Communications in Mathematical Sciences

Physical and numerical viscosity for quantum hydrodynamics

Ansgar Jüngel and Jospa Pina Milišić

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Viscous stabilizations of the quantum hydrodynamic equations are studied. The quantum hydrodynamic model consists of the conservation laws for the particle density, momen- tum, 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 tunnelling diode show the influence of the viscosity on the solution.

Article information

Commun. Math. Sci., Volume 5, Issue 2 (2007), 447-471.

First available in Project Euclid: 9 July 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 76Y05: Quantum hydrodynamics and relativistic hydrodynamics [See also 82D50, 83C55, 85A30] 35J40: Boundary value problems for higher-order elliptic equations 65N05

quantum hydrodynamics viscous quantum hydrodynamics existence of stationary solutions numerical dispersion numerical viscosity resonant tunnelling diode semiconductors


Jüngel, Ansgar; Milišić, Jospa Pina. Physical and numerical viscosity for quantum hydrodynamics. Commun. Math. Sci. 5 (2007), no. 2, 447--471.

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