A Propagator-based Multi-level Monte Carlo Method for Kinetic Neutral Species in Edge Plasmas

TL;DR

The paper introduces a propagator-based MLMC approach to the kinetic Boltzmann equation for neutral species in edge plasmas, exploiting the velocity-localizing effect of frequent collisions to compress evolution into real-space propagator matrices. It develops a generalized semi-group framework with source-correction and standard estimators, enabling equilibrium reconstruction via linear algebra and fixed-point iterations, while maintaining velocity information only for initial and final steps. Numerical experiments show that propagator MLMC reproduces standard MC results and dramatically improves trajectory correlation in coupled plasma-neutral simulations, achieving machine-precision convergence in several tested regimes. The work suggests strong potential for fully implicit Jacobian-free Newton-Krylov solvers and highlights sparsity-based opportunities for future ML surrogates and higher-order extensions.

Abstract

We propose and investigate a new multi-level Monte Carlo scheme for numerical solutions of the kinetic Boltzmann equation for neutral species in edge plasmas. In particular, this method explicitly exploits a key structural property of neutral particle dynamics: the prevalence of frequent collisions for which the outgoing velocity is determined by local plasma parameters. Using this property, we derive a multi-level algorithm based on collision event propagator and show, both analytically and through numerical experiments, that it reproduces the results of standard Monte Carlo methods. We further demonstrate that, in the context of coupled plasma-neutral edge simulations employing correlated Monte Carlo, the proposed scheme retains trajectory correlation to machine precision as the system evolves, whereas conventional methods exhibit rapid decorrelation. These results indicate that the propagator-based multi-level Monte Carlo scheme is a promising candidate for use in fully implicit Jacobian-free Newton-Krylov (JFNK) solvers for coupled plasma-neutral systems.

Figures (4)

• Figure 1: (A) 1-dimensional slices of the 2-dimensional propagator profiles converging as collision number $\sigma\to \infty$ to the steady-state propagator $K_\infty$ (solid blue). (B) The difference of the same profiles from the limit $\underline K_\infty$. (C) The $L^2$-convergence of the propagator iteration as a function of number of collisions $\sigma$ for a variety of total particle trajectories $N$. Colors fading from magenta to dark blue mirror those in (A)-(B), and converged equilibria are indicated by squares or diamonds for each fixed $N$. (D) $L^2$ residual difference between the equilibrium obtained by propagator MLMC and the equilibrium obtained by standard MC, as a function of total flight number. Datapoints are the converged equilibria obtained in (C) indicated by the same markers. The expected convergence rate $O(N^{-1/2})$ is indicated by the dotted black line.
• Figure 2: (A) Comparison of propagator-based MLMC and standard MC for sampling of tails in the fluid limit in one spatial dimension, where the analytic solution is given by Eq. (\ref{['fluidlimit']}). (Bottom) Sparsity patterns of the matrices (B) $\underline K_\infty(x,x')$ and (C) $\underline K_{10}(x,x')$ for the numerical parameters used in Figure \ref{['Fig1']}.
• Figure 3: Relative successive $L^2$ residuals (Eq. \ref{['successiveres']}) for plasma-neutral simulation using explicit coupling. (Red) Standard MC residuals, without correlation. (Magenta) Standard MC residuals with current correlation methods. (Blue) Propagator MLMC residuals with correlation.
• Figure 4: (A) The successive residual for coupled simulations using standard MC and standard correlation methods as parameters vary. The average of the final 100 $L^2$-convergence residuals is displayed on a log scale heatmap. (B) The successive residual for coupled simulations using the Propagator MLMC method. (Insets, left and right) Coupled simulation plasma (I1, I2) and neutral (N1, N2) equilibrium state for two particular parameter pairs indicated by orange arrows, obtained by the propagator MLMC method (B). Equilibria for the two methods (A) and (B) have good visual agreement. (R1, R2) The relative successive residuals as a function of iteration for these two parameter pairs; colors match the final values in (A) and (B).