This project is devoted to the mathematical and numerical analysis in statistical physics with a special interest to applications in Plasma Physics and nanotechnology with Micro Electro Mechanical Systems (MEMS). We propose to achieve numerical simulations in plasma physics by fully deterministic methods. Using super-computers, a non stationary collisional plasma can be modelled taking into account Coulombian interactions and self-consistent electromagnetic fields to study different regimes and instabilities. These methods are based on high order and conservative finite volume schemes for the transport and fast multi-grid methods for the treatment of collisions. The first application is the simulation of fast ignition or Inertial Confinement Fusion, which is an important issue in plasma physics. Here, the main difficulty concerns the modelling of collisions of relativistic particles and the development of new algorithms for their treatment. Another part is devoted to the derivation of moments models which require less computational effort but keep the main properties of the initial models. The second application concerns micro and nanotechnologies, which are expected to play a very important role in the development of MEMS. Since the scale of micro flows is often comparable with the molecular mean free path, it is necessary to adopt the point of view of kinetic theory. Then applications of kinetic theory methods to micro flows are becoming very important and an accurate approximation of the Boltzmann equation is a key issue. Even nowadays a deterministic numerical solution of the Boltzmann equation still represents a challenge for scientific computing. Recently, a new class of algorithms based on spectral techniques in the velocity space has been been developed for the trend to equilibrium. The next important step is to treat applications for MEMS in nanotechnology for which the main difficulty is to treat complex geometries and moving boundary problems.
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