Transcript Document

Analytical and numerical issues
for non-conservative
non-linear Boltzmann transport equation
Irene M. Gamba
Department of Mathematics and ICES
The University of Texas at Austin
In collaboration with:
Alexandre Bobylev , Karlstad University, Sweden, and
Carlo Cercignani, Politecnico di Milano, Italy, on selfsimilar
asymptotics
and decay rates to generalized models for multiplicative stochastic
interactions.
Sri Harsha Tharkabhushanam , ICES- UT Austin, on Deterministic-Spectral
solvers for non-conservative, non-linear Boltzmann transport equation
• Rarefied ideal gases-elastic: conservative Boltzmann Transport eq.
• Energy dissipative phenomena: Gas of elastic or inelastic interacting systems in the
presence of a thermostat with a fixed background temperature өb or Rapid granular flow
dynamics: (inelastic hard sphere interactions): homogeneous cooling states,
randomly heated states, shear flows, shockwaves past wedges, etc.
•(Soft) condensed matter at nano scale: Bose-Einstein condensates models and
charge transport in solids: current/voltage transport modeling semiconductor.
•Emerging applications from stochastic dynamics for multi-linear Maxwell type
interactions : Multiplicatively Interactive Stochastic Processes:
Pareto tails for wealth distribution, non-conservative dynamics: opinion dynamic
models, particle swarms in population dynamics, etc (Fujihara, Ohtsuki, Yamamoto’
06,Toscani, Pareschi, Caceres 05-06…).
Goals:
• Understanding of analytical properties: large energy tails
•long time asymptotics and characterization of asymptotics states
•A unified approach for Maxwell type interactions.
•Development of deterministic schemes: spectral-Lagrangian methods
A general form for Boltzmann equation for binary interactions
with external ‘heating’ sources
For a Maxwell type model: a linear equation for the kinetic energy
Time irreversibility is expressed in this inequality
stability
In addition:
The Boltzmann Theorem:
there are only N+2 collision invariants
(
)
asymptotics
An important application:
The homogeneous BTE in Fourier space
Boltzmann Spectrum
A benchmark case:
Deterministic numerical method: Spectral Lagrangian solvers
Numerical simulations
Comparisons of energy conservation vs dissipation
For a same initial state, we test the energy
Conservative scheme and the scheme for the
energy dissipative Maxwell-Boltzmann Eq.
Numerical simulations
Moments calculations:
Thank you
very much
for your attention !!