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

TL;DR

This paper introduces a positivity-preserving numerical scheme for two-dimensional advection-diffusion equations with mixed derivatives, enhancing accuracy and ensuring physical properties like positivity and conservation in complex simulations.

Contribution

The work presents a novel positivity-preserving scheme specifically designed for 2D advection-diffusion equations with mixed derivatives, addressing a gap in existing methods.

Findings

The scheme accurately solves a 2D diffusion equation with known analytical solution.

It conserves particle number and energy in the Fokker-Planck collision operator application.

Positivity preservation and correct steady-state (Maxwellian) distribution are demonstrated.

Abstract

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.