  A recent proof-of-principle study proposes a nonlinear electrostatic implicit
particle-in-cell (PIC) algorithm in one dimension (Chen, Chacon, Barnes, J.
Comput. Phys. 230 (2011) 7018). The algorithm employs a kinetically enslaved
Jacobian-free Newton-Krylov (JFNK) method, and conserves energy and charge to
numerical round-off. In this study, we generalize the method to electromagnetic
simulations in 1D using the Darwin approximation of Maxwell's equations, which
avoids radiative aliasing noise issues by ordering out the light wave. An
implicit, orbit-averaged time-space-centered finite difference scheme is
applied to both the 1D Darwin field equations (in potential form) and the 1D-3V
particle orbit equations to produce a discrete system that remains exactly
charge- and energy-conserving. Furthermore, enabled by the implicit Darwin
equations, exact conservation of the canonical momentum per particle in any
ignorable direction is enforced via a suitable scattering rule for the magnetic
field. Several 1D numerical experiments demonstrate the accuracy and the
conservation properties of the algorithm.
