Analytical and numerical issues

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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 MAMOS workshop UT Austin October
07
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• 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
  ?b 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

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A general form for Boltzmann equation for binary
interactions with external heating sources
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For a Maxwell type model a linear equation for
the kinetic energy
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Time irreversibility is expressed in this
inequality
stability
In addition
The Boltzmann Theorem there are only N2
collision invariants
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asymptotics
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An important application
The homogeneous BTE in Fourier space
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Boltzmann Spectrum
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A benchmark case
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Deterministic numerical method Spectral
Lagrangian solvers
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Numerical simulations
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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.
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Numerical simulations
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Moments calculations
Thank you very much for your attention !!