90 research outputs found
An Asymptotic Preserving Scheme for the ES-BGK model
In this paper, we study a time discrete scheme for the initial value problem
of the ES-BGK kinetic equation. Numerically solving these equations are
challenging due to the nonlinear stiff collision (source) terms induced by
small mean free or relaxation time. We study an implicit-explicit (IMEX) time
discretization in which the convection is explicit while the relaxation term is
implicit to overcome the stiffness. We first show how the implicit relaxation
can be solved explicitly, and then prove asymptotically that this time
discretization drives the density distribution toward the local Maxwellian when
the mean free time goes to zero while the numerical time step is held fixed.
This naturally imposes an asymptotic-preserving scheme in the Euler limit. The
scheme so designed does not need any nonlinear iterative solver for the
implicit relaxation term. Moreover, it can capture the macroscopic fluid
dynamic (Euler) limit even if the small scale determined by the Knudsen number
is not numerically resolved. We also show that it is consistent to the
compressible Navier-Stokes equations if the viscosity and heat conductivity are
numerically resolved. Several numerical examples, in both one and two space
dimensions, are used to demonstrate the desired behavior of this scheme
Cauchy problem for the Boltzmann-BGK model near a global Maxwellian
In this paper, we are interested in the Cauchy problem for the Boltzmann-BGK
model for a general class of collision frequencies. We prove that the
Boltzmann-BGK model linearized around a global Maxwellian admits a unique
global smooth solution if the initial perturbation is sufficiently small in a
high order energy norm. We also establish an asymptotic decay estimate and
uniform -stability for nonlinear perturbations.Comment: 26 page
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