We propose a new fully-discretized finite difference scheme for a quantum diffusion equation, in both one and two dimensions. This is the first fully-discretized scheme with proven positivity-preserving and energy stable properties using only standard finite difference discretization. The difficulty in proving the positivity-preserving property lies in the lack of a maximum principle for fourth order partial differential equations. To overcome this difficulty, we reformulate the scheme as an optimization problem based on a variational structure and use the singular nature of the energy functional near the boundary values to exclude the possibility of non-positive solutions. The scheme is also shown to be mass conservative and consistent.
|Original language||English (US)|
|Journal||Numerical Methods for Partial Differential Equations|
|State||Published - Jul 17 2021|
ASJC Scopus subject areas
- Computational Mathematics
- Applied Mathematics
- Numerical Analysis