Implementation of higher-order velocity mapping between marker particles and grid in the particle-in-cell code XGC

To exchange particle distribution function between coupled codes in the ECP-WDM project and to evaluate the dissipative operations, such as plasma collisions, the total-$f$ gyrokinetic particle-in-cell code {\sc xgc}~[S.~Ku et al. {\em Phys. Plasmas} {\bf 25} 056107 (2018)] implements a velocity grid and a bilinear mapping between marker particles (continuous in velocity space) and velocity grid [Yoon, Chang {\em Phys. Plasmas} {\bf 21} 032503 (2014)]. The bilinear operation ensures conservation of particle density and momentum, but fails to conserve particle energy with enough accuracy. In the present work we have updated {\sc xgc}~to instead use a novel mapping technique, recently included in the {{\sc pets}c}~library [S.~Balay et al. {{{\sc pets}c}~Users Manual} {ANL-95/11~-~Revision~3.13} (2020)], which employs a pseudo-inverse to preserve moments up to the order of the discretization space [Hirvijoki et al. arXiv:1802.05263]. We demonstrate the functionality and that $2^{\mathrm{nd}}$-order elements, in addition to particle and momentum conservation, also conserve energy.