Verification of a fully implicit particle-in-cell method for the $v∥$-formalism of electromagnetic gyrokinetics in the XGC code

A fully implicit particle-in-cell method for handling the $v∥$-formalism of electromagnetic gyrokinetics has been implemented in XGC. By choosing the $v∥$-formalism, we avoid introducing the nonphysical skin terms in Ampère's law, which are responsible for the well-known “cancellation problem” in the $p∥$-formalism. The $v∥$-formalism, however, is known to suffer from a numerical instability when explicit time integration schemes are used due to the appearance of a time derivative in the particle equations of motion from the inductive component of the electric field. Here, using the conventional δf scheme, we demonstrate that our implicitly discretized algorithm can provide numerically stable simulation results with accurate dispersive properties. We verify the algorithm using a test case for shear Alfvén wave propagation in addition to a case demonstrating the ion temperature gradient-kinetic ballooning mode (ITG-KBM) transition. The ITG-KBM transition case is compared to results obtained from other δf gyrokinetic codes/schemes, whose verification has already been archived in the literature.