Friday,
October 15, 4:30 p.m. PSF 123
Shi Jin
Department of Mathematics, University of Wisconsin
Asymptotic-preserving Schemes for the Boltzmann Equation and Related Problems
(this talk is also a colloquium)
Abstract
We propose a general framework to design asymptotic
preserving schemes for the Boltzmann kinetic kinetic and related
equations. Numerically solving these equations is challenging
due to the nonlinear stiff collision (source)
terms induced by small mean free or relaxation time.
We propose to penalize the nonlinear collision term
by a BGK-type relaxation term, which can be solved explicitly even if
discretized implicitly in time. Moreover, the BGK-type relaxation operator
helps to drive the density distribution toward the local Maxwellian, thus
natually imposes an asymptotic-preserving scheme in the Euler limit.
The scheme so designed does not need any nonlinear iterative solver or
the use of Wild Sum. It
is uniformly stable in terms of the (possibly small)
Knudsen number, and can capture the macroscopic fluid dynamic (Euler)
limit
even if the small scale determined by the Knudsen number is not
numerically
resolved. It is also consistent to the compressible Navier-Stokes
equations
if the viscosity and heat conductivity are numerically resolved.
The method is applicable to many other related problems, such as
hyperbolic systems with stiff relaxation, and high order parabolic
equations.
