Physical and numerical viscosity for quantum hydrodynamics

Abstract

Abstract. 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 tunnelling diode show the influence of the viscosity on the solution.Key words. quantum hydrodynamics, viscous quantum hydrodynamics, existence of stationary solutions, numerical dispersion, numerical viscosity, resonant tunnelling diode, semiconductors AMS subject classifications. 76Y05, 35J40, 65N05 IntroductionThe fluiddynamical formulation of the Schrödinger equation, which has been known since the early years of quantum mechanics [32], has been used in many applications for quantum plasmas [34] or quantum semiconductor devices [12,14]. Quantum effects appear through terms involving the Bohm potential and containing third-order derivatives which express the dispersive structure of the equations. These terms are mathematically difficult to handle, and in fact there are only partial analytical and numerical results. This work is devoted to an analytical and numerical study of these equations when they contain additional viscosity terms originating either from particle collisions or from an upwind numerical discretization of the equations.More specifically, we consider the scaled stationary quantum hydrodynamic (QHD) equations for the particle density n, the current density J, and the energy density ne,

Diffusive semiconductor moment equations using Fermi–Dirac statistics

“…Boundary conditions satisfying the Shapiro-Lopatinskii criterion have been examined in [12]. Furthermore, in [32,35] Dirichlet-type conditions have been employed in the analyzed, but only for the (simpler) one-dimensional equations.…”

“…For instance, in the Brenner-Navier-Stokes model with constant viscosity, Feireisl and Vasseur [17] proved that the density is positive except on a set of Lebesgue measure zero. Furthermore, in [35] it is shown that the solution of the one-dimensional stationary viscous quantum Euler model admits strictly positive particle densities.…”

“…For the viscous quantum Euler system, the existence of one-dimensional solutions to the stationary problem [35] and the time-dependent problem [11,19] has been achieved. Concerning the multidimensional transient system, there exist only localin-time existence theorems [11,15].…”

“…The Brenner-Navier-Stokes system has been analyzed in [17]. The variable nw = nu + ν∇n was also employed in [35] to prove the existence of solutions to the one-dimensional stationary viscous quantum Euler problem with physical boundary conditions. In fact, in this case, nw is constant and it can be shown that the density n is strictly positive.…”

“…Indeed, it has been derived from a Wigner-Fokker-Planck equation by a moment method [21,35]. The viscous terms ν∆n and ν∆(nu) arise from the moments of the Fokker-Planck collision operator.…”