A new formulation is presented for the implicit moment method applied to the time-dependent relativistic Vlasov-Maxwell system. The new approach is based on a specific formulation of the implicit moment method that allows us to retain the same formalism that is valid in the classical case despite the formidable complication introduced by the nonlinear nature of the relativistic equations of motion. To demonstrate the validity of the new formulation, an implicit finite difference algorithm is developed to solve the Maxwell’s equations and equations of motion. A number of benchmark problems are run: two stream instability, ion acoustic wave damping, Weibel instability, and Poynting flux acceleration. The numerical results are all in agreement with analytical solutions.

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A coupled continuum-discrete numerical model is proposed to study complex physical systems composed by a set of clusters of different chemical species immersed in a matrix with which they interact. The overall model describes the transient of the basic mechanisms governing the processes of interaction in a two-dimensional micrometer size system. At each time step, the continuum (micrometer scale) model computes the macroscopic temperature field according to the prescribed boundary conditions. The continuum system is discretized with a desired number of uniform computational cells. Each cell contains a number of computational particles which represent the actual particles mixture. The particle-incell (discrete) model maps the macroscopic fields from the (continuum) cells to the particles. Chemical reactions and particles dynamics are followed using a molecular dynamics approach. We present results of a recent application of this approach to the simulation of nanoparticles formation in SHS reactors 1 . To investigate the physical conditions under which this phenomenon takes place, we propose a statistical analysis based on the pair distribution function for the particles formed during the reaction. In particular, the dependence of this function on the interaction potential between particles has been investigated. As a demonstration of the effectiveness of the method some paradigmatic examples will be showed.