A fast solver for the spatially homogeneous electron Boltzmann equation

Abstract

We present a numerical method for the velocity-space, spatially homogeneous, collisional Boltzmann equation for electron transport in low-temperature plasma (LTP) conditions. Modeling LTP plasmas is useful in many applications, including advanced manufacturing, material processing, semiconductor processing, and hypersonics, to name a few. Most state-of-the-art methods for electron kinetics are based on Monte-Carlo sampling for collisions combined with Lagrangian particle-in-cell methods. We discuss an Eulerian solver that approximates the electron velocity distribution function using spherical harmonics (angular components) and B-splines (energy component). Our solver supports electron-heavy elastic and inelastic binary collisions, electron-electron Coulomb interactions, steady-state and transient dynamics, and an arbitrary nmber of angular terms in the electron distribution function. We report convergence results and compare our solver to two other codes: an in-house particle Monte-Carlo ethod; and Bolsig+, a state-of-the-art Eulerian solver for electron transport in LTPs. Furthermore, we use our solver to study the relaxation time scales of the higher-order anisotropic correction terms. Our code is open-source and provides an interface that allows coupling to multiphysics simulations of low-temperature plasmas.

Figures (7)

• Figure 1: Rate of convergence for several selected runs from \ref{['tab:self_convergence']}. The above shows the rate of convergence of the relative $L_2$ error (i.e., taking $N_r$=256 as the exact solution) of the electron energy density function (i.e., $f_0$) computed with fixed two-term approximation with increasing resolution in the radial direction with cubic B-splines. As expected, the $L_2$ norm shows fourth order convergence rate for the cubic B-splines.
• Figure 2: Convergence of steady-state EDF radial components $f_l\mleft( v \mright)$ with respect to the number of angular terms used in the approximation. The input parameters for the above run is given by $E/N$=100Td and $n_e/n_0=0$.
• Figure 3: Computed higher order $l$-modes (i.e., $f_2$ and $f_3$ modes) using four-term approximation with increasing resolution in the radial coordinate for input parameters given by $E/n_0$=100Td, and $n_e/n_0=0$. Increased radial resolution helps to capture the high energy tails of the computed $l$-modes.
• Figure 4: Computed QoIs from the developed Eulerian steady-state solver (i.e., denoted by A and the • marker) compared against the Bolsig+ solver (i.e., denoted by B and the $\times$ marker) for varying $E/n_0$ and $n_e/n_0$ values.
• Figure 5: In the above simulation, we compare the computed $l$-modes between the DSMC, proposed Eulerian approach and the Bolsig+ code for the case with $E/n_0$=500Td and $n_e/n_0=0$. Note that, Bolsig+ code uses the two-term expansion, the proposed Eulerian approach uses multi-term expansion, and DSMC code does not inherit a term-expansion.