A positivity-preserving and energy stable scheme for a quantum diffusion equation

Xiaokai Huo, And Hailiang Liu

Research output: Contribution to journalArticlepeer-review

2 Scopus citations


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 languageEnglish (US)
JournalNumerical Methods for Partial Differential Equations
StatePublished - Jul 17 2021

ASJC Scopus subject areas

  • Computational Mathematics
  • Analysis
  • Applied Mathematics
  • Numerical Analysis


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