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Numerical simulations and analysis of kinetic models - Applications to plasma physics and Nanotechnology

Final Report Summary - NUSIKIMO (Numerical simulations and analysis of kinetic models - Applications to plasma physics and Nanotechnology)

Our research project concerns the mathematical and numerical analysis of partial differential equations occurring in Kinetic theory (Boltzmann equation or Vlasov-Poisson type systems) with a special interest for applications in plasma physics, astrophysics and micro electro mechanical systems. Most of numerical methods used for these simulations are neither accurate nor robust enough to study the correct qualitative behavior of the solution for large time asymptotic or for multi-scale problems. Indeed, for the long time behavior of the solution (development of instabilities, blows-up) as well as for boundary or multi-scale effects, numerical artefacts are difficult to control and may generate instabilities.

Here we propose and analyse specific algorithms (spectral methods, finite volume schemes, asymptotic preserving schemes), specially designed to increase accuracy and stability of numerical solutions. The key idea here is to preserve at the discrete level some important properties of the continuous model in order to improve the long time behavior of the solution when instabilities or small scale phenomena occur. These properties are based on Lyapunov functional, energy conservation, entropy production...

Therefore, we focus on practical application from plasma physics (nuclear fusion reaction) or rarefied gas dynamics (mems) and develop numerical softwares for kinetic models taking into account transports phenomena, boundary effects, and collisions.