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Linear Stability of Hyperbolic Moment Models for Boltzmann Equation

Part of: Time-dependent statistical mechanics (dynamic and nonequilibrium) Partial differential equations Partial differential equations on manifolds; differential operators

Published online by Cambridge University Press:  09 May 2017

Yana Di ,
Yuwei Fan ,
Ruo Li  and
Lingchao Zheng
Yana Di*
Affiliation:
LSEC, Institute of Computational Mathematics and Scientific/Engineering Computing, NCMIS, AMSS, Chinese Academy of Sciences, Beijing 100190, China
Yuwei Fan*
Affiliation:
School of Mathematical Sciences, Peking University, Beijing 100871, China
Ruo Li*
Affiliation:
HEDPS & CAPT, LMAM & School of Mathematical Sciences, Peking University, Beijing 100871, China
Lingchao Zheng*
Affiliation:
School of Mathematical Sciences, Peking University, Beijing 100871, China
*
*Corresponding author. Email addresses:yndi@lsec.cc.ac.cn (Y. Di), ywfan@pku.edu.cn (Y. Fan), rli@math.pku.edu.cn (R. Li), lczheng@pku.edu.cn (L. Zheng)
*Corresponding author. Email addresses:yndi@lsec.cc.ac.cn (Y. Di), ywfan@pku.edu.cn (Y. Fan), rli@math.pku.edu.cn (R. Li), lczheng@pku.edu.cn (L. Zheng)
*Corresponding author. Email addresses:yndi@lsec.cc.ac.cn (Y. Di), ywfan@pku.edu.cn (Y. Fan), rli@math.pku.edu.cn (R. Li), lczheng@pku.edu.cn (L. Zheng)
*Corresponding author. Email addresses:yndi@lsec.cc.ac.cn (Y. Di), ywfan@pku.edu.cn (Y. Fan), rli@math.pku.edu.cn (R. Li), lczheng@pku.edu.cn (L. Zheng)
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Abstract

Grad's moment models for Boltzmann equation were recently regularized to globally hyperbolic systems and thus the regularized models attain local well-posedness for Cauchy data. The hyperbolic regularization is only related to the convection term in Boltzmann equation. We in this paper studied the regularized models with the presentation of collision terms. It is proved that the regularized models are linearly stable at the local equilibrium and satisfy Yong's first stability condition with commonly used approximate collision terms, and particularly with Boltzmann's binary collision model.

Keywords

Boltzmann equationGrad's moment methodhyperbolic moment equationlinear stability

MSC classification

Secondary: 82C40: Kinetic theory of gases 58J45: Hyperbolic equations 35B35: Stability
Type
Research Article
Information
Numerical Mathematics: Theory, Methods and Applications , Volume 10 , Issue 2 , May 2017 , pp. 255 - 277
Copyright
Copyright © Global-Science Press 2017 

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References

[1] Bhatnagar, P. L., Gross, E. P. and Krook, M., A model for collision processes in gases. I. small amplitude processes in charged and neutral one-component systems, Phys. Rev., 94(3) (1954), pp. 511–525.Google Scholar
[2] Bobylev, A. V., The Chapman-Enskog and Grad methods for solving the Boltzmann equation, Sov. Phys. Dokl., 27(1) (1982), pp. 29–31.Google Scholar
[3] Boltzmann, L., Weitere studien über das wärmegleichgewicht unter gas-molekülen, Wiener Berichte, 66 (1872), pp. 275–370.Google Scholar
[4] Cai, Z., Fan, Y. and Li, R., Globally hyperbolic regularization of Grad's moment system in one dimensional space, Commun. Math. Sci., 11(2) (2013), pp. 547–571.Google Scholar
[5] Cai, Z., Fan, Y. and Li, R., Globally hyperbolic regularization of Grad's moment system, Commun. Pure Appl. Math., 67(3) (2014), pp. 464–518.Google Scholar
[6] Cai, Z., Fan, Y. and Li, R., On hyperbolicity of 13-moment system, Kinetic and Related Models, 7(3) (2014), pp. 415–432.Google Scholar
[7] Cai, Z., Fan, Y. and Li, R., A framework on moment model reduction for kinetic equation, SIAM J. Appl. Math., 75(5) (2015), pp. 2001–2023.Google Scholar
[8] Cai, Z., Fan, Y., Li, R. and Qiao, Z., Dimension-reduced hyperbolic moment method for the Boltzmann equation with BGK-type collision, Commun. Comput. Phys., 15(5) (2014), pp. 1368–1406.Google Scholar
[9] Chapman, S. and Cowling, T. G., The Mathematical Theory of Non-Uniform Gases, 3rd Edition, Cambridge University Press, 1990.Google Scholar
[10] Fan, Y., Koellermeier, J., Li, J., Li, R. and Torrilhon, M., Model reduction of kinetic equations by operator projection, J. Stat. Phys., 161(4) (2015).Google Scholar
[11] Fan, Y. W., Development and Application of Moment Method in Gas Kinetic Theory (in Chinese), PhD thesis, Peking University, June 2016.Google Scholar
[12] Grad, H., Note on N-dimensional Hermite polynomials, Commun. Pure Appl. Math., 2(4) (1949), pp. 325–330.Google Scholar
[13] Grad, H., On the kinetic theory of rarefied gases, Commun. Pure Appl. Math., 2(4) (1949), pp. 331–407.Google Scholar
[14] Holway, L. H., New statistical models for kinetic theory: Methods of construction, Phys. Fluids, 9(1) (1966), pp. 1658–1673.Google Scholar
[15] Müller, I. and Ruggeri, T., Rational Extended Thermodynamics, 2nd Edition, vol. 37 of Springer Tracts in Natural Philosophy, Springer-Verlag, New York, 1998.Google Scholar
[16] Rosenau, P., Extending hydrodynamics via the regularization of the Chapman-Enskog expansion, Phys. Rev. A, 40(12) (1989), pp. 7193–7196.Google Scholar
[17] Shakhov, E. M., Generalization of the Krook kinetic relaxation equation, Fluid Dyn., 3(5) (1968), pp. 95–96.CrossRefGoogle Scholar
[18] Struchtrup, H., Macroscopic Transport Equations for Rarefied Gas Flows: Approximation Methods in Kinetic Theory, Springer, 2005.CrossRefGoogle Scholar
[19] Struchtrup, H. and Torrilhon, M., Regularization of Grad's 13 moment equations: Derivation and linear analysis, Phys. Fluids, 15(9) (2003), pp. 2668–2680.Google Scholar
[20] Torrilhon, M., Convergence study of moment approximations for boundary value problems of the Boltzmann-BGK equation, Commun. Comput. Phys., 18(3) (2015), pp. 529–557.Google Scholar
[21] Yong, W. A., Singular Perturbation of First-order Hyperbolic Systems, PhD Thesis, Universität Heidelbery, 1992.Google Scholar
[22] Yong, W. A., Singular perturbations of first-order hyperbolic systems with stiff source terms, J. Diff. Equations, 155(1) (1999), pp. 89–132.Google Scholar