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A positivity-preserving and energy stable scheme for a quantum diffusion equation
Numerical Methods for Partial Differential Equations ( IF 3.9 ) Pub Date : 2021-07-17 , DOI: 10.1002/num.22809
Xiaokai Huo 1, 2 , And Hailiang Liu 3
Affiliation  

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.

中文翻译:

量子扩散方程的一种保正能量稳定格式

我们为一维和二维的量子扩散方程提出了一种新的完全离散化的有限差分格式。这是第一个完全离散化的方案,仅使用标准的有限差分离散化就证明了正性保持和能量稳定的特性。证明保正性的难点在于缺乏四阶偏微分方程的极大值原理。为了克服这个困难,我们将该方案重新表述为基于变分结构的优化问题,并使用边界值附近能量泛函的奇异性来排除非正解的可能性。该方案还被证明是大规模保守和一致的。
更新日期:2021-09-20
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