# Conservative spectral methods for Fokker-Planck-Landau type equations : simulations, long-time behaviour and error estimates

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The focus of this thesis is to investigate a conservative spectral method for solving Fokker-Planck-Landau type (F. P. L.) equations as a model for plasmas, when coupled to the Vlasov-Poisson equation in the mean-field limit, modelling particle interactions extending from Coulomb to hard sphere potentials. This study will range from numerical examples, that emphasise the strength and accuracy of the method, to a rigorous proof showing that approximations from the numerical scheme converge to analytical solutions. In particular, two sets of novel simulations are included. The first presents benchmark results of decay rates to statistical equilibrium in transport plasma models for Coulomb particle interactions, as well as with Maxwell type and hard sphere interactions. The other studies the essentially unexplored phenomenon of the plasma sheath for Coulomb interactions, exhibiting the formation of a strong field due to charge separation. These topics will be arranged in three major projects: 1. Entropy decay rates for the conservative spectral scheme modelling Fokker-Planck-Landau type flows in the mean field limit. Benchmark simulations of decay rates to statistical equilibrium are created for F.P.L. equations associated to Coulomb particle interactions, as well as with Maxwell type and hard sphere interactions. The qualitative decay to the equilibrium Maxwell-Boltzmann distribution is studied in detail through relative entropy for all three types of particle interactions by means of a conservative hybrid spectral and discontinuous Galerkin scheme, adapted from Chenglong Zhang’s thesis in 2014. More precisely, the Coulomb case shows that there is a degenerate spectrum, with a decay rate close to the law of two thirds predicted by upper estimates in work by Strain and Guo in 2006, while the Maxwell type and hard sphere examples both exhibit a spectral gap as predicted by Desvillettes and Villani in 2000. Such decay rate behaviour indicates that the analytical estimates for the Coulomb case is sharp while, still to this date, there is no analytical proof of sharp degenerate spectral behaviour for the F.P.L. operator. Simulations are presented, both for the space-homogeneous case of just particle potential interactions and the space-inhomogeneous case with the mean field coupling through the Poisson equation for total charges in periodic domains. New explicit derivations of spectral collisional weights are presented in the case of Maxwell type and hard sphere interactions and the stability of all three scenarios, including Coulomb interactions, is investigated. 2. Convergence and error estimates for the conservative spectral method for Fokker-Planck-Landau equations. Error estimates are rigorously derived for a semi-discrete version of the conservative spectral method for approximating the space-homogeneous F.P.L. equation associated to hard potentials. The analysis included shows that the semi-discrete problem has a unique solution with bounded moments. In addition, the derivatives of such a solution up to any order also remain bounded in L² spaces globally time, under certain conditions. These estimates, combined with spectral projection control, are enough to obtain error estimates to the analytical solution and convergence to equilibrium states. It should be noted that this is the first time that an error estimate has been produced for any numerical method which approximates F.P.L. equations associated to any range of potentials. 3. Modelling charge separation with the Landau equation. A model for the plasma sheath is investigated using the space-inhomogeneous linear Landau equation (namely, the F.P.L. equation associated Coulomb interactions), modelling interactions between positive and negative particles. Some theory has been established for the plasma sheath, but this is the first time that an attempt has been made to simulate it with the Landau equation. The particular design of the kinetic model is described and an attempt made to capture physical phenomena associated to separation of charges. Several parameters within the model are varied to try and explain the creation of the sheaths.