Meshfree Methods for Plasma Kinetic Theory

We revisit a meshfree particle method for kinetic theory. We first treat a 1D electrostatic plasma with immobile ions. We find the electron density by kernel density estimation and a similar method for the electric field E. The kernel K(x - y) represents the charge distribution within the macroparticles. Two length scales enter, the width w of K and the interparticle spacing s. This model conserves momentum and energy. Similarly, continuity is satisfied exactly. A unified analysis is used for numerical stability and noise properties. E(x) is computed via a kernel G(x-y), which is related to the kernel K. The force is integrated over the particle and is associated with a new kernel K2(x-y), the correlation of K with itself. K2 is symmetric and positive definite (SPD) even if K is neither. Using a single SPD kernel Kp is related to the ‘kernel trick’ of machine learning. Numerical instability can occur unless Kp is SPD, related to a breakdown in energy conservation. The covariance matrix for the electric field shows a plasma dispersion function modified by w and s. The noise is characterized by the number of particles per kernel width, i.e. w/s. We present the bias-variance optimization (BVO) for E or the force, and compare it to the density BVO[1]. We numerically verify our theory using three methods, a meshfree particle code, a PIC code, and a meshfree Fourier method[2]. The 2nd second and third methods eliminate the Np
^{^}
2 problem, adding some overhead. These approaches have the same conservation properties. Method 3 does not employ spectral solve for E(x), which is never computed. In the limit of fine mesh in Method 2, these three formulations are equivalent and can be obtained from a variational arguments.