Entropic approximation in kinetic theory

Jacques Schneider

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique (2004)

  • 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 - Modélisation Mathématique et Analyse Numérique 38.3 (2004): 541-561. <http://eudml.org/doc/245932>.

@article{Schneider2004,
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 - Modélisation Mathématique et Analyse Numérique},
keywords = {kinetic entropy; convex analysis; nonlinear approximation; moments systems; maxwellian molecules; Entropic approximation; moment closure; kinetic theory},
language = {eng},
number = {3},
pages = {541-561},
publisher = {EDP-Sciences},
title = {Entropic approximation in kinetic theory},
url = {http://eudml.org/doc/245932},
volume = {38},
year = {2004},
}

TY - JOUR
AU - Schneider, Jacques
TI - Entropic approximation in kinetic theory
JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
PY - 2004
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/245932
ER -

References

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  1. [1] P. Andries, P. Le Tallec, J.P. Perlat and B. Perthame, The Gaussian-BGK model of Boltzmann equation with small Prandtl number. Eur. J. Mech. B Fluids 19 (2000) 813–830. Zbl0967.76082
  2. [2] L. Arkeryd, On the Boltzmann equation. Arch. Rational Mech. Anal. 45 (1972) 1–34. Zbl0245.76059
  3. [3] F. Bouchut, C. Bourdarias and B. Perthame, An example of MUSCL method satisfying all the entropy inequalities. C.R. Acad Sc. Paris, Serie I 317 (1993) 619–624. Zbl0779.65061
  4. [4] F. Coquel and P. LeFloch, An entropy satisfying muscl scheme for systems of conservation laws. Numerische Math. 74 (1996) 1–34. Zbl0860.65076
  5. [5] I. Csiszár, I-divergence geometry of probability distributions and minimization problems Sanov property. Ann. Probab. 3 (1975) 146–158. Zbl0318.60013
  6. [6] R. DiPerna and P.-L. Lions, On the Cauchy problem for Boltzmann equations: Global existence and weak stability. Ann. Math. 130 (1989) 321–366. Zbl0698.45010
  7. [7] H. Grad, On the kinetic theory of rarefied gases. Comm. Pure Appl. Math. 2 (1949) 331–407. Zbl0037.13104
  8. [8] M. Junk, Domain of definition of Levermore’s five moments system. J. Stat. Phys. 93 (1998) 1143-1167. Zbl0952.82024
  9. [9] M. Junk, Maximum entropy for reduced moment problems. M3AS 10 (2000) 1001–1025. Zbl1012.44005
  10. [10] C. Léonard, Some results about entropic projections, in Stochastic Analysis and Mathematical Analysis, Vol. 50, Progr. Probab., Birkhaüser, Boston, MA (2001) 59–73. Zbl1020.60014
  11. [11] C.D. Levermore, Moment closure hierarchies for kinetic theories. J. Stat. Phys. 83 (1996) 1021–1065. Zbl1081.82619
  12. [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. Zbl1078.82526
  13. [13] A.J. Povzner, The Boltzmann equation in the kinetic theory of gases. Amer. Math. Soc. Trans. 47 (1965) 193–214. Zbl0188.21204
  14. [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. Zbl0811.76050
  15. [15] C. Villani, Fisher information bounds for Boltzmann’s collision operator. J. Math. Pures Appl. 77 (1998) 821–837. Zbl0918.60093

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