The Stochastic Weighted Particle Method for the Computation of the low Probability Tail of the Velocity Distribution in Spatially Homogeneous Plasmas
Accurate and efficient modeling of the low probability tails of the particle velocity probability density function (pdf) is important to study plasmas, especially since the reaction rates in plasmas are determined by the overlap between the electron velocity pdf and the electronimpact cross section of the various species. To study these low probability tails, we focus on implementing stochastic methods, since the stochastic methods are more flexible for the modeling of a diverse range of plasma processes. In particular, we focus on the stochastic weighted particle method (SWPM). We describe the details of the SWPM, improve its various components, and implement it to approximate the higher order moments and the low probability tails of the velocity pdf accurately and efficiently in the spatially homogeneous case. The Stochastic Weighted Particle Method (SWPM) is a Monte Carlo technique developed by Rjasanow and Wagner that generalizes Bird’s Direct Simulation Monte Carlo (DSMC) method for solving the Boltzmann equation. In Bird’s method, each stochastic particle represents an equal number of physical particles, which results in a fewer number of stochastic particles in the tail and hence in large statistical errors. This issue is resolved in the SWPM. In SWPM, each stochastic particle represents a varying proportion of physical particles which allows the stochastic method to have more stochastic particles in the tail for better accuracy. However, the SWPM has its own computational challenges. One of the major challenges is the gradual increase in the number of stochastic particles after each collision. To reduce the increase in computational cost as a result of this issue, Rjasanow and Wagner proposed a reduction method based on a particle reduction scheme in combination with a clustering technique. They proposed several particle reduction schemes designed to preserve specified moments of the velocity distribution. To improve upon the existing particle reduction schemes, we introduce a novel particle reduction scheme that preserves all moments of the velocity distribution up to the second order, as well as the raw and central heat flux both within each group of particles to be reduced and for the entire system. Furthermore, we demonstrate that with the new reduction scheme the scalar fourth-order moment can be computed more accurately at a reduced computational cost. In addition to reduction schemes that conserve the higher order moments, group formation techniques also play an important role in the computation of the low probability tails. It is crucial to group stochastic particles that are close together in the phase space for convergence of the numerical solution. The closeness of the velocity samples in a group is quantified by the maximum relative velocity, which is also called the diameter of a group. We present an improved group splitting technique that leads to the reduction of the maximum diameter of groups faster than the existing group splitting technique by Rjasanow and Wagner. Based on this new group splitting technique, we design a case study to define the parameters required for the convergence of the numerical solution. With the implementation of these parameters, we present for the first time a study of the rate of convergence of tail functionals as the initial number of samples increases.