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|
Bibliographical noteKAUST Repository Item: Exported on 2021-08-06
Acknowledgements: The first author is funded by King Abdullah University of Science and Technology. The second author is funded by the National Science Foundation under Grant DMS1812666. The authors are grateful to Athanasios E. Tzavaras for valuable suggestions and comments. The authors acknowledge TU Wien Bibliothek for financial support through its Open Access Funding Program.
ASJC Scopus subject areas
- Computational Mathematics
- Applied Mathematics
- Numerical Analysis