Seminar by John M. Finn – Tibbar Plasma Technologies (USA)

Abstract: We reconsider a meshfree approach to plasma kinetic theory, specialized to 1D electrostatic plasmas. This method uses kernel density estimation for the charge density and a related Green’s function method, from Gauss’s law, for the electric field. The kernel K(x−y) represents the the charge distribution within each macroparticle, both for computing the electric field E(x) and for using E(x) to compute the force on each macroparticle. This method has good conservation properties, conserving momentum and energy exactly. Similarly, the continuity equation is satisfied exactly, and this Vlasov-Gauss system is exactly equivalent to the Vlasov-Ampere and the Vlasov-Poisson systems. The use of the same kernel above leads to a symmetric / positive definite kernel, the correlation of the original kernel with itself, and allows an analog of the kernel trick in Machine Learning: a single positive definite kernel can be substituted for this correlation. We show how the positive definiteness of the kernel guarantees numerical stability. This analysis uncovers a connection between kernels used for density estimation and positive definite (reproducing) kernels. This analysis is useful for constructing meshfree codes and for analyzing PIC codes, which have a grid. For the latter, I will discuss how the numerical stability in the meshfree formulation can break down or be preserved with a grid, depending on the discretization.