Charge-conserving, variational particle-in-cell method for the drift-kinetic Vlasov-Maxwell system

TL;DR

This paper introduces a novel charge-conserving, variational discretization method for the drift-kinetic Vlasov-Maxwell system that preserves physical laws, is computationally efficient, and suitable for simulating plasma turbulence with strong magnetic fields.

Contribution

It presents the first structure-preserving, electromagnetic particle-in-cell scheme for the mixed kinetic-ion and drift-kinetic-electron Vlasov-Maxwell model, incorporating polarization and magnetization effects.

Findings

Automatically satisfies Gauss law at each step.

Explicit magnetic field update involving only local operations.

Linearly implicit electric field update with sparse matrix inversion.

Abstract

This paper proposes a charge-conserving, variational, spatio-temporal discretization for the drift-kinetic Vlasov-Maxwell system, utilizing finite-elements for the electromagnetic fields and the particle-in-cell approach for the Vlasov distribution. The proposed scheme is fully electromagnetic, dealing with fields instead of potentials, and includes the effects of polarization and magnetization in the Gauss and Amp\`ere-Maxwell laws, a consequence of reducing the full particle dynamics to drift-center dynamics. There is, however, no need to invert the Gauss law: it is satisfied automatically at every time-step as a result of a discrete Noether symmetry, and the electric field is updated directly from the Amp\`ere-Maxwell equation. The method provides an update for the magnetic field that is fully explicit, involving only local operations. The update for particles is implicit for each particle individually, also leading to local operations only. The update for the electric field is linearly implicit due to the presence of a finite-element mass matrix and polarization and magnetization effects in the Amp\`ere-Maxwell equation, hence involving a sparse matrix inversion once at every time step. Because the scheme deals with the electromagnetic fields and not the potentials, it also provides the first serious attempt at constructing a structure-preserving numerical scheme for the mixed kinetic-ion--drift-kinetic-electron Vlasov-Maxwell model. Consequently, the proposed method could be used to simulate electromagnetic turbulence in fusion experiments or space plasmas that exhibit a strong background magnetic field while retaining all of the ion physics, most of the necessary electron physics, yet eliminating perhaps the biggest obstacle in reaching macroscopic transport time scales in kinetic simulations, namely the electron cyclotron time scale.