An Energy- and Charge-conserving, Implicit, Electrostatic Particle-in-Cell Algorithm in curvilinear geometry

A recent proof-of-principle study proposes an energy- and charge-conserving, fully implicit particle-in-cell algorithm in one dimension [1], which is able to use timesteps comparable to the dynamical timescale of interest. Here, we generalize the method to employ non-uniform meshes via a curvilinear map. The key enabling technology is a hybrid particle pusher [2], with particle positions updated in logical space and particle velocities updated in physical space. The self-adaptive, charge-conserving particle mover of Ref. [1] is extended to the non-uniform mesh case. The fully implicit implementation, using a Jacobian-free Newton-Krylov iterative solver, remains exactly charge- and energy-conserving. The extension of the formulation to multiple dimensions will be discussed. We present numerical experiments of 1D electrostatic, long-timescale ion-acoustic wave and ion-acoustic shock wave simulations, demonstrating that charge and energy are conserved to round-off for arbitrary mesh non-uniformity, and that the total momentum remains well conserved.\\[4pt] [1] Chen, Chac\'{o}n, Barnes, \emph{J. Comput. Phys.} \textbf{230} (2011). \\[0pt] [2] Camporeale and Delzanno, \textit{Bull. Am. Phys. Soc. }\textbf{56}(6) (2011); Wang, et al., \textit{J. Plasma Physics}, \textbf{61} (1999).