Author: John M Finn
Requested Type: Poster
Submitted: 2022-02-24 12:14:09
Tibbar Plasma Technologies, LLC
274 DP Road
Los Alamos, NM 87544
We revisit a meshfree method for kinetic theory for a 1D electrostatic plasma. This method uses kernel density estimation for the density and a similar method for finding the electric field $E$. With the kernel $K(x-y)$ representing the the macroparticle charge distribution, both for computing $E$ and for using $E$ to compute the force on each macroparticle, the model represents the exact dynamics of the macroparticles. This model is has two length scales, the width $w$ of $K$ and the interparticle spacing $lambda$. This method has good conservation properties, conserving momentum and energy (for time step $hto 0$.) Similarly, continuity is satisfied exactly, and the Gauss's law formulation is exactly equivalent to the Amp`ere's or the Poisson approach.
We analyze in a unified manner the numerical stability properties of this method and its noise properties. Since $K$ is used to compute the both the electric field and the force on each macroparticle, the force can be computed directly using the convolution $K_2=K*K$, and $K_2$ is positive definite. We can, instead, specify a single positive definite kernel $K_p$, related to the `kernel trick' of machine learning. A novel feature is numerical instability that can occur if $K_p$ is not positive definite; it is shown how this is related to a breakdown in energy conservation. For the noise analysis, the covariance matrix for the electric field is computed in space and time, showing denominators (propagators) proportional to the plasma dispersion function modified by $w$ and $lambda$, and also the spacing in velocity space. In this model, the number of particles per cell does not enter, and the noise is characterized by the number of particles per kernel width, i.e. w/lambda. For non-uniform density, we present the bias-variance optimization (BVO) for the electric field, and compare it to the density BVO.