Nonlinear full-$f$ simplified Fokker-Planck multi-species collisions in (gyro)kinetics

We present a simplified, nonlinear collision model of like-particle and multi-species collisions for full-$f$ kinetic plasma studies. This Fokker-Planck-like model is a generalization of the popular Dougherty operator, formulated to more rigorously support arbitrary mass ratios, non-quasineutral pairs, and flexible collision frequencies. In particular, this approach can be extended to support velocity-dependent collisionalities. The proposed operator preserves the conservative properties of the Fokker-Planck operator and, in the case of velocity-independent collisionality, can be shown to obey the H-theorem. Non-decreasing entropy can be proved as long as the cross-species temperature remains positive, even in the non-equilibrium case of unequal temperatures. Benchmarks, like Landau damping of Langmuir waves, show the effect of this simplified model to be comparable to that of the full Fokker-Planck operator. These features make it an attractive approach for direct numerical simulation of plasmas where using the full Fokker-Planck operator may be prohibitively expensive. We will present the formulation, discontinuous Galerkin implementation within Gkeyll, and various tests carried out for testing and validation.