Physical and numerical viscosity for quantum hydrodynamics. (English) Zbl 1130.76088

Jüngel, Ansgar; Milišić, Josipa Pina

Commun. Math. Sci. 5, No. 2, 447-471 (2007).

Summary: We study viscous stabilizations of quantum hydrodynamic equations. The quantum hydrodynamic model consists of conservation laws for particle density, momentum, and energy density, including quantum corrections from Bohm potential. Two different stabilizations are analyzed. First, viscous terms are derived using a Fokker-Planck collision operator in Wigner equation. We show the existence of solutions (with strictly positive particle density) to the isothermal, stationary, one-dimensional viscous model for general data and nonhomogeneous boundary conditions. The estimates depend on viscosity and do not allow to perform the inviscid limit. Second, we compute the numerical viscosity of the second upwind finite difference discretization of the inviscid quantum hydrodynamic model. Finally, numerical simulations using the non-isothermal stationary one-dimensional model of a resonant tunnelling diode show the influence of the viscosity on the solution.

MSC:

76Y05	Quantum hydrodynamics and relativistic hydrodynamics
76M20	Finite difference methods applied to problems in fluid mechanics
82D37	Statistical mechanics of semiconductors

Keywords:

conservation laws; Fokker-Planck collision operator; Wigner equation; upwind finite difference discretization