Implicit Particle in Cell Plasma Simulation Codes
The space plasmas of the solar wind, of the planetary magnetospheres and other hot and tenuous environnments are governed by two sets of equations : the Vlasov equation that govern the evolution of the particle distribution function (PDF), and the Maxwell equations that govern the electromagnetic field. Particle in cell (PIC) simulation codes describe the particle distribution function, through a set of macro particles whose trajectories evolve according to the Lorentz force. The charge and current densities are interpollated on a rectangular grid. The Maxwell equations are solved on this grid. The electric and the magnetic field are then interpollated on the macroparticle positions to evaluate the Lorentz force. When simulating an electromagnetic plasma with an explicit PIC code, several time scales must be taken into account. They depend on the propagation of light waves, on the electrostatic electron plasma oscilliations, on the electron gyromotion into the magnetic field... This is not necessarily relevant if we are interested in low frequency phenomenons such as ion-cyclotron or Alfvn waves. The aim of an implicit code is to solve the equations of evolution of a dynamical system without keeping the high frequency fluctuations. For instance, time decentered schemes are made implicit by a time-discretisation method, in which the intermediate time level is slightly decentered. In the direct implicit scheme, a few variables of the system (at time step N) are replaced by time averaged implicit variables. These implicit variables depend both on the state of the system in the past (time N-1) and in the future (time N+1). A rough classification of time scales in a plasma can help us to understand which physical parameters must be made implicit, according to the problem to be treated. I will present how implicit variables are chosen, how to deal with them, what are the consequences concerning the other equations, a few pitfalls, and a few ways of avoiding problems. A few examples of applications of implicit PIC codes will illustrate this talk.