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Étude théorique et numérique d'équations cinétiques stochastiques multi-échelles

Abstract : In this thesis, we study a class of slow-fast systems modeled by kinetic linear Stochastic Partial Differential Equations (SPDEs) or Stochastic Differential Equations (SDEs). We study these systems from theoretical and a numerical points of view in two asymptotic regimes: the averaging regime and the diffusion approximation regime.The first two chapters state the main theoretical contributions of this work. We prove the convergence of the slow component of the considered SPDEs to the solution of a diffusion equation with a source term depending on the asymptotic regime. The first chapter focuses on the diffusion approximation regime, where the source term of the limiting equation is a stochastic diffusive term (Wiener process). The second chapter focuses on the averaging regime, where the limiting source term is the average of the original source term.The last two chapters are devoted to the numerical part of this work. In general, a numerical scheme which is consistent with a multiscale system for a fixed parameter epsilon can perform badly in the asymptotic regime when epsilon tends to 0 due to the presence of stiff terms in the model. On the contrary, some schemes are asymptotic preserving: they are consistent for fixed epsilon, converge to some limiting schemes when epsilon tends to 0 and the limiting scheme is consistent with the limiting equation. The goal of the last two chapters is to design asymptotic preserving schemes, respectively for the class of SDEs and SPDEs we consider. We also analyze these schemes and illustrate numerically their efficiency
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Submitted on : Tuesday, July 27, 2021 - 3:13:09 PM
Last modification on : Wednesday, July 28, 2021 - 4:06:58 AM
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  • HAL Id : tel-03301561, version 1


Shmuel Rakotonirina-Ricquebourg. Étude théorique et numérique d'équations cinétiques stochastiques multi-échelles. Equations aux dérivées partielles [math.AP]. Université de Lyon, 2021. Français. ⟨NNT : 2021LYSE1142⟩. ⟨tel-03301561⟩



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