# 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

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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 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 -

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