Theoretical and numerical methods for kinetic simulation of plasmas

Understanding and simulating the dynamics of plasmas in Tokamak devices is a crucial aspect of the plasma physics research, especially with the upcoming ITER device. The development of numerical schemes that possess conservation laws over the vast time scale that covers the dynamics of charged particles in fusion plasmas is an intimidating yet a very important task. This thesis presents novel numerical and theoretical techniques to tackle this problem. First, an overview of the kinetic theory, in particular the derivations of the Vlasov equation, the Fokker-Planck equation and the Vlasov-Maxwell equation in a variational setting, is given. The Euler-Poincar\'{e} reduction, which is a powerful mathematical tool that allows to derive the the Vlasov-Maxwell equations in a straightforward way, is presented as well. A multi-species, marker based, structure-preserving numerical code for the Landau equation is presented. The code is able to preserve energy and momentum to machine precision and leverages GPU-computing to efficiently scale with the dimension of the system. The scheme was validated against relaxation, isotropization and thermalization theoretical estimates for different mass-ratio of the species, including a real electron-deuteron case, showing good agreement in all performed tests. Finally, the problem of fast ions is tackled by introducing the Backward Monte Carlo (BMC) scheme. The approach aims at increasing the poor statistics of current Forward Monte Carlo simulations by integrating the probability of fast ions backward in time and taking into account deterministically the spread of the Monte Carlo collision operator. The scheme was implemented as a module of the orbit following code ASCOT5, enabling high performance simulations especially with modern supercomputers, and test cases with realistic plasma profiles, magnetic fields and wall geometries. The BMC scheme was applied to a realistic ASDEX Upgrade configuration of beam-ion distributions, with a Fast-Ion Loss Detector (FILD) placed near the divertor. The results shows a substantial increase of wall hits compared to a standard Forward Monte Carlo simulation.