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SUMMARY:A Numerical Scheme for the Quantum Boltzmann Equation Efficient in
  the Fluid Regime - Filbert\, F (Claude Bernard Lyon 1)
DTSTART:20101216T100000Z
DTEND:20101216T110000Z
UID:TALK28423@talks.cam.ac.uk
CONTACT:Mustapha Amrani
DESCRIPTION:Numerically solving the Boltzmann kinetic equations with the s
 mall Knudsen number is challenging due to the stiff nonlinear collision te
 rm. A class of asymptotic preserving schemes was introduced in [5] to hand
 le this kind of problems. The idea is to penalize the stiff collision term
  by a BGK type operator. This method\, however\, encounters its own diffic
 ulty when applied to the quantum Boltzmann equation. To define the quantum
  Maxwellian (Bose- Einstein or Fermi-Dirac distribution) at each time step
  and every mesh point\, one has to invert a nonlinear equation that connec
 ts the macroscopic quantity fugacity with density and internal energy. Set
 ting a good initial guess for the iterative method is troublesome in most 
 cases because of the complexity of the quantum functions (Bose-Einstein or
  Fermi- Dirac function). In this paper\, we propose to penalize the quantu
 m collision term by a 'classical' BGK operator instead of the quantum one.
  This is based on the observation that the classical Maxwellian\, with the
  temperature replaced by the internal energy\, has the same first five mom
 ents as the quantum Maxwellian. The scheme so designed avoids the aforemen
 tioned difficulty\, and one can show that the density distribution is stil
 l driven toward the quantum equilibrium. Numerical results are present to 
 illustrate the efficiency of the new scheme in both the hydrodynamic and k
 inetic regimes. We also develop a spectral method for the quantum collisio
 n operator.
LOCATION:Seminar Room 1\, Newton Institute
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