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

Francis Filbet; Jingwei Hu; Shi Jin

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

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topFilbet, 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 -

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