Positivity-preserving scheme for two-dimensional advection-diffusion equations including mixed derivatives

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. Keywords: Advection-diusion, Fokker-Planck equation, low-order positivity-preserving scheme