A deterministic Gaussian-Mixtures Coulomb-collision algorithm for particle-in-cell methods

Coulomb-collision modules in PIC simulations are typically Monte-Carlo-based. Monte Carlo (MC) is attractive for its simplicity, efficiency in high dimensions, and conservation properties. However, it is noisy, of low temporal order (typically O(√∆t), and has to resolve the collision frequency for accuracy [1]. In this study, we explore a machine-learning- based, multiscale alternative to MC for PIC. The approach is based on the reconstruction of the particles’ velocity distribution function (VDF) using a Gaussian Mixtures Model (GMM) via the Maximum Likelihood Estimation principle [2,3]. A key element of our algorithm is to decompose each Gaussian in the GMM into a convex linear combination of isotropic Maxwellians for which an exact set of evolution equations can be de- rived according to the Landau-Fokker-Planck collision operator [4]. The proposed method is deterministic, free of instability, positivity-preserving, and strictly conservative, and is orders of magnitude faster than either MC or Eulerian Fokker-Planck solvers. We will illustrate the accuracy and performance of the proposed method with several examples of varying complexity.

References:

[1] Dimits, et. al., JCP, 228 p.4881 (2009)

[2] Redner, Walker, SIAM Review, 26, (1984)

[3] Chen, Chacon, Nguyen, JCP, 436 (2021)

[4] Echim, Lemaire, Svendsen, Surv Geophys, 32, p. 1–70 (2011)