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A. Jüngel, J. Milisic:
"Physical and numerical viscosity for quantum hydrodynamics";
Commun. Mathematical Sciences, 5 (2007), 447 - 471.

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

Siehe das englische Abstract.

Quantum hydrodynamics, viscous quantum hydrodynamics, existence of stationary solutions, numerical dispersion, numerical viscosity, resonant tunneling diode, semiconductors