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Numerically solving the Boltzmann kinetic equations with the small Knudsennumber is challenging due to the stiff nonlinear collision term. A class ofasymptotic preserving schemes was introduced in [6] to handle this kind ofproblems. The idea is to penalize the stiff collision term by a BGK typeoperator. This method, however, encounters its own difficulty when applied tothe quantum Boltzmann equation. To define the quantum Maxwellian (Bose-Einsteinor Fermi- Dirac distribution) at each time step and every mesh point, one hasto invert a nonlinear equation that connects the macroscopic quantity fugacitywith density and internal energy. Setting a good initial guess for theiterative method is troublesome in most cases because of the complexity of thequantum functions (Bose-Einstein or Fermi-Dirac function). In this paper, wepropose to penalize the quantum collision term by a 'classical' BGK operatorinstead of the quantum one. This is based on the observation that the classicalMaxwellian, with the temperature replaced by the internal energy, has the samefirst five moments as the quantum Maxwellian. The scheme so designed avoids theaforementioned difficulty, and one can show that the density distribution isstill driven toward the quantum equilibrium. Numerical results are present toillustrate the efficiency of the new scheme in both the hydrodynamic andkinetic regimes. We also develop a spectral method for the quantum collisionoperator.

A Numerical Scheme for the Quantum Boltzmann Equation Efficient in the Fluid Regime