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A numerical scheme for the quantum Boltzmann equation withstiff collision terms⋆

Published online by Cambridge University Press:  24 October 2011

Francis Filbet ,
Jingwei Hu  and
Shi Jin
Francis Filbet
Affiliation:
Universitéde Lyon, Université Lyon I, CNRS UMR 5208, Institut Camille Jordan, 43 boulevard du 11 Novembre 1918, 69622 Villeurbanne cedex, France. [email protected]
Jingwei Hu
Affiliation:
Department of Mathematics, University of Wisconsin-Madison, 480 Lincoln Drive, Madison, 53706 WI, USA; [email protected] ; [email protected] ; Institute for Computational Engineering and Sciences (ICES), The University of Texas at Austin, 1 University Station C0200, Austin, 78712 TX, USA
Shi Jin
Affiliation:
Department of Mathematics, University of Wisconsin-Madison, 480 Lincoln Drive, Madison, 53706 WI, USA; [email protected] ; [email protected] ;
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Abstract

Numerically solving the Boltzmann kinetic equations with the small Knudsen number ischallenging due to the stiff nonlinear collision terms. A class of asymptotic-preservingschemes was introduced in [F. Filbet and S. Jin,J. Comput. Phys. 229 (2010)7625–7648] to handle this kind of problems. The idea is to penalize the stiff collisionterm by a BGK type operator. This method, however, encounters its own difficulty whenapplied to the quantum Boltzmann equation. To define the quantum Maxwellian (Bose-Einsteinor Fermi-Dirac distribution) at each time step and every mesh point, one has to invert anonlinear equation that connects the macroscopic quantity fugacity with density andinternal energy. Setting a good initial guess for the iterative method is troublesome inmost cases because of the complexity of the quantum functions (Bose-Einstein orFermi-Dirac function). In this paper, we propose to penalize the quantum collision term bya ‘classical’ BGK operator instead of the quantum one. This is based on the observationthat the classical Maxwellian, with the temperature replaced by the internal energy, hasthe same first five moments as the quantum Maxwellian. The scheme so designed avoids theaforementioned difficulty, and one can show that the density distribution is still driventoward the quantum equilibrium. Numerical results are presented to illustrate theefficiency of the new scheme in both the hydrodynamic and kinetic regimes. We also developa spectral method for the quantum collision operator.

Keywords

Quantum Boltzmann equationBose/Fermi gasasymptotic-preserving schemesfluid dynamic limit
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 46 , Issue 2 , March 2012 , pp. 443 - 463
Copyright
© EDP Sciences, SMAI, 2011

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References

Arlotti, L. and Lachowicz, M., Euler and Navier-Stokes limits of the Uehling-Uhlenbeck quantum kinetic equations. J. Math. Phys. 38 (1997) 3571–3588. Google Scholar
Carleman, T., Sur la théorie de l’équation intégrodifférentielle de Boltzmann. Acta Math. 60 (1933) 91–146. Google Scholar
C. Cercignani, The Boltzmann Equation and Its Applications. Springer-Verlag, New York (1988).
G. Dimarco and L. Pareschi, Exponential Runge-Kutta methods for stiff kinetic equations. arXiv:1010.1472.
Escobedo, M. and Mischler, S., On a quantum Boltzmann equation for a gas of photons. J. Math. Pures Appl. 80 (2001) 471–515. Google Scholar
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
Filbet, F., Mouhot, C. and Pareschi, L., Solving the Boltzmann equation in NlogN. SIAM J. Sci. Comput. 28 (2006) 1029–1053. Google Scholar
T. Goudon, S. Jin, J.-G. Liu and B. Yan, Asymptotic-Preserving schemes for kinetic-fluid modeling of disperse two-phase flows. Preprint.
Hu, J.W. and Jin, S., On kinetic flux vector splitting schemes for quantum Euler equations. KRM 4 (2011) 517–530. Google Scholar
J.W. Hu and L. Ying, A fast spectral algorithm for the quantum Boltzmann collision operator. Preprint.
R.J. LeVeque, Numerical Methods for Conservation Laws, 2nd edition. Birkhäuser Verlag, Basel (1992).
Lu, X., A modified Boltzmann equation for Bose-Einstein particles: isotropic solutions and long-time behavior. J. Statist. Phys. 98 (2000) 1335–1394. Google Scholar
Lu, X., On spatially homogeneous solutions of a modified Boltzmann equation for Fermi-Dirac particles. J. Statist. Phys. 105 (2001) 353–388. Google Scholar
Lu, X. and Wennberg, B., On stability and strong convergence for the spatially homogeneous Boltzmann equation for Fermi-Dirac particles. Arch. Ration. Mech. Anal. 168 (2003) 1–34. Google Scholar
Markowich, P. and Pareschi, L., Fast, conservative and entropic numerical methods for the Bosonic Boltzmann equation. Numer. Math. 99 (2005) 509–532. Google Scholar
Mouhot, C. and Pareschi, L., Fast algorithms for computing the Boltzmann collision operator. Math. Comput. 75 (2006) 1833–1852. Google Scholar
Nordheim, L.W., On the kinetic method in the new statistics and its application in the electron theory of conductivity. Proc. R. Soc. Lond. Ser. A 119 (1928) 689–698. Google Scholar
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. Google Scholar
Pareschi, L. and Russo, G., Implicit-Explicit Runge-Kutta methods and applications to hyperbolic systems with relaxation. J. Sci. Comput. 25 (2005) 129–155. Google Scholar
W.H. Press, S.A. Teukolsky, W.T. Vetterling and B.P. Flannery, Numerical Recipes: The Art of Scientific Computing, 3th edition. Cambridge University Press, Cambridge (2007).
Uehling, E.A. and Uhlenbeck, G.E., Transport phenomena in Einstein-Bose and Fermi-Dirac gases. I. Phys. Rev. 43 (1933) 552–561. Google Scholar
C. Villani, A review of mathematical topics in collisional kinetic theory, in Handbook of Mathematical Fluid Mechanics I. edited by S. Friedlander and D. Serre, North-Holland (2002) 71–305.