Fully implicit particle-in-cell algorithms for kinetic simulation of plasmas

Particle-in-cell (PIC) simulation techniques have been wildly successful in the firstprinciples simulation of plasma dynamics. However, the fundamental algorithmic underpinnings of standard PIC algorithms have not changed in decades. Classical PIC employs an explicit approach (leap-frog) to advance the Vlasov-Poisson system using particles coupled to a grid. Explicit PIC is subject to both temporal stability constraints (requiring a minimum temporal resolution) and spatial stability constraints (requiring a minimum spatial resolution), which makes it unsuitable for realistic multi-scale kinetic problems, even with the aid of modern super-computers. Implicit algorithms can potentially eliminate both spatial and temporal stability constraints, thus becoming orders of magnitude more efficient than explicit PIC methods. This motivated much exploration of these algorithms in the literature since the 1980's. However, the lack of efficient nonlinear solvers for the resulting system of equations required approximations that resulted in intolerable accumulation of numerical error in long-term simulations. In this talk, we will present the first successful (i.e., efficient and accurate) fully implicit, nonlinear PIC algorithm using a Jacobian-Free-Newton-Krylov method on a one-dimensional electrostatic model [1]. The formulation conserves local charge and total energy exactly. Momentum is not exactly conserved, but errors are kept small by an adaptive particle substepping orbit integrator. The formulation survives when one considers mapped meshes, thus opening the possibility of accurate spatially adaptive PIC computations [2]. The superior accuracy and efficiency properties of the scheme will be demonstrated with challenging numerical examples. We will also discuss the extension of the approach to electromagnetic PIC in the non-relativistic regime via a Darwin model, which avoids noise issues associated with numerical Cherenkov radiation [3] while remaining exactly energy conserving.