The aim of this work is to introduce and to analyze new algorithms for solving the transport neutronique equation in 2D geometry. These algorithms present the duplicate favors to be, on the one hand faster than some classic algorithms and easily to be implemented and naturally deviced for parallelisation on the other hand. They are based on a splitting of the collision operator holding amount of caracteristics of the transport operator. Some numerical results are given at the end of this work.

We study a finite volume method, used to approximate the solution of the linear two dimensional convection diffusion equation, with mixed Dirichlet and Neumann boundary conditions, on Cartesian meshes refined by an automatic technique (which leads to meshes with hanging nodes). We propose an analysis through a discrete variational approach, in a discrete H 1 finite volume space. We actually prove the convergence of the scheme in a discrete H 1 norm, with an error estimate of order O(h) (on meshes of size h).

Consider the domain $Z_\epsilon=\{x\in\mathbb{R}^n ; {dist}(x,\epsilon\mathbb{Z}^n)> \epsilon^\gamma\}$ and let the free path length be defined as $\tau_\epsilon(x,v)=\inf\{t> 0 ; x-tv\in Z_\epsilon\}.$ In the Boltzmann-Grad scaling corresponding to $\gamma=\frac{n}{n-1}$ , it is shownthat the limiting distribution $\phi_\epsilon$ of $\tau_\epsilon$ is bounded from belowby an expression of the form C/t, for some C> 0. A numerical study seems toindicate that asymptotically for large t, $\phi_\epsilon\sim C/t$ .This is an extension of a previous work [J. Bourgain et al., Comm. Math. Phys.190 (1998) 491-508]. As aconsequence, it is proved that the linear Boltzmann type transport equation is inappropriate to describethe Boltzmann-Grad limit of the periodic Lorentz gas, at variance with the usualcase of a Poisson distribution of scatterers treated in [G. Gallavotti (1972)].

The paper is devoted to the convergence analysis of a well-knowncell-centered Finite Volume Method (FVM) for aconvection-diffusion problem in $\mathbb{R}^2$ . This FVM is based on Voronoi boxes andexponential fitting. To prove the convergence of the FVM, we usea new nonconforming Petrov-Galerkin Finite Element Method (FEM)for which the system of linear equations coincides completely withthat of the FVM. Thus, by proving convergence properties of theFEM we obtain similar ones for the FVM. For the error estimationof the FEM well-known statements have to be modified.

The accuracy of the domain embedding method from [A. Rieder, Modél. Math. Anal. Numér.32 (1998) 405-431] for the solution of Dirichlet problemssuffers under a coarse boundary approximation. To overcome this drawback the methodis furnished withan a priori (static) strategy for an adaptive approximation space refinement near the boundary. This is done by selecting suitable wavelet subspaces. Error estimates and numerical experiments validate the proposed adaptive scheme.In contrast to similar, but rather theoretical, concepts already described in theliterature our approach combines a high generality with an easy-to-implement algorithm.

In order to accommodate solutions with multiple phases, corresponding to crossing rays, weformulate geometrical optics for the scalar wave equation as a kinetic transport equation set in phase space. If the maximum number of phases is finite and known a priori we can recover the exact multiphase solution from anassociated system of moment equations, closed by an assumption on the form of the density function in the kinetic equation. We consider two different closure assumptions based ondelta and Heaviside functions and analyze the resultingequations. They form systems of nonlinear conservation laws with source terms. In contrast to the classicaleikonal equation, theseequations will incorporate a "finite" superposition principle in the sense that while the maximum number of phasesis not exceeded a sum of solutions is also a solution. We present numerical results for a varietyof homogeneous and inhomogeneous problems.

An optimal shape control problem for the stationary Navier-Stokessystem is considered. An incompressible, viscous flow in atwo-dimensional channel is studied to determine the shape of part ofthe boundary that minimizes the viscous drag. Theadjoint method and the Lagrangian multiplier method are used to derivethe optimality system for the shapegradient of the design functional.

We introduce a family of new second-order Godunov-type central schemes for one-dimensional systems of conservation laws. They are a less dissipative generalization of the central-upwind schemes,proposed in [A. Kurganov et al., submitted to SIAM J. Sci. Comput.], whose construction is based on the maximal one-sided local speeds of propagation. We also present a recipe, which helps to improve the resolution of contact waves. This is achieved by using the partial characteristic decomposition, suggested by Nessyahu and Tadmor [J. Comput. Phys.87 (1990) 408-463], which is efficiently applied in the context of the new schemes.The method is tested on the one-dimensional Euler equations, subject to differentinitial data, and the results are compared to the numericalsolutions, computed by other second-order central schemes.The numerical experiments clearly illustrate the advantages of theproposed technique.

We introduce and discuss a one-dimensional kinetic model of the Boltzmann equation with dissipative collisions and variable coefficient of restitution. Then, the behavior of the Boltzmann equation in the quasi elastic limit is investigated for a wide range of the rate function. By this limit procedure we obtain a class of nonlinear equations classified as nonlinear friction equations. The analysis of the cooling process shows that the nonlinearity on the relative velocity is of paramount importance for the finite time extinction of the solution.