Development of a charge- and energy-conserving implicit moment-acceleration method for the Vlasov-Darwin Particle-in-Cell System

The coupled set of Vlasov-Maxwell equations describes the dynamic evolution of a collisionless plasma. Explicit particle-in-cell (PIC) approaches are attractive because they are simple to implement, highly parallelizable, and inexpensive per timestep. However, explicit PIC schemes are subject to numerical instabilities when the Debye length is not resolved or the Courant–Friedrichs–Lewy (CFL) condition is not met. This becomes a significant challenge when the dynamics of interest occur on large spatial and temporal scales compared to these parameters. Conversely, implicit schemes can bypass both of these limitations at the cost of being more expensive per timestep by solving the evolution iteratively. Previous work by Taitano et al. [1] developed a method that uses an auxiliary but self-consistent set of moment-field equations, called the lower order (LO) equations, which act as an algorithmic accelerator for an underlying iterative nonlinear solver for the electrostatic, one-dimensional (1D) implicit PIC, i.e., the higher order (HO) system. In the present work, we discuss preliminary results of extending the HOLO framework to a 1D electromagnetic Darwin system. The solver performance sensitivity to different choices of LO equations is studied on various test cases, such as the electron/ion Weibel instabilities.