Boltzsim: A fast solver for the 1D-space electron Boltzmann equation with applications to radio-frequency glow discharge plasmas

TL;DR

This work introduces Boltzsim, an Eulerian solver for the 1D3V electron Boltzmann transport equation in low-temperature plasmas, featuring multi-term EDF expansions and both semi-implicit and fully implicit time integration. It advances a self-consistent hybrid framework that treats heavy species with fluid equations while solving the electron transport kinetically, enabling accurate RF glow discharge simulations. The authors validate the approach through self-convergence tests and cross-verification with PIC-DSMC, and provide comprehensive performance analyses demonstrating scalability on CPU and GPU hardware. The findings emphasize the necessity of kinetic electron modeling at low pressures and deliver an open-source, high-fidelity tool for RF GDPs and related plasma applications.

Abstract

We present an algorithm for solving the one-dimensional space collisional Boltzmann transport equation (BTE) for electrons in low-temperature plasmas (LTPs). Modeling LTPs is useful in many applications, including advanced manufacturing, material processing, and hypersonic flows, to name a few. The proposed BTE solver is based on an Eulerian formulation. It uses Chebyshev collocation method in physical space and a combination of Galerkin and discrete ordinates in velocity space. We present self-convergence results and cross-code verification studies compared to an in-house particle-in-cell (PIC) direct simulation Monte Carlo (DSMC) code. Boltzsim is our open source implementation of the solver. Furthermore, we use Boltzsim to simulate radio-frequency glow discharge plasmas (RF-GDPs) and compare with an existing methodology that approximates the electron BTE. We compare these two approaches and quantify their differences as a function of the discharge pressure. The two approaches show an 80x, 3x, 1.6x, and 0.98x difference between cycle-averaged time periodic electron number density profiles at 0.1 Torr, 0.5 Torr, 1 Torr, and 2 Torr discharge pressures, respectively. As expected, these differences are significant at low pressures, for example less than 1 Torr.

Figures (10)

• Figure 1: An illustrative schematic for the glow discharge plasma with Argon.
• Figure 1: Electron transport using BTE with specified electric field $E(x,t)=10^4 \sin(2\pi \zeta t)$ with $\zeta=13.56$MHz. \ref{['fig:case1_1_1', 'fig:case1_1_2', 'fig:case1_1_3']} show the electron number density $n_e(x)=\int_{\boldsymbol{v}} f\mleft( x, \boldsymbol{v} \mright)\, d\boldsymbol{x}$ evolution in time for runs $r_0$, $r_1$, and $r_2$ specified in \ref{['tab:refinement_params']}. \ref{['fig:case1_2_1', 'fig:case1_2_2', 'fig:case1_2_3']} show relative error in $n_e(x)$ with respect to the highest resolution run $r_2$. \ref{['fig:case1_3_1', 'fig:case1_3_2', 'fig:case1_3_3']} show the electron energy density function at spatial location $\mleft( 2x/L-1 \mright)=\hat{x} = -0.988$ for subsequent runs $r_0$, $r_1$ and $r_2$.
• Figure 2: We consider the electron BTE given in \ref{['eq:hybrid_cnt_b', 'eq:hybrid_cnt_b_bdy']} with electric field $E(x,t)=10^4 \sin(2\pi \zeta t)$ with $\zeta=13.56$MHz. The figure shows solutions after a one period 1/f. The blue line is Boltzsim and the broken green line is PIC-DSMC.
• Figure 3: The overall runtime cost breakdown for a single RF-GDP timestep using the hybrid modeling approach. The left and right plots show the cost breakdown using semi-implicit and fully-implicit schemes for evolving the electron BTE. In both cases, $\boldsymbol{E}$-field solve cost is negligible compared to others, hence omitted in the plot.
• Figure 4: An illustrative diagram that describes the process of computing tabulated kinetic coefficients for electrons to be used in the fluid model (Equation \ref{['eq:fluid_model_eqs']}). For a fixed $n_0$, $T_0$, we compute the steady-state EDF solutions for the spatially homogeneous BTEs for a given electric field values i.e., $\{E_0,...,E_n\}$. For each $E_j$, we compute kinetic coefficients {$k_i^j$, $\mu_e^j$, $D_e^j$} and corresponding temperature $T_e^j$. Then for example, for $\mu_e$, we build an interpolant $\mu_e(T_e)$ using the $\{\mu_e^j, T_e^j\}$ pairs.