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

TL;DR

This paper presents a novel deterministic particle-in-cell method that accurately incorporates Landau collisional effects into plasma simulations, conserving key physical quantities and applicable in multiple dimensions.

Contribution

It introduces a regularised variational PIC scheme for the Vlasov-Maxwell-Landau equations that conserves mass, charge, momentum, and energy without splitting collision and transport.

Findings

Successfully simulates Landau damping, two-stream, and Weibel instabilities.

Conserves physical quantities during simulations.

Effective in arbitrary dimensions and for Coulomb interactions.

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.