Positivity-preserving scheme for two-dimensional advection-diffusion equations including mixed derivatives
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.
Cite
@article{arxiv.1802.01015,
title = {Positivity-preserving scheme for two-dimensional advection-diffusion equations including mixed derivatives},
author = {Erasmus J. du Toit and Martin R. O'Brien and Roddy G. L. Vann},
journal= {arXiv preprint arXiv:1802.01015},
year = {2018}
}