A Fully Implicit Particle-in-Cell Method for Gyrokinetic Electromagnetic Modes in XGC

Electromagnetic gyrokinetic particle-in-cell methods are known to suffer from numerical difficulties, limiting their applicability to low-$\beta$ or short wavelength regimes. The $v_{\parallel}$ formulation with explicit time discretization suffers from a severe time step constraint, and the $p_{\parallel}$ formulation suffers from an inexact cancellation of two large, non-physical terms appearing in Ampère’s law that emerge from the choice of coordinates. Here, we describe our implementation of a fully-implicit time integration scheme based on the work of Chen and Chacón [1-2] for a gyrokinetic ion, drift kinetic electron electromagnetic model employing the $v_{\parallel}$ formulation in the full volume fusion plasma code XGC1. By using an implicit discretization scheme, we overcome the previous time stepping difficulties associated with the $v_{\parallel}$ formulation and avoid introducing non-physical terms in Ampère’s law. The resulting system of nonlinear equations is solved iteratively using a preconditioner derived from an electron fluid model. We consider kinetic ballooning and micro-tearing modes to verify the scheme. [1] G. Chen, L. Chacón, and D.C. Barnes, J. Comput. Phys. 230 (2011) 7018. [2] G. Chen, L. Chacón, Comput. Phy. Comm. 197, (2015) 73-87.