  In this work, we propose a positivity-preserving scheme for solving
two-dimensional advection-diffusion equations including mixed derivative terms,
in order to improve the accuracy of lower-order methods. The solution to these
equations, in the absence of mixed derivatives, has been studied in detail,
while positivity-preserving solutions to mixed derivative terms have received
much less attention. A two-dimensional diffusion equation, for which the
analytical solution is known, is solved numerically to show the applicability
of the scheme. It is further applied to the Fokker-Planck collision operator in
two-dimensional cylindrical coordinates under the assumption of local thermal
equilibrium. For a thermal equilibration problem, it is shown that the scheme
conserves particle number and energy, while the preservation of positivity is
ensured and the steady-state solution is the Maxwellian distribution.
