Numerical Analysis and Simulation for the Boltzmann's equation
October 22, 2014
3:00pm PGH 646
Abstract
The development of accurate and fast algorithms for the Boltzmann collision
integral and their analysis represent a challenging problem in scientific
computing and numerical analysis. Recently, several works were devoted to
the derivation of spectrally accurate schemes for the Boltzmann equation,
but very few of them were concerned with the stability analysis of the
method. In particular, there was no result of stability except when the
method is modified in order to enforce the positivity preservation, which
destroys the accuracy. We propose a new method to study the stability of
homogeneous Boltzmann equations perturbed by smoothed balanced operators
which do not preserve positivity of the distribution. This method takes
advantage of the "spreading" property of the collision, together with
estimates on regularity and entropy production. As an application we prove
stability and convergence of spectral methods for the Boltzmann equation,
when the discretization parameter is large enough (with explicit bound).
Finally, several numerical simulations will be presented with applications
in rarefied gas dynamics, and computational fluid mechanics.
Note: the talk will be accessible to a general
math audience.
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Last modified: April 11 2016 - 18:14:43