On the Convergence of Particle Methods for Multidimensional Vlasov–Poisson Systems

For Vlasov–Poisson systems, particle methods are numerical techniques that simulate the behavior of a plasma by a large set of charged superparticles, which obey the classical laws of electrostatics. The trajectories of these charged particles are then followed. Estimates for the errors incurred for a “semi-discrete” approximation to the underlying Vlasov–Poisson system are given by first superimposing a rectangular grid or mesh on all of phase space and then replacing the initial continuous distribution of charges or masses by discrete charges or masses located at the centroid of each grid cell. Our analysis, on one hand, generalizes that of G. H. Cottet and P. A. Raviart (SIAM J. Numer. Anal., 21 (1984), pp. 52–76) to higher-dimensional Vlasov–Poisson systems, and, on the other, those fundamental results of Ole Hald (SIAM J. Numer. Anal., 16 (1979), pp. 726–755) and of J. T. Beale and A. Majda (Math. Comp., 39 (1982), pp. 1–52) on vortex methods for two- and three-dimensional Euler equations, to particle-in-cell methods for multidimensional Vlasov–Poisson settings.