Etude mathématique et numérique de quelques modèles cinétiques et de leurs asymptotiques : limites de diffusion et de diffusion anormale

Abstract : In this thesis, we construct numerical schemes for kinetic equations in some anomalous diffusion regimes. As the model becomes stiff when reaching the asymptotic model, the standard numerical methods become costly in this regime. Asymptotic Preserving (AP) schemes have been designed to overcome this difficulty. Indeed, they are uniformly stable along the transition from the mesoscopic regime to the microscopic one. In the first chapter, we study the case of a heavy-tailed equilibrium distribution, with infinite second order moment. The importance of the high velocities in the equilibrium makes the classical diffusion limit fail, and one can prove that the asymptotic model is a fractional diffusion equation. We construct three AP schemes for this problem, based on a formal asymptotic analysis of the convergence towards the limit model. The discretization of the velocities is then discussed to take into account the high velocities. Moreover, we prove that the third scheme enjoys the stronger property of being uniformly accurate along the convergence towards the microscopic regime. In chapter 2, we extend these results to the case of a degenerated collision frequency, also leading to a fractional diffusion limit. In chapter 3, these methods are then adapted to the case of a classical diffusion limit with anomalous time scale in the kinetic equation. In this case, an adaptation of the AP approach of the previous chapter is needed, because of the slow convergence rate of the kinetic equation towards the limit model. Eventually, a AP scheme for kinetic equation with heavy-tailed equilibria and non local collision operator is presented in chapter 4.
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Hélène Hivert. Etude mathématique et numérique de quelques modèles cinétiques et de leurs asymptotiques : limites de diffusion et de diffusion anormale. Equations aux dérivées partielles [math.AP]. Université Rennes 1, 2016. Français. ⟨NNT : 2016REN1S025⟩. ⟨tel-01393275⟩

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