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A positivity‐preserving and energy stable scheme for a quantum diffusion equation
Author(s) -
Huo Xiaokai,
Liu Hailiang
Publication year - 2021
Publication title -
numerical methods for partial differential equations
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.901
H-Index - 61
eISSN - 1098-2426
pISSN - 0749-159X
DOI - 10.1002/num.22809
Subject(s) - mathematics , discretization , partial differential equation , scheme (mathematics) , energy functional , energy (signal processing) , finite difference method , mathematical analysis , boundary value problem , order (exchange) , boundary (topology) , diffusion equation , property (philosophy) , finite difference , philosophy , statistics , economy , finance , epistemology , economics , service (business)
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.