Noise and error analysis and optimization in particle-based kinetic plasma simulations

We revisit a meshfree particle model for kinetics of a 1D

electrostatic plasma, using kernel density estimation and a similar

method for the electric field E. The kernel K(x − y) represents the

macroparticle charge distribution. Two length scales enter, the width

w of K and the interparticle spacing λ. This model conserves momentum

and energy. Similarly, continuity is satisfied exactly, and the

Gauss’s law and Ampere’s law formulations are exactly equivalent. A

unified analysis is used for numerical stability and noise

properties. The force can be computed directly using the correlation

K2 = K ∗ K, and K2 is symmetric and positive definite. We discuss the

analogy in the presence of a grid. We can specify a single kernel Kp ,

related to the `kernel trick’ of machine learning. Numerical

instability can occur unless Kp is positive definite, related to a

breakdown in energy conservation. For the noise analysis, the

covariance matrix for the electric field shows a plasma dispersion

function modified by w and λ. 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/λ. We present the bias-variance optimization

(BVO) for the electric field, and compare it to the density BVO[1].