunoury, C. Besse, V. Mouysset, S. Pernet,
, submitted, 2018, Link*Accurate, Automatic and Compressed Visualization of Radiated Helmholtz Fields from Boundary Element Solutions*

**Abstract:**We consider a general conservation law on the circle, in the presence of a sublinear damping. If the damping acts on the whole circle, then the solution becomes identically zero in finite time, following the same mechanism as the corresponding ordinary differential equation. When the damping acts only locally in space, we show a dichotomy: if the flux function is not zero at the origin, then the transport mechanism causes the extinction of the solution in finite time, as in the first case. On the other hand, if zero is a non-degenerate critical point of the flux function, then the solution becomes extinct in finite time only inside the damping zone, decays algebraically uniformly in space, and we exhibit a boundary layer, shrinking with time, around the damping zone. Numerical illustrations show how similar phenomena may be expected for other equations.

**Abstract:**While high order methods became very popular as they allow to perform very accurate solutions with low computational time and memory cost, there is a lack of tools to visualize and post-treat the solutions given by these methods. Originally, visualization softwares were developed to post-process results from methods such that finite differences or usual finite elements and therefore process linear primitives. In this paper, we present a methodology to visualize results of high order methods. Our approach is based on the construction of an optimized affine approximation of the high order solution which can therefore be handled by any visualization software. A representation mesh is constructed and the process is guided by an a posteriori estimate which control the error between the numerical solution and its representation pointwise. This point by point control is crucial as under their picture form, data correspond to values mapped on elements where anyone can pick up a pointwise information. A strategy is established to ensure that discontinuities are well represented. These discontinuities come either from the physical problem (material change) or the numerical method (discontinuous Galerkin method) and are pictured accurately. Several numerical examples are presented to demonstrate the potential of the method.

**Abstract:**We propose a methodology to generate an accurate and efficient reconstruction of radiated fields based on high order interpolation. As the solution is obtained with the convolution by a smooth but potentially high frequency oscillatory kernel, our basis functions therefore incorporate plane waves. Directional interpolation is shown to be efficient for smart directions. An adaptive subdivision of the domain is established to limit the oscillations of the kernel in each element. The new basis functions , combining high order polynomials and plane waves, provide much better accuracy than low order ones. Finally, as standard visualization softwares are generally unable to represent such fields, a method to have a well-suited visualization of high order functions is used. Several numerical results confirm the potential of the method.

### ANR

- Membre de l'ANR Nabuco , (2018-2021)
- Membre de l'ANR Moonrise , (2015-2019)
- Membre de l'ANR Bond , (2013-2017)
- Coordinateur local de l'ANR BECASIM (2013-2016), programme Méthodes Numérique
- 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