Skip to main content Accessibility help

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

  • Jingwei Hu (a1), Shi Jin (a2) (a3) and Bokai Yan (a2)


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 QqFPL 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 QqFPL 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.


Corresponding author

Corresponding author.Email


Hide All
[1]Bagland, V., Well-posedness for the spatially homogeneous Landau-Fermi-Dirac equation for hard potentials, Proc. R. Soc. Edinburgh Ser. A, 134 (2004), 415447.
[2]Bagland, V. and Lemou, M., Equilibrium states for the Landau-Fermi-Dirac equation, Banach Center Publ., 66 (2004), 2937.
[3]Carrillo, J. A., Laurencot, P. and Rosado, J., Fermi-Dirac-Fokker-Planck equation: well posed- ness and long-time asymptotics, J. Differ. Equ., 247 (2009), 22092234.
[4]Carrillo, J. A., Rosado, J. and Salvarani, F., 1D nonlinear Fokker-Planck equations for fermions and bosons, Appl. Math. Lett., 21 (2008), 148154.
[5]Chen, Y., Analytic regularity for solutions of the spatially homogeneous Landau-Fermi-Dirac equation for hard potentials, Kinet. Relat. Models, 3 (2010), 645667.
[6]Coron, F. and Perthame, B., Numerical passage from kinetic to fluid equations, SIAM J. Numer. Anal., 28 (1991), 2642.
[7]Danielewicz, P., Nonrelativistic and relativistic Landau/Fokker-Planck equation for arbitrary statistics, Phys. A, 100 (1980), 167182.
[8]Filbet, F., Hu, J. W. and Jin, S., A numerical scheme for the quantum Boltzmann equation with stiff collision terms, ESAIM: M2AN, 46 (2012), 443463.
[9]Filbet, F. and Jin, S., A class of asymptotic-preserving schemes for kinetic equations and related problems with stiff sources, J. Comput. Phys., 229 (2010), 76257648.
[10]Filbet, F. and Pareschi, L., A numerical method for the accurate solution of the Fokker-Planck-Landau equation in the nonhomogeneous case, J. Comput. Phys., 179 (2002), 126.
[11]Gosse, L. and Toscani, G., Space localization and well-balanced schemes for discrete kinetic models in diffusive regimes, SIAM J. Numer. Anal., 41 (2004), 641658.
[12]Greenberg, J. M., LeRoux, A. Y., Baraille, R. and Noussair, A., Analysis and approximation of conservation laws with source terms, SIAM J. Numer. Anal., 34 (1997), 19802007.
[13]Hu, J. W. and Jin, S., On kinetic flux vector splitting schemes for quantum Euler equations, Kinet. Relat. Models, 4 (2011), 517530.
[14]Jin, S., Runge-Kutta methods for hyperbolic conservation laws with stiff relaxation terms, J. Comput. Phys., 122 (1995), 5167.
[15]Jin, S., Efficient asymptotic-preserving (AP) schemes for some multiscale kinetic equations, SIAM J. Sci. Comput., 21 (1999), 441454.
[16]Jin, S., Asymptotic preserving (AP) schemes for multiscale kinetic and hyperbolic equations: a review (Lecture Notes for Summer School on “Methods and Models of Kinetic Theory”, Porto Ercole, June 2010). Rivista di Matematica della Università di Parma, to appear.
[17]Jin, S. and Yan, B., A class of asymptotic-preserving schemes for the Fokker-Planck-Landau equation, J. Comput. Phys., 230 (2011), 64206437.
[18]Kaniadakis, G., Generalized Boltzmann equation describing the dynamics of bosons and fermions, Phys. Lett. A, 203 (1995), 229234.
[19]Kaniadakis, G. and Quarati, P., Kinetic equation for classical particles obeying an exclusion principle, Phys. Rev. E, 48 (1993), 42634270.
[20]Kaniadakis, G. and Quarati, P., Classical model of bosons and fermions, Phys. Rev. E, 49 (1994), 51035110.
[21]Landau, L. D., Die kinetische Gleichung fur den Fall Coulombscher Wechselwirkung, Phys. Z. Sowjetunion, 10 (1936), 154164.
[22]Landau, L. D., The kinetic equation in the case of Coulomb interaction. Zh. Eksper. I Teoret. Fiz., 7 (1937), 203209.
[23]Lemou, M., Linearized quantum and relativistic Fokker-Planck-Landau equations, Math. Meth. Appl. Sci., 23 (2000), 10931119.
[24]LeVeque, R. J., Numerical Methods for Conservation Laws, Birkhauser Verlag, Basel, second edition, 1992.
[25]Pareschi, L. and Russo, G., Numerical solution of the Boltzmann equation I: spectrally accurate approximation of the collision operator, SIAM J. Numer. Anal., 37 (2000), 12171245.
[26]Pareschi, L., Russo, G. and Toscani, G., Fast spectral methods for the Fokker-Planck-Landau collision operator, J. Comput. Phys., 165 (2000), 216236.
[27]Press, W. H., Teukolsky, S. A., Vetterling, W. T. and Flannery, B. P., Numerical Recipes: The Art of Scientific Computing, Cambridge University Press, Cambridge, third edition, 2007.
[28]Toscani, G., Finite time blow up in Kaniadakis-Quarati model of Bose-Einstein particles, Commun. Part. Diff. Eqns., 37 (2012), 7787.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Communications in Computational Physics
  • ISSN: 1815-2406
  • EISSN: 1991-7120
  • URL: /core/journals/communications-in-computational-physics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *



Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed