Entropic Discretization of a Quantum Drift-Diffusion Model

This paper is devoted to the discretization and numerical simulation of a new quantum drift-diffusion model that was recently derived. In a first step, we introduce an implicit semi-discretization in time which possesses some interesting properties: this system is well-posed, it preserves the positivity of the density, the total charge is conserved, and it is entropic (a free energy is dissipated). Then, after a discretization of the space variable, we define a numerical scheme which has the same properties and is equivalent to a convex minimization problem. Moreover, we show that this discrete solution converges for long times to the solution of a discrete Schrödinger-Poisson system. These results are illustrated by some numerical simulations.