A positivity-preserving energy stable scheme for a quantum diffusion equation
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We propose a new fully-discretized finite difference scheme for a quantum diffusion equation, both in 1D and 2D dimensions. The scheme is a first proven positivity-preserving energy stable fully-discretized scheme using standard finite difference discretizations. The difficulty in proving the positivity-preserving property is because the equation is of fourth order in space and maximum principle fails to hold. To overcome this difficulty, we reformulate the scheme as an optimization problem using the variational structure and use the singularity of the energy functional at zero to prove positivenesss of the numerical scheme. The proposed scheme is also shown to be mass conservative and consistent.
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