A finite volume Fokker-Planck collision operator in constants-of-motion coordinates

TEMPEST is a 5D gyrokinetic continuum code for edge plasmas. Constants of motion, namely, the total energy $E$ and the magnetic moment $\mu$, are chosen as coordinate s because of their advantage in minimizing numerical diffusion in advection operato rs. Most existing collision operators are written in other coordinates; using them by interpolating is shown to be less satisfactory in maintaining overall numerical accuracy and conservation. Here we develop a Fokker-Planck collision operator directly in $(E,\mu)$ space usin g a finite volume approach. The $(E, \mu)$ grid is Cartesian, and the turning point boundary represents a straight line cutting through the grid that separates the ph ysical and non-physical zones. The resulting cut-cells are treated by a cell-mergin g technique to ensure a complete particle conservation. A two dimensional fourth or der reconstruction scheme is devised to achieve good numerical accuracy with modest number of grid points. The new collision operator will be benchmarked by numerical examples.