A Numerical Scheme for the Quantum Fokker-Planck-Landau Equation Efficient in the Fluid Regime (Communications in Computational Physics, 2012)
We construct an efficient numerical scheme for the quantum Fokker-Planck-Landau (FPL) equation that works uniformly from kinetic to fluid regimes. Such ascheme inevitably needs an implicit discretization of the nonlinear collision operator,which is difficult to invert. Inspired by work [9] we seek a linear operator to penalize the quantum FPL collision term QqFPL in order to remove the stiffness induced bythe small Knudsen number. However, there is no suitable simple quantum operatorserving the purpose and for this kind of operators one has to solve the complicatedquantum Maxwellians (Bose-Einstein or Fermi-Dirac distribution). In this paper, wepropose to penalize QqFPL by the "classical" linear Fokker-Planck operator. It is basedon the observation that the classical Maxwellian, with the temperature replaced by theinternal energy, has the same first five moments as the quantum Maxwellian. Numerical results for Bose and Fermi gases are presented to illustrate the efficiency of thescheme in both fluid and kinetic regimes.