Entropic approximation in kinetic theory

Jacques Schneider

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

  • Volume: 38, Issue: 3, page 541-561
  • ISSN: 0764-583X

Abstract

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Approximation theory in the context of probability density function turns out to go beyond the classical idea of orthogonal projection. Special tools have to be designed so as to respect the nonnegativity of the approximate function. We develop here and justify from the theoretical point of view an approximation procedure introduced by Levermore [Levermore, J. Stat. Phys.83 (1996) 1021–1065] and based on an entropy minimization principle under moment constraints. We prove in particular a global existence theorem for such an approximation and derive as a by-product a necessary and sufficient condition for the so-called problem of moment realizability. Applications of the above result are given in kinetic theory: first in the context of Levermore's approach and second to design generalized BGK models for Maxwellian molecules.

How to cite

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Schneider, Jacques. "Entropic approximation in kinetic theory." ESAIM: Mathematical Modelling and Numerical Analysis 38.3 (2010): 541-561. <http://eudml.org/doc/194227>.

@article{Schneider2010,
abstract = { Approximation theory in the context of probability density function turns out to go beyond the classical idea of orthogonal projection. Special tools have to be designed so as to respect the nonnegativity of the approximate function. We develop here and justify from the theoretical point of view an approximation procedure introduced by Levermore [Levermore, J. Stat. Phys.83 (1996) 1021–1065] and based on an entropy minimization principle under moment constraints. We prove in particular a global existence theorem for such an approximation and derive as a by-product a necessary and sufficient condition for the so-called problem of moment realizability. Applications of the above result are given in kinetic theory: first in the context of Levermore's approach and second to design generalized BGK models for Maxwellian molecules. },
author = {Schneider, Jacques},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Kinetic entropy; convex analysis; nonlinear approximation; moments systems; Maxwellian molecules.; Entropic approximation; moment closure; kinetic theory},
language = {eng},
month = {3},
number = {3},
pages = {541-561},
publisher = {EDP Sciences},
title = {Entropic approximation in kinetic theory},
url = {http://eudml.org/doc/194227},
volume = {38},
year = {2010},
}

TY - JOUR
AU - Schneider, Jacques
TI - Entropic approximation in kinetic theory
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2010/3//
PB - EDP Sciences
VL - 38
IS - 3
SP - 541
EP - 561
AB - Approximation theory in the context of probability density function turns out to go beyond the classical idea of orthogonal projection. Special tools have to be designed so as to respect the nonnegativity of the approximate function. We develop here and justify from the theoretical point of view an approximation procedure introduced by Levermore [Levermore, J. Stat. Phys.83 (1996) 1021–1065] and based on an entropy minimization principle under moment constraints. We prove in particular a global existence theorem for such an approximation and derive as a by-product a necessary and sufficient condition for the so-called problem of moment realizability. Applications of the above result are given in kinetic theory: first in the context of Levermore's approach and second to design generalized BGK models for Maxwellian molecules.
LA - eng
KW - Kinetic entropy; convex analysis; nonlinear approximation; moments systems; Maxwellian molecules.; Entropic approximation; moment closure; kinetic theory
UR - http://eudml.org/doc/194227
ER -

References

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  11. C.D. Levermore, Moment closure hierarchies for kinetic theories. J. Stat. Phys.83 (1996) 1021–1065.  
  12. L. Mieussens, Discrete velocity model and implicit scheme for the BGK equation of rarefied gas dynamics. Math. Models Methods Appl. Sci. 10 (2000) 1121–1149.  
  13. A.J. Povzner, The Boltzmann equation in the kinetic theory of gases. Amer. Math. Soc. Trans. 47 (1965) 193–214.  
  14. F. Rogier and J. Schneider, A Direct Method for Solving the Boltzmann Equation. Proc. Colloque Euromech n0287 Discrete Models in Fluid Dynamics, Transport Theory Statist. Phys. 23 (1994) 1–3.  
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