Physics-based preconditioners for two-fluid electrostatic and electromagnetic models with charge separation

Recently, fluid acceleration of a fully implicit kinetic particle-in-cell (PIC) simulation has been successfully demonstrated.\footnote{Taitano et al., SISC, 2013}$^,$\footnote{Chen et al., JCP, submitted.} Central to these algorithms is robust preconditioning of the fluid system. In the context of kinetic simulations, the fluid system features conservation equations for both ions and electrons, plus field evolution equations, and must allow for charge separation effects. In this work, we concern ourselves with electrostatic and electromagnetic two-fluid models in multiple dimensions. Electromagnetic fields are prescribed via the Darwin approximation to project out spurious light-wave time scales.\footnote{Nielson and Lewis, Meth. Comput. Phys., 1976.} Disparate time scales remain among the abundance of supported plasma waves. The resulting nonlinear, stiff hyperbolic PDE systems are effectively preconditioned using physics-based preconditioning ideas,\footnote{Knoll and Keyes., JCP 2004.} whereby their linearized form is transformed into parabolic PDEs that target the fast wave behavior. These elliptic systems can be efficiently inverted by multigrid methods.\footnote{Brandt., Math. Comp., 1977} We will demonstrate the effectiveness of the approach via numerical experiments.