In this talk we describe two approaches to preconditioning an
electromagnetic two-fluid plasma model. In the first, a physics-based
preconditioning (PBP) approach is used to extract a parabolic PDE that
targets stiff plasma waves and can be efficiently inverted using
multigrid methods. In the second approach we draw from the first-order
system least squares (FOSLS) methodology and manipulate the full PDE
system into being 
-elliptic.
Recently, fluid acceleration of a fully implicit particle-in-cell (PIC) simulation was successfully demonstrated. The fluid system features conservation equations for both ions and electrons coupled to field evolution equations. We concern ourselves with the electromagnetic two-fluid model in multiple dimensions. Electromagnetic fields are prescribed via the Darwin approximation in order to project out spurious lightwave time scales. Even with this removal, disparate time scales still exist among the supported plasma waves. The resulting nonlinear, stiff hyperbolic PDE system is solved using preconditioned Jacobian-free Newton-Krylov, for which the availability of a robust preconditioner is crucial.
Complimentary ideas from both PBP and FOSLS will be explored. Physics-based preconditioning promises to capture the correct fast wave physics, but care must be taken to generate a system compatible with fast iterative methods. In contrast, FOSLS generates a system suitable for multigrid iteration, but attention must be given to designing a system that picks up the correct physics. We present the benefits and difficulties that arise in both the PBP and FOSLS contexts.