Hamiltonian Particle-in-Cell methods for Vlasov–Poisson equations

In this paper, Particle-in-Cell algorithms for the Vlasov–Poisson system are presented based on its Poisson bracket structure. The Poisson equation is solved by finite element methods, in which the appropriate finite element spaces are taken to guarantee that the semi-discretized system possesses a well defined discrete Poisson bracket structure. Then, splitting methods are applied to the semi-discretized system by decomposing the Hamiltonian function. The resulting discretizations are proved to be Poisson bracket preserving. Moreover, the conservative quantities of the system are also well preserved. In numerical experiments, we use the presented numerical methods to simulate various physical phenomena. Due to the huge computational effort of the practical computations, we employ the strategy of parallel computing. The numerical results verify the efficiency of the new derived numerical discretizations.