  Numerically solving the Boltzmann kinetic equations with the small Knudsen
number is challenging due to the stiff nonlinear collision term. A class of
asymptotic preserving schemes was introduced in [6] to handle this kind of
problems. The idea is to penalize the stiff collision term by a BGK type
operator. This method, however, encounters its own difficulty 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 connects the macroscopic quantity fugacity
with density and internal energy. Setting 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 quantum 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 moments as the quantum Maxwellian. The scheme so designed avoids the
aforementioned difficulty, and one can show that the density distribution is
still driven toward the quantum equilibrium. Numerical results are present to
illustrate the efficiency of the new scheme in both the hydrodynamic and
kinetic regimes. We also develop a spectral method for the quantum collision
operator.
