Commun. Comput. Phys., 12 (2012), pp. 1541-1561.


A Numerical Scheme for the Quantum Fokker-Planck-Landau Equation Efficient in the Fluid Regime

Jingwei Hu 1, Shi Jin 2*, Bokai Yan 3

1 Institute for Computational Engineering and Sciences (ICES), The University of Texas at Austin, 1 University Station C0200, Austin, TX 78712, USA.
2 Department of Mathematics, University of Wisconsin-Madison, 480 Lincoln Drive, Madison, WI 53706, USA; and Department of Mathematics and Institute of Natural Sciences, Shanghai Jiao Tong University, Shanghai 200240, China.
3 Department of Mathematics, University of Wisconsin-Madison, 480 Lincoln Drive, Madison, WI 53706, USA.

Received 22 April 2011; Accepted (in revised version) 9 January 2012
Available online 4 June 2012
doi:10.4208/cicp.220411.090112a

Abstract

We construct an efficient numerical scheme for the quantum Fokker-Planck-Landau (FPL) equation that works uniformly from kinetic to fluid regimes. Such a scheme 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 $Q_{qFPL}$ in order to remove the stiffness induced by the small Knudsen number. However, there is no suitable simple quantum operator serving the purpose and for this kind of operators one has to solve the complicated quantum Maxwellians (Bose-Einstein or Fermi-Dirac distribution). In this paper, we propose to penalize $Q_{qFPL}$ by the "classical" linear Fokker-Planck operator. It 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. Numerical results for Bose and Fermi gases are presented to illustrate the efficiency of the scheme in both fluid and kinetic regimes.

AMS subject classifications: 35Q84, 65L04, 76Y05

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Key words: Quantum Fokker-Planck-Landau equation, fluid limit, asymptotic-preserving scheme.

*Corresponding author.
Email: hu@ices.utexas.edu (J. Hu), jin@math.wisc.edu (S. Jin), yan@math.wisc.edu (B. Yan)
 

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