Istituto di Matematica Applicata e Tecnologie Informatiche     
Pietra P., Pohl C. Weak limits of the quantum hydrodynamic model. Preprint ercim.cnr.ian//1997-1068, 1997.
A numerical study of the dispersive limit of the quantum hydrodynamic (QHD) equations for semiconductor is presented. Mathematically, the QHD system is a dispersive regularization of the hydrodynamic equations (HD). The regularization depends on the scaled Planck constant and, formally, vanishes in the classical limit. Thus, in the formal limit, the QHD model tends to the HD model. However, due to the non-linearity and the dispersive nature of the regularization, one cannot expect that the formal limit describes the correct limiting behavior in general. Due to the dispersive term, the solutions of the QHD system may develop fast oscillations which are not damped as the scaled Planck constant goes to zero and in that case the limiting system is not expected to be the HD system. Here we present numerical evidence of the fact that the solution of the QHD system develops dispersive oscillations when the corresponding HD system exhibits a shock wave and that the weak limit of the QHD solution is not a solution of the HD equations. The understanding of the dispersive nature of the problem and of its dispersive limit is an important issue in semiconductor applications, since the scaled Planck constant is often small. In particular, it gives important hints on the choice of reliable numerical schemes.
Subject Quantun hydrodynamic equations, semiconductors

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