We found a match
Your institution may have rights to this item. Sign in to continue.
- Title
A positivity‐preserving and energy stable scheme for a quantum diffusion equation.
- Authors
Huo, Xiaokai; Liu, Hailiang
- Abstract
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.
- Subjects
HEAT equation; PARTIAL differential equations; MAXIMUM principles (Mathematics); FINITE differences
- Publication
Numerical Methods for Partial Differential Equations, 2021, Vol 37, Issue 6, p2973
- ISSN
0749-159X
- Publication type
Article
- DOI
10.1002/num.22809