THE CONVERGENCE THEORY OF PARTICLE-IN-CELL METHODS FOR MULTIDIMENSIONAL VLASOV-POISSON SYSTEMS

In this work, a convergence theory of particle-in-cell methods for multidimensional Vlasov-Poisson systems is provided. Such methods replace the continuous distribution of masses or charges by discrete particles in phase (spatial and momentum) space with appropriate weights. Additionally, Poisson's equation is solved using a mollified mass or charge density at each timestep. This procedure is equivalent to using a mollified Poisson kernel convolved with the density to obtain the approximate field quantities. In the analysis here, Poisson's equation is solved by a finite element method which can be seen to require less operations to compute the field quantities than required by the usual convolution formula.