Particle-in-cell method for numerical simulation of beam and plasma dynamics

The finite particle methods are Euler or Lagrange methods for mathematical description of a continuous medium. These methods are widely applied for the numerical simulation of a continuous medium including continuum electrodynamics and simulation of charged particle beams. The finite particle method allows studying the detailed beam and plasma characteristics, the distribution function of particles, the nonlinear self and external fields. The so-called 'particle-in-cells' method is a combination of both Euler and Lagrange methods, and it is most effective for simulations in the physics of plasma and charged particle beams. The basement and recent development of the particle-in-cells method are presented. Two examples illustrate the application of the particle-in-cells method in beam dynamics and plasma physics.