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

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. Furthermore, the preservation of the Coulomb gauge for the vector potential (mandatory for Darwin) proved difficult numerically, often resulting in high-order diffential systems that required ad-hoc boundary conditions. Here, we explore fully implicit PIC algorithms for the Vlasov-Darwin model. Fully implicit electrostatic PIC algorithms have been recently developed in 1D [1], featuring exact discrete energy and charge conservation. In this study, we develop a Vlasov-Darwin PIC algorithm in single [2] and multiple dimensions. Employing the scalar and vector potentials in the Darwin field equations, and consistent B-spline interpolations between field and particle quantities, the algorithm conserves total energy, local charge, canonical-momentum in the ignorable direction, and the Coulomb gauge exactly. A simple fluid preconditioner allows efficient use of large timesteps, orders of magnitude larger than the explicit CFL of a standard electromagnetic PIC simulation. We demonstrate the accuracy and efficiency properties of the algorithm with various numerical experiments in 1D-2V and 2D-3V.