Homogenization of the Vlasov Equation and of the Vlasov - Poisson System with a Strong External Magnetic Field

TL;DR

This paper develops a homogenized model for the Vlasov and Vlasov-Poisson equations under strong magnetic fields, simplifying numerical simulations of charged particle dynamics by capturing their mean behavior.

Contribution

It introduces a homogenization approach using two-scale convergence to derive approximate equations for particle motion in strong magnetic and electric fields.

Findings

Derived homogenized equations for Vlasov and Vlasov-Poisson systems.

Proved convergence of the homogenization process.

Extended analysis to include orthogonal electric fields.

Abstract

Motivated by the difficulty arising in the numerical simulation of the movement of charged particles in presence of a large external magnetic field, which adds an additional time scale and thus imposes to use a much smaller time step, we perform in this paper a homogenization of the Vlasov equation and of the Vlasov-Poisson system which yield approximate equations describing the mean behavior of the particles. The convergence proof is based on the two scale convergence tools introduced by N'Guetseng and Allaire. We also consider the case where, in addition to the magnetic field, a large external electric field orthogonal to the magnetic field and of the same magnitude is applied.