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

Francis Filbet; Jingwei Hu; Shi Jin

ESAIM: Mathematical Modelling and Numerical Analysis (2011)

  • 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 46.2 (2011): 443-463. <http://eudml.org/doc/222127>.

@article{Filbet2011,
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},
keywords = {Quantum Boltzmann equation; Bose/Fermi gas; asymptotic-preserving schemes; fluid dynamic limit; quantum Boltzmann equation},
language = {eng},
month = {10},
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/222127},
volume = {46},
year = {2011},
}

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
DA - 2011/10//
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; quantum Boltzmann equation
UR - http://eudml.org/doc/222127
ER -

References

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