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

Published online by Cambridge University Press:  20 August 2015

Jingwei Hu ,
Shi Jin  and
Bokai Yan
Jingwei Hu*
Affiliation:
Institute for Computational Engineering and Sciences (ICES), The University of Texas at Austin, 1 University Station C0200, Austin, TX 78712, USA
Shi Jin*
Affiliation:
Department of Mathematics, University of Wisconsin-Madison, 480 Lincoln Drive, Madison, WI 53706, USA Department of Mathematics and Institute of Natural Sciences, Shanghai Jiao Tong University, Shanghai 200240, China
Bokai Yan*
Affiliation:
Department of Mathematics, University of Wisconsin-Madison, 480 Lincoln Drive, Madison, WI 53706, USA
*
Email address:[email protected]
Corresponding author.Email address:[email protected]
Email address:[email protected]
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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 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.

Keywords

35Q8465L0476Y05Quantum Fokker-Planck-Landau equationfluid limitasymptotic-preserving scheme
Type
Research Article
Information
Communications in Computational Physics , Volume 12 , Issue 5 , November 2012 , pp. 1541 - 1561
Copyright
Copyright © Global Science Press Limited 2012

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References

[1]Bagland, V., Well-posedness for the spatially homogeneous Landau-Fermi-Dirac equation for hard potentials, Proc. R. Soc. Edinburgh Ser. A, 134 (2004), 415–447.Google Scholar
[2]Bagland, V. and Lemou, M., Equilibrium states for the Landau-Fermi-Dirac equation, Banach Center Publ., 66 (2004), 29–37.Google Scholar
[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), 2209–2234.Google Scholar
[4]Carrillo, J. A., Rosado, J. and Salvarani, F., 1D nonlinear Fokker-Planck equations for fermions and bosons, Appl. Math. Lett., 21 (2008), 148–154.Google Scholar
[5]Chen, Y., Analytic regularity for solutions of the spatially homogeneous Landau-Fermi-Dirac equation for hard potentials, Kinet. Relat. Models, 3 (2010), 645–667.Google Scholar
[6]Coron, F. and Perthame, B., Numerical passage from kinetic to fluid equations, SIAM J. Numer. Anal., 28 (1991), 26–42.CrossRefGoogle Scholar
[7]Danielewicz, P., Nonrelativistic and relativistic Landau/Fokker-Planck equation for arbitrary statistics, Phys. A, 100 (1980), 167–182.Google Scholar
[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), 443–463.Google Scholar
[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), 7625–7648.Google Scholar
[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), 1–26.CrossRefGoogle Scholar
[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), 641–658.Google Scholar
[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), 1980–2007.CrossRefGoogle Scholar
[13]Hu, J. W. and Jin, S., On kinetic flux vector splitting schemes for quantum Euler equations, Kinet. Relat. Models, 4 (2011), 517–530.CrossRefGoogle Scholar
[14]Jin, S., Runge-Kutta methods for hyperbolic conservation laws with stiff relaxation terms, J. Comput. Phys., 122 (1995), 51–67.Google Scholar
[15]Jin, S., Efficient asymptotic-preserving (AP) schemes for some multiscale kinetic equations, SIAM J. Sci. Comput., 21 (1999), 441–454.Google Scholar
[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.Google Scholar
[17]Jin, S. and Yan, B., A class of asymptotic-preserving schemes for the Fokker-Planck-Landau equation, J. Comput. Phys., 230 (2011), 6420–6437.Google Scholar
[18]Kaniadakis, G., Generalized Boltzmann equation describing the dynamics of bosons and fermions, Phys. Lett. A, 203 (1995), 229–234.Google Scholar
[19]Kaniadakis, G. and Quarati, P., Kinetic equation for classical particles obeying an exclusion principle, Phys. Rev. E, 48 (1993), 4263–4270.Google Scholar
[20]Kaniadakis, G. and Quarati, P., Classical model of bosons and fermions, Phys. Rev. E, 49 (1994), 5103–5110.CrossRefGoogle ScholarPubMed
[21]Landau, L. D., Die kinetische Gleichung fur den Fall Coulombscher Wechselwirkung, Phys. Z. Sowjetunion, 10 (1936), 154–164.Google Scholar
[22]Landau, L. D., The kinetic equation in the case of Coulomb interaction. Zh. Eksper. I Teoret. Fiz., 7 (1937), 203–209.Google Scholar
[23]Lemou, M., Linearized quantum and relativistic Fokker-Planck-Landau equations, Math. Meth. Appl. Sci., 23 (2000), 1093–1119.Google Scholar
[24]LeVeque, R. J., Numerical Methods for Conservation Laws, Birkhauser Verlag, Basel, second edition, 1992.Google Scholar
[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), 1217–1245.CrossRefGoogle Scholar
[26]Pareschi, L., Russo, G. and Toscani, G., Fast spectral methods for the Fokker-Planck-Landau collision operator, J. Comput. Phys., 165 (2000), 216–236.Google Scholar
[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.Google Scholar
[28]Toscani, G., Finite time blow up in Kaniadakis-Quarati model of Bose-Einstein particles, Commun. Part. Diff. Eqns., 37 (2012), 77–87.Google Scholar