On hyperbolicity of 13-moment system

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September 2014, 7(3): 415-432. doi: 10.3934/krm.2014.7.415

On hyperbolicity of 13-moment system

Zhenning Cai ^1, , Yuwei Fan ^2, and Ruo Li ^3,

1.	CAPT & School of Mathematical Sciences, Peking University, Yiheyuan Road 5, 100871 Beijing, China
2.	School of Mathematical Sciences, Peking University, Yiheyuan Road 5, 100871 Beijing, China
3.	HEDPS, CAPT & School of Mathematical Sciences, Peking University, Yiheyuan Road 5, 100871 Beijing, China

Received July 2013 Revised March 2014 Published July 2014

Abstract
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We point out that the thermodynamic equilibrium is not an interior point of the hyperbolicity region of Grad's 13-moment system. With a compact expansion of the phase density, which is more compact than Grad's expansion, we derive a modified 13-moment system. The new 13-moment system admits the thermodynamic equilibrium as an interior point of its hyperbolicity region. We deduce a concise criterion to ensure the hyperbolicity, and thus the hyperbolicity region can be quantitatively depicted.

Keywords: modified 13-moment system, 1D-reduced moment system., Grad's moment system, hyperbolicity, real diagnoalizability.

Mathematics Subject Classification: 82C40, 35L6.

Citation: Zhenning Cai, Yuwei Fan, Ruo Li. On hyperbolicity of 13-moment system. Kinetic and Related Models, 2014, 7 (3) : 415-432. doi: 10.3934/krm.2014.7.415

References:

[1]	G. A. Bird, Molecular Gas Dynamics and the Direct Simulation of Gas Flows, Clarendon Press, Oxford, 1995.
[2]	S. Chapman and T. G. Cowling, The Mathematical Theory of Non-uniform Gases, $3^{rd}$ edition, Cambridge University Press, Cambridge, 1990.
[3]	H. Grad, Note on $N$-dimensional Hermite polynomials, Comm. Pure Appl. Math., 2 (1949), 325-330. doi: 10.1002/cpa.3160020402.
[4]	H. Grad, On the kinetic theory of rarefied gases, Comm. Pure Appl. Math., 2 (1949), 331-407. doi: 10.1002/cpa.3160020403.
[5]	S. Jin, L. Pareschi and M. Slemrod, A relaxation scheme for solving the Boltzmann equation based on the Chapman-Enskog expansion, Acta Math. Appl. Sin.-E., 18 (2002), 37-62. doi: 10.1007/s102550200003.
[6]	S. Jin and M. Slemrod, Regularization of the Burnett equations via relaxation, J. Stat. Phys., 103 (2001), 1009-1033. doi: 10.1023/A:1010365123288.
[7]	G. Karniadakis, A. Beskok and N. Aluru, Microflows and Nanoflows: Fundamentals and Simulation, Springer, New York, 2005.
[8]	I. Müller and T. Ruggeri, Rational Extended Thermodynamics, $2^{nd}$ edition, Springer Tracts in Natural Philosophy, Vol. 37, Springer-Verlag, New York, 1998. doi: 10.1007/978-1-4612-2210-1.
[9]	L. Mieussens and H. Struchtrup, Numerical comparison of Bhatnagar-Gross-Krook models with proper Prandtl number, Phys. Fluids, 16 (2004), 2797-2813. doi: 10.1063/1.1758217.
[10]	H. Struchtrup and M. Torrilhon, Regularization of Grad's 13 moment equations: Derivation and linear analysis, Phys. Fluids, 15 (2003), 2668-2680. doi: 10.1063/1.1597472.
[11]	H. Struchtrup, Derivation of 13 moment equations for rarefied gas flow to second order accuracy for arbitrary interaction potentials,, Multiscale Model. Simul., 3 (): 221. doi: 10.1137/040603115.
[12]	M. Torrilhon, Regularized 13-moment-equations, in Proceedings of Rarefied Gas Dynamics: 25th International Symposium (eds. M. S. Ivanov and A. K. Rebrov), 2006, 167-172.
[13]	M. Torrilhon, Hyperbolic moment equations in kinetic gas theory based on multi-variate Pearson-IV-distributions, Commun. Comput. Phys., 7 (2010), 639-673. doi: 10.4208/cicp.2009.09.049.
[14]	Wolfram Research, Mathematica 9,, , ().

show all references

References:

[1]	G. A. Bird, Molecular Gas Dynamics and the Direct Simulation of Gas Flows, Clarendon Press, Oxford, 1995.
[2]	S. Chapman and T. G. Cowling, The Mathematical Theory of Non-uniform Gases, $3^{rd}$ edition, Cambridge University Press, Cambridge, 1990.
[3]	H. Grad, Note on $N$-dimensional Hermite polynomials, Comm. Pure Appl. Math., 2 (1949), 325-330. doi: 10.1002/cpa.3160020402.
[4]	H. Grad, On the kinetic theory of rarefied gases, Comm. Pure Appl. Math., 2 (1949), 331-407. doi: 10.1002/cpa.3160020403.
[5]	S. Jin, L. Pareschi and M. Slemrod, A relaxation scheme for solving the Boltzmann equation based on the Chapman-Enskog expansion, Acta Math. Appl. Sin.-E., 18 (2002), 37-62. doi: 10.1007/s102550200003.
[6]	S. Jin and M. Slemrod, Regularization of the Burnett equations via relaxation, J. Stat. Phys., 103 (2001), 1009-1033. doi: 10.1023/A:1010365123288.
[7]	G. Karniadakis, A. Beskok and N. Aluru, Microflows and Nanoflows: Fundamentals and Simulation, Springer, New York, 2005.
[8]	I. Müller and T. Ruggeri, Rational Extended Thermodynamics, $2^{nd}$ edition, Springer Tracts in Natural Philosophy, Vol. 37, Springer-Verlag, New York, 1998. doi: 10.1007/978-1-4612-2210-1.
[9]	L. Mieussens and H. Struchtrup, Numerical comparison of Bhatnagar-Gross-Krook models with proper Prandtl number, Phys. Fluids, 16 (2004), 2797-2813. doi: 10.1063/1.1758217.
[10]	H. Struchtrup and M. Torrilhon, Regularization of Grad's 13 moment equations: Derivation and linear analysis, Phys. Fluids, 15 (2003), 2668-2680. doi: 10.1063/1.1597472.
[11]	H. Struchtrup, Derivation of 13 moment equations for rarefied gas flow to second order accuracy for arbitrary interaction potentials,, Multiscale Model. Simul., 3 (): 221. doi: 10.1137/040603115.
[12]	M. Torrilhon, Regularized 13-moment-equations, in Proceedings of Rarefied Gas Dynamics: 25th International Symposium (eds. M. S. Ivanov and A. K. Rebrov), 2006, 167-172.
[13]	M. Torrilhon, Hyperbolic moment equations in kinetic gas theory based on multi-variate Pearson-IV-distributions, Commun. Comput. Phys., 7 (2010), 639-673. doi: 10.4208/cicp.2009.09.049.
[14]	Wolfram Research, Mathematica 9,, , ().

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