analytical and numerical issues for non-conservative non-linear boltzmann transport equation
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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 - PowerPoint PPT PresentationTRANSCRIPT
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Analytical and numerical issues Analytical and numerical issues for non-conservativefor non-conservative
non-linear Boltzmann transport equationnon-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, SwedenKarlstad University, Sweden, and
Carlo Cercignani, Politecnico di Milano, ItalyPolitecnico di Milano, Italy, on selfsimilar asymptotics and decay rates to generalized models for multiplicative stochastic interactions.
Sri Harsha Tharkabhushanam , ICES- UT Austin, 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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Goals: • Understanding of analytical properties: large energy tails Understanding of analytical properties: large energy tails •long time asymptotics and characterization of asymptotics stateslong time asymptotics and characterization of asymptotics states
•A unified approach for Maxwell type interactions.A unified approach for Maxwell type interactions.
•Development of deterministic schemes: spectral-Lagrangian methodsDevelopment of deterministic schemes: spectral-Lagrangian methods
• Rarefied ideal gases-elastic: conservativeconservative Boltzmann Transport eq.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…).
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A general form for Boltzmann equation for binary interactionsA general form for Boltzmann equation for binary interactionswith external ‘heating’ sourceswith external ‘heating’ sources
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For a Maxwell type model: a linear equation for the kinetic energy
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The Boltzmann Theorem:The Boltzmann Theorem: there are only N+2 collision invariants
Time irreversibility is expressed in this inequality stability
In addition:
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( )
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asymptotics
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An important application:An important application:
The homogeneous BTE in Fourier space
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Boltzmann Spectrum
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A benchmark case: A benchmark case:
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Deterministic numerical method: Spectral Lagrangian solversDeterministic numerical method: Spectral Lagrangian solvers
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Numerical simulationsNumerical simulations
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Comparisons of energy conservation vs dissipationComparisons 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 simulationsNumerical simulations
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Moments calculations:Moments calculations:
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