  A conservative discontinuous Galerkin scheme for a nonlinear Dougherty
collision operator in full-f long-wavelength gyrokinetics is presented.
Analytically this model operator has the advective-diffusive form of
Fokker-Planck operators, it has a non-decreasing entropy functional, and
conserves particles, momentum and energy. Discretely these conservative
properties are maintained exactly as well, independent of numerical resolution.
In this work the phase space discretization is performed using a novel version
of the discontinuous Galerkin scheme, carefully constructed using concepts of
weak equality and recovery. Discrete time advancement is carried out with an
explicit time-stepping algorithm, whose stability limits we explore. The
formulation and implementation within the long-wavelength gyrokinetic solver of
Gkeyll are validated with relaxation tests, collisional Landau-damping
benchmarks and the study of 5D gyrokinetic turbulence on helical, open field
lines.
