We consider the three-dimensional compressible Navier-Stokes-Poisson system where the electric field of the internal electrostatic potential force is governed by the self-consistent Poisson equation.If the Fourier modes of the initial data are degenerate at the low frequency or the initial data decay fast at spatial infinity,we show that the density converges to its equilibrium state at the L 2-rate (1+t)(-7/4) or L ∞-rate (1+t)(-5/2),and the momentum decays at the L 2-rate (1+t)(-5/4) or L ∞-rate (1+t)(-2).These convergence rates are shown to be optimal for the compressible Navier-Stokes-Poisson system.
We propose a deterministic solver for the time-dependent multi-subband Boltzmann transport equation(MSBTE)for the two dimensional(2D)electron gas in double gate metal oxide semiconductor field effect transistors(MOSFETs)with flared out source/drain contacts.A realistic model with six-valleys of the conduction band of silicon and both intra-valley and inter-valley phonon-electron scattering is solved.We propose a second order finite volume method based on the positive and flux conservative(PFC)method to discretize the Boltzmann transport equations(BTEs).The transport part of the BTEs is split into two problems.One is a 1D transport problem in the position space,and the other is a 2D transport problem in the wavevector space.In order to reduce the splitting error,the 2D transport problem in the wavevector space is solved directly by using the PFC method instead of splitting into two 1D problems.The solver is applied to a nanoscale double gate MOSFET and the current-voltage characteristic is investigated.Comparison of the numerical results with ballistic solutions show that the scattering influence is not ignorable even when the size of a nanoscale semiconductor device goes to the scale of the electron mean free path.
A globally hyperbolic moment system upto arbitrary order for the Wigner equation was derived in[6].For numerically solving the high order hyperbolic moment system therein,we in this paper develop a preliminary numerical method for this system following the NRxx method recently proposed in[8],to validate the moment system of the Wigner equation.The method developed can keep both mass and momentum conserved,and the variation of the total energy under control though it is not strictly conservative.We systematically study the numerical convergence of the solution to the moment system both in the size of spatial mesh and in the order of the moment expansion,and the convergence of the numerical solution of the moment system to the numerical solution of the Wigner equation using the discrete velocity method.The numerical results indicate that the high order moment system in[6]is a valid model for the Wigner equation,and the proposed numerical method for the moment system is quite promising to carry out the simulation of the Wigner equation.
