Adaptive Multi-Dimensional Particle In Cell
  Kinetic Particle In Cell (PIC) methods can extend greatly their range of applicability if implicit time differencing and spatial adaption are used to address the wide range of time and length scales typical of plasmas. For implicit differencing, we refer the reader to our recent summary of the implicit moment PIC method implemented in our CELESTE3D code [G. Lapenta, Phys. Plasmas, 13, 055904 (2006)]. Instead, the present document deals with the issue of PIC spatial adaptation. Adapting a kinetic PIC code requires two tasks: adapting the grid description of the fields and moments and adapting the particle description of the distribution function. Below we address both issues. First, we describe how grid adaptation can be guided by appropriate measures of the local accuracy of the solution. Based on such information, grid adaptation can be obtained by moving grid points from regions of lesser interest to regions of higher interest or by adding and removing points. We discuss both strategies. Second, we describe how to adapt the local number of particles to reach the required statistical variance in the description of the particle population. Finally two typical applications of adaptive PIC are shown: collisionless shocks and charging of small bodies immersed in a plasma. 
Giovanni Lapenta