In the rarefied gas dynamics, the classic kinetic models are more accurate and

complicated, while the fluid models are much simpler but fail in some cases. In this

thesis, we propose a new local up-scaling model to couple Euler equations with the

kinetic model when the previous up-scaling model in [19] does not apply, e.g. when

the Boltzmann equation is solved by the particle method, like DSMC. By means

of the first order Chapman-Enskog expansion we propose a new NSLU model to

couple the Navier-Stokes equations with the kinetic models. We also propose the

zero-moment projection based on the macro-micro decomposition ([34]) to correct

the non-fluid part in the up-scaling models.

  Numerical tests of these local up-scaling models have been done in various

multi-scale problems, including the Jin-Xin relaxation model for the traveling shock,

1D1D BGK model for the dynamics of a small perturbation of an equilibrium, 1D3D

BGK model for the stationary shock and the simulation of a planar Couette flow by

direct simulation of Monte Carlo (DSMC) for the Boltzmann equation.

  The implicit-explicit scheme for the relaxation models is applied, which is

shown to preserve the positiveness of the distribution function, the conservation

laws and entropy inequality. Numerical results show that the zero-projection is

necessary to ensure the stability and accuracy for the up-scaling models, especially

when non-kinetic schemes are applied in the moment equations. NSLU model must

be applied to replace the up-scaling model in [19] if the macroscopic approximation

is the viscous fluid.

  The similar scaling exists in the relaxation-time model for the semiconductor

device when electric field is low. The DrDiLU model based on drift-diffusion model

for the diode is proposed which is similar to NSLU model for the rarefied gas.

Numerical experiments show it is stable and accurate compared with the results

from the relaxation-time model.