A multi-dimensional, energy- and charge-conserving, nonlinearly implicit, electromagnetic Vlasov-Darwin particle-in-cell algorithm

TL;DR

This paper introduces a fully implicit, energy- and charge-conserving electromagnetic Vlasov-Darwin PIC algorithm that enables large time steps and cell sizes, improving efficiency and accuracy in plasma simulations.

Contribution

It develops a novel implicit PIC algorithm for the Vlasov-Darwin model that overcomes stability issues and conserves key physical quantities.

Findings

Conserves total energy, charge, and canonical momentum.

Enables large time steps and cell sizes without stability loss.

Demonstrates high accuracy and efficiency in 2D-3V simulations.

Abstract

For decades, the Vlasov-Darwin model has been recognized to be attractive for particle-in-cell (PIC) kinetic plasma simulations in non-radiative electromagnetic regimes, to avoid radiative noise issues and gain computational efficiency. However, the Darwin model results in an elliptic set of field equations that renders conventional explicit time integration unconditionally unstable. Here, we explore a fully implicit PIC algorithm for the Vlasov-Darwin model in multiple dimensions, which overcomes many difficulties of traditional semi-implicit Darwin PIC algorithms. The finite-difference scheme for Darwin field equations and particle equations of motion is space-time-centered, employing particle sub-cycling and orbit-averaging. The algorithm conserves total energy, local charge, canonical-momentum in the ignorable direction, and preserves the Coulomb gauge exactly. An asymptotically well-posed fluid preconditioner allows efficient use of large time steps and cell sizes, which are determined by accuracy considerations, not stability, and can be orders of magnitude larger than required in a standard explicit electromagnetic PIC simulation. We demonstrate the accuracy and efficiency properties of the algorithm with various numerical experiments in 2D-3V.