Modélisation Mathématique et Simulation Numérique de Systèmes Fluides Quantiques. (Mathematical modeling and Numerical Simulation of Quantum Fluid Systems)

This chapter 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. These results are illustrated by some numerical simulations.