The particle method for the Vlasov–Poisson system using equally spaced initial data points

Some results are given on the convergence of particle methods for the Vlasov–Poisson system of equations. This system describes the time evolution of a collisionless plasma in the presence of an internally consistent electrostatic field. Particle methods are a means for numerically approximating the solutions. For initial data points that are equally spaced a proof of convergence is given in some cases not contained in previous analysis of this problem. It is shown that for once differentiable initial data the particle method converges and the rate is close to linear in the spatial small parameter. For twice differentiable initial data the rate of convergence can be better than linear, and for three times differentiable initial data the rate can be quadratic in the spatial parameter. The proof is based on some different methods than in previous papers on the subject.