Abstract:The numerical resolution of kinetic equations and in particular of Vlasov type equations is most of the time performed using PIC (Particle In Cell) methods which consist in describing the time evolution of the equation through a finite number of particles which follow the characteristic curves of the equation, the interaction with the external and self consistent fields being resolved using a grid. Another approach consists in computing directly the distribution function on a grid by following the characteristics backward in time for one time step and interpolating the value at the feet of the characteristics using the grid points values of the distribution function at the previous time step. In this report we introduce this last method and its use for different types of Vlasov equations. Key-words:

The form of topological derivatives for an integral shape functional is derived for a class of semilinear elliptic equations. The convergence of finite element approximation for the topological derivatives is shown and the error estimates in the L ∞ norm are obtained. The results of numerical experiments which confirm the theoretical convergence rate are presented.

International audienceThe inverse electromagnetic casting problem consists in looking for a suitable set of electric wires such that the electromagnetic field induced by an alternating current passing through them, makes a given mass of liquid metal acquire a predefined shape. In this paper we propose a method for the topol-ogy design of such inductors. The inverse electromagnetic casting problem is formulated as an optimization problem, and topological derivatives are considered in order to locate new wires in the right position. Several numerical examples are presented showing that the proposed technique is effective to design suitable inductors

SUMMARYA ÿnite di erence solution for a system of non-linear integro-di erential equations modelling the steadystate combined radiative-conductive heat transfer is proposed. A new backward-forward ÿnite di erence scheme is formulated for the Radiative Transfer Equation. The non-linear heat conduction equation is solved using the Kirchho transformation associated with a centred ÿnite di erence scheme. The coupled system of equations is solved using a ÿxed-point method, which relates to the temperature ÿeld. An application on a real insulator composed of silica ÿbres is illustrated. The results show that the method is very e cient.

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This article deals with a numerical solution for combined radiation and conduction heat transfer in a grey absorbing and emitting medium applied to a two-dimensional domain using triangular meshes. The radiative transfer equation was solved using the high order Discontinuous Galerkin method with an upwind numerical flux. The energy equation was discretized using a high order finite element method. Stability and error analysis were performed for the Discontinuous Galerkin method to solve radiative transfer equation. A new algorithm to solve the nonlinear radiative-conductive heat transfer systems was introduced and different types of boundary conditions were considered in numerical simulations. The proposed technique's high performance levels in terms of accuracy and stability are discussed in this paper with numerical examples given.

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We consider a model for the propagation and absorption of electromagnetic waves (in the time-harmonic regime) in a magnetised plasma. We present a rigorous derivation of the model and several boundary conditions modelling wave injection into the plasma. Then we propose several variational formulations, mixed and non-mixed, and prove their well-posedness thanks to a theorem by Sébelin et al. Finally, we propose a non-overlapping domain decomposition framework, show its well-posedness and equivalence with the one-domain formulation. These results appear strongly linked to the spectral properties of the plasma dielectric tensor. Résumé. Nous considérons un modèle de propagation et d'absorption d'ondes électromagnétiques(en régime harmonique) dans un plasma magnétique. Nous présentons une justification rigoureuse du modèle et diverses conditions aux limites modélisant l'injection de l'onde dans le plasma. Puis nous proposons plusieurs formulations variationnelles, mixtes ou non, et montrons qu'elles sont bien posées grâce à un théorème de Sébelin et al. Enfin, nous décrivons le principe d'une décomposition de domaine sans recouvrement, et établissons le caractère bien posé de la formulation décomposée et l'équivalence avec la formulation à un seul domaine. Ces résultats paraissent intimement liés aux propriétés spectrales du tenseur diélectrique du plasma.