7. The Fokker-Planck collision operator in Gkeyll

PDF of note.

Also see the note: Dimensionally independent indexing

There are several texts in which the Fokker-Planck operator (FPO) is described and studied in detail. See, for example, the original paper of Rosenbluth, MacDonald and Judd from 1957 (RMJ) or Chapter 3 of “Collisional Transport in Magnetized Plasmas” by Helander and Dieter. For the list of equation see this page from the NRL Plasma Formulary.

Historically, the first derivation of the FPO in the case of Coulomb potential (inverse square law) was by Lev Landau in 1936. However, the 1957 paper by RMJ does not mention Landau’s work at all. As is usually the case, the original papers by these Masters of the field remain highly readable and still provide the best derivations.

My goal in this note is to list the FPO equations as implemented in Gkeyll, with especial emphasis on properties needed to discretize them using finite-volume and discontinuous Galerkin schemes. Note that we use the Rosenbluth form of the equations and not the Landau form. The FPO is a very complicated equation: it is a nonlinear integro-differential equation in 3D velocity space and has a rich structure, specially when combined with the particle motion in self-consistent electromagnetic fields. Designing a general production solver for the case of multi-species plasmas poses a formidable challenge. (This is an understatement). In fact, I am not aware of any widely usable production implementation of the operator for general, multi-species problems. (Though there are some excellent special-purpose solvers that are used in fusion physics).