A numerical scheme for the quantum Boltzmann equation with stiff collision terms

Francis Filbet; Jingwei Hu; Shi Jin

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique (2012)

  • Volume: 46, Issue: 2, page 443-463
  • ISSN: 0764-583X

Abstract

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Numerically solving the Boltzmann kinetic equations with the small Knudsen number is challenging due to the stiff nonlinear collision terms. A class of asymptotic-preserving schemes 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 collision term by a BGK type operator. This method, however, encounters its own difficulty when applied to the quantum Boltzmann equation. To define the quantum Maxwellian (Bose-Einstein or Fermi-Dirac distribution) at each time step and every mesh point, one has to invert a nonlinear equation that connects the macroscopic quantity fugacity with density and internal energy. Setting a good initial guess for the iterative method is troublesome in most cases because of the complexity of the quantum functions (Bose-Einstein or Fermi-Dirac function). In this paper, we propose to penalize the quantum collision term by a ‘classical’ BGK operator instead of the quantum one. This 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. The scheme so designed avoids the aforementioned difficulty, and one can show that the density distribution is still driven toward the quantum equilibrium. Numerical results are presented to illustrate the efficiency of the new scheme in both the hydrodynamic and kinetic regimes. We also develop a spectral method for the quantum collision operator.

How to cite

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Filbet, Francis, Hu, Jingwei, and Jin, Shi. "A numerical scheme for the quantum Boltzmann equation with stiff collision terms." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 46.2 (2012): 443-463. <http://eudml.org/doc/273149>.

@article{Filbet2012,
abstract = {Numerically solving the Boltzmann kinetic equations with the small Knudsen number is challenging due to the stiff nonlinear collision terms. A class of asymptotic-preserving schemes 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 collision term by a BGK type operator. This method, however, encounters its own difficulty when applied to the quantum Boltzmann equation. To define the quantum Maxwellian (Bose-Einstein or Fermi-Dirac distribution) at each time step and every mesh point, one has to invert a nonlinear equation that connects the macroscopic quantity fugacity with density and internal energy. Setting a good initial guess for the iterative method is troublesome in most cases because of the complexity of the quantum functions (Bose-Einstein or Fermi-Dirac function). In this paper, we propose to penalize the quantum collision term by a ‘classical’ BGK operator instead of the quantum one. This 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. The scheme so designed avoids the aforementioned difficulty, and one can show that the density distribution is still driven toward the quantum equilibrium. Numerical results are presented to illustrate the efficiency of the new scheme in both the hydrodynamic and kinetic regimes. We also develop a spectral method for the quantum collision operator.},
author = {Filbet, Francis, Hu, Jingwei, Jin, Shi},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},
keywords = {quantum Boltzmann equation; Bose/Fermi gas; asymptotic-preserving schemes; fluid dynamic limit},
language = {eng},
number = {2},
pages = {443-463},
publisher = {EDP-Sciences},
title = {A numerical scheme for the quantum Boltzmann equation with stiff collision terms},
url = {http://eudml.org/doc/273149},
volume = {46},
year = {2012},
}

TY - JOUR
AU - Filbet, Francis
AU - Hu, Jingwei
AU - Jin, Shi
TI - A numerical scheme for the quantum Boltzmann equation with stiff collision terms
JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
PY - 2012
PB - EDP-Sciences
VL - 46
IS - 2
SP - 443
EP - 463
AB - Numerically solving the Boltzmann kinetic equations with the small Knudsen number is challenging due to the stiff nonlinear collision terms. A class of asymptotic-preserving schemes 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 collision term by a BGK type operator. This method, however, encounters its own difficulty when applied to the quantum Boltzmann equation. To define the quantum Maxwellian (Bose-Einstein or Fermi-Dirac distribution) at each time step and every mesh point, one has to invert a nonlinear equation that connects the macroscopic quantity fugacity with density and internal energy. Setting a good initial guess for the iterative method is troublesome in most cases because of the complexity of the quantum functions (Bose-Einstein or Fermi-Dirac function). In this paper, we propose to penalize the quantum collision term by a ‘classical’ BGK operator instead of the quantum one. This 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. The scheme so designed avoids the aforementioned difficulty, and one can show that the density distribution is still driven toward the quantum equilibrium. Numerical results are presented to illustrate the efficiency of the new scheme in both the hydrodynamic and kinetic regimes. We also develop a spectral method for the quantum collision operator.
LA - eng
KW - quantum Boltzmann equation; Bose/Fermi gas; asymptotic-preserving schemes; fluid dynamic limit
UR - http://eudml.org/doc/273149
ER -

References

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  1. [1] L. Arlotti and M. Lachowicz, Euler and Navier-Stokes limits of the Uehling-Uhlenbeck quantum kinetic equations. J. Math. Phys.38 (1997) 3571–3588. Zbl0885.35102MR1455570
  2. [2] T. Carleman, Sur la théorie de l’équation intégrodifférentielle de Boltzmann. Acta Math.60 (1933) 91–146. Zbl0006.40002MR1555365JFM59.0404.02
  3. [3] C. Cercignani, The Boltzmann Equation and Its Applications. Springer-Verlag, New York (1988). Zbl0646.76001MR1313028
  4. [4] G. Dimarco and L. Pareschi, Exponential Runge-Kutta methods for stiff kinetic equations. arXiv:1010.1472. Zbl1298.76150MR2861709
  5. [5] M. Escobedo and S. Mischler, On a quantum Boltzmann equation for a gas of photons. J. Math. Pures Appl.80 (2001) 471–515. Zbl1134.82318MR1831432
  6. [6] F. Filbet and S. Jin, A class of asymptotic-preserving schemes for kinetic equations and related problems with stiff sources. J. Comput. Phys.229 (2010) 7625–7648. Zbl1202.82066MR2674294
  7. [7] F. Filbet, C. Mouhot and L. Pareschi, Solving the Boltzmann equation in NlogN. SIAM J. Sci. Comput.28 (2006) 1029–1053. Zbl1174.82012MR2240802
  8. [8] T. Goudon, S. Jin, J.-G. Liu and B. Yan, Asymptotic-Preserving schemes for kinetic-fluid modeling of disperse two-phase flows. Preprint. MR3066185
  9. [9] J.W. Hu and S. Jin, On kinetic flux vector splitting schemes for quantum Euler equations. KRM4 (2011) 517–530. Zbl1220.35128MR2786396
  10. [10] J.W. Hu and L. Ying, A fast spectral algorithm for the quantum Boltzmann collision operator. Preprint. Zbl1260.82068MR2911206
  11. [11] R.J. LeVeque, Numerical Methods for Conservation Laws, 2nd edition. Birkhäuser Verlag, Basel (1992). Zbl0723.65067MR1153252
  12. [12] X. Lu, A modified Boltzmann equation for Bose-Einstein particles: isotropic solutions and long-time behavior. J. Statist. Phys.98 (2000) 1335–1394. Zbl1005.82026MR1751703
  13. [13] X. Lu, On spatially homogeneous solutions of a modified Boltzmann equation for Fermi-Dirac particles. J. Statist. Phys.105 (2001) 353–388. Zbl1156.82380MR1861208
  14. [14] X. Lu and B. Wennberg, On stability and strong convergence for the spatially homogeneous Boltzmann equation for Fermi-Dirac particles. Arch. Ration. Mech. Anal.168 (2003) 1–34. Zbl1044.76058MR2029003
  15. [15] P. Markowich and L. Pareschi, Fast, conservative and entropic numerical methods for the Bosonic Boltzmann equation. Numer. Math.99 (2005) 509–532. Zbl1204.82031MR2117737
  16. [16] C. Mouhot and L. Pareschi, Fast algorithms for computing the Boltzmann collision operator. Math. Comput.75 (2006) 1833–1852. Zbl1105.76043MR2240637
  17. [17] L.W. Nordheim, On the kinetic method in the new statistics and its application in the electron theory of conductivity. Proc. R. Soc. Lond. Ser. A119 (1928) 689–698. Zbl54.0988.05JFM54.0988.05
  18. [18] L. Pareschi and G. Russo, Numerical solution of the Boltzmann equation I. Spectrally accurate approximation of the collision operator. SIAM J. Numer. Anal. 37 (2000) 1217–1245. Zbl1049.76055MR1756425
  19. [19] L. Pareschi and G. Russo, Implicit-Explicit Runge-Kutta methods and applications to hyperbolic systems with relaxation. J. Sci. Comput.25 (2005) 129–155. Zbl1203.65111MR2231946
  20. [20] 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). Zbl0587.65003
  21. [21] E.A. Uehling and G.E. Uhlenbeck, Transport phenomena in Einstein-Bose and Fermi-Dirac gases. I. Phys. Rev.43 (1933) 552–561. Zbl0006.33404
  22. [22] 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. Zbl1170.82369MR1942465

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