Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The Collisional Particle-In-Cell Method for the Vlasov-Maxwell-Landau Equations

Rafael Bailo, José A. Carrillo, Jingwei Hu

TL;DR

The paper addresses simulating collisional plasmas described by the Vlasov-Maxwell-Landau equations in a fully deterministic, structure-preserving manner. It introduces CPIC, a collisional particle-in-cell method that regularises the Landau operator via a variationally motivated entropic framework, yielding a discretisation that conserves mass, charge, momentum, and energy while increasing a regularised entropy. The method avoids transport-collision splitting by incorporating collisions as a deterministic effective force and supports arbitrary dimensions and interaction types, including Coulomb, using regularised splines and a three-step collision evaluation, along with optimisations like cell lists and random batching. Numerical experiments validate convergence, Landau damping, and collisional effects in the two-stream and Weibel instabilities, demonstrating improved energy conservation and physically correct entropy dynamics. The work provides a scalable, physics-faithful tool for simulating collisional plasmas and offers pathways to higher-dimensional, multi-species, and uncertainty-quantified extensions in plasma modeling.

Abstract

We introduce an extension of the particle-in-cell (PIC) method that captures the Landau collisional effects in the Vlasov-Maxwell-Landau equations. The method arises from a regularisation of the variational formulation of the Landau equation, leading to a discretisation of the collision operator that conserves mass, charge, momentum, and energy, while increasing the (regularised) entropy. The collisional effects appear as a fully deterministic effective force, thus the method does not require any transport-collision splitting. The scheme can be used in arbitrary dimension, and for a general interaction, including the Coulomb case. We validate the scheme on scenarios such as the Landau damping, the two-stream instability, and the Weibel instability, demonstrating its effectiveness in the numerical simulation of plasma.

The Collisional Particle-In-Cell Method for the Vlasov-Maxwell-Landau Equations

TL;DR

The paper addresses simulating collisional plasmas described by the Vlasov-Maxwell-Landau equations in a fully deterministic, structure-preserving manner. It introduces CPIC, a collisional particle-in-cell method that regularises the Landau operator via a variationally motivated entropic framework, yielding a discretisation that conserves mass, charge, momentum, and energy while increasing a regularised entropy. The method avoids transport-collision splitting by incorporating collisions as a deterministic effective force and supports arbitrary dimensions and interaction types, including Coulomb, using regularised splines and a three-step collision evaluation, along with optimisations like cell lists and random batching. Numerical experiments validate convergence, Landau damping, and collisional effects in the two-stream and Weibel instabilities, demonstrating improved energy conservation and physically correct entropy dynamics. The work provides a scalable, physics-faithful tool for simulating collisional plasmas and offers pathways to higher-dimensional, multi-species, and uncertainty-quantified extensions in plasma modeling.

Abstract

We introduce an extension of the particle-in-cell (PIC) method that captures the Landau collisional effects in the Vlasov-Maxwell-Landau equations. The method arises from a regularisation of the variational formulation of the Landau equation, leading to a discretisation of the collision operator that conserves mass, charge, momentum, and energy, while increasing the (regularised) entropy. The collisional effects appear as a fully deterministic effective force, thus the method does not require any transport-collision splitting. The scheme can be used in arbitrary dimension, and for a general interaction, including the Coulomb case. We validate the scheme on scenarios such as the Landau damping, the two-stream instability, and the Weibel instability, demonstrating its effectiveness in the numerical simulation of plasma.
Paper Structure (35 sections, 2 theorems, 72 equations, 15 figures)

This paper contains 35 sections, 2 theorems, 72 equations, 15 figures.

Table of Contents

  1. Introduction
  2. The Collisional Particle-In-Cell Method
  3. The Vlasov-Maxwell-Landau Equations
  4. Description of the Method
  5. Regularisations
  6. Lorentz Term
  7. Collision Term
  8. Step I
  9. Step II
  10. Step III
  11. Choice of Regularisation Parameters
  12. Computational Optimisation of the Method
  13. Cell List Optimisation
  14. Random Batch Optimisation
  15. Properties of the Regularised Collision Operator
  16. ...and 20 more sections

Key Result

Theorem 2.3

The functions $g=1$, $g=v$, and $g=\frac{1}{2}|v|^2$ are collision invariants of the CPIC collision operator:

Figures (15)

  • Figure 1: Typical solution in the Maxwellian case of the validation test of \ref{['sec:BKW']}. $C=2^{-4}$, $t\in(*){0,15}$. $N_{v}=64$, $N_c=8$, $\Delta t=10^{-2}$. $R=16$.
  • Figure 2: Order of convergence for the validation test of \ref{['sec:BKW']}. Relative ${L^{2}}$ errors of the regularised solutions $\tilde{f}^N$. $C=2^{-4}$, $t\in(*){0,15}$.
  • Figure 3: Order of convergence for the validation test of \ref{['sec:BKW']}. Relative ${L^{2}}$ errors of the regularised solutions $\tilde{f}^N$. $C=2^{-4}$, $t\in(*){0,15}$.
  • Figure 4: Numerical Landau damping in the validation test of \ref{['sec:LandauDamping']}. $t\in(*){0,10}$. $N_{x}=128$, $N_{v}=32$, $N_c=8$, $\Delta t=50^{-1}$. $R=32$. Total of $1\,048\,576$ particles. Constants $\gamma_l$ and $\gamma_{l,c}$ given in \ref{['sec:LandauDamping']}.
  • Figure 5: Entropy and entropy transport error in the Landau damping validation test of \ref{['sec:LandauDamping']}. $t\in(*){0,10}$. $N_{x}=128$, $N_{v}=32$, $N_c=8$, $\Delta t=50^{-1}$. $R=32$. Total of $1\,048\,576$ particles.
  • ...and 10 more figures

Theorems & Definitions (2)

  • Theorem 2.3: Discrete collision invariants
  • Theorem 2.4: Discrete H-theorem