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Contribution à l'analyse et la simulation numérique de l'équation de Vlasov

Abstract : This work is devoted to the numerical simulation of transport of particles and collisions. We first propose a finite volume scheme to discretize the Vlasov-Poisson system we prove the convergence of the approximation to the couple (E.f), which is the unique weak solution of the Vlasov-Poisson system. Then, a new scheme for solving the Vlasov equation using a phase spacegrid is proposed. Several numerical results are presented in two and four dimensional phase space and the scheme is compared with the semi-lagrangian method. We alsa give numerical results of beam propagation and halo formation in the transverse plan.The next part is devoted to the comparison of deterministic schemes to approximate the Fokker-Planck-Landau equation. Numerical results for both space homogeneous and space non homogeneous situations show the efficiency and the accuracy of the present method. ln the last chapter, we prove global existence of weak solutions for the Vlasov-Darwin system with a restriction on the initial energy. From classical estimates on kinetic and elliptic equations, we finally prove that when the characteristic velocity of particles is negligeable in front of the light velocity, the solution of Vlaso-Darwin system converges to the weak solution of the Vlasov-Poisson system.
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Submitted on : Thursday, March 29, 2018 - 10:46:19 AM
Last modification on : Tuesday, March 2, 2021 - 5:12:06 PM
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  • HAL Id : tel-01746860, version 1



Francis Filbet. Contribution à l'analyse et la simulation numérique de l'équation de Vlasov. Mathématiques générales [math.GM]. Université Henri Poincaré - Nancy 1, 2001. Français. ⟨NNT : 2001NAN10068⟩. ⟨tel-01746860⟩



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