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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

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 & 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, (1995).   Google Scholar

[2]

S. Chapman and T. G. Cowling, The Mathematical Theory of Non-uniform Gases,, $3^{rd}$ edition, (1990).   Google Scholar

[3]

H. Grad, Note on $N$-dimensional Hermite polynomials,, Comm. Pure Appl. Math., 2 (1949), 325.  doi: 10.1002/cpa.3160020402.  Google Scholar

[4]

H. Grad, On the kinetic theory of rarefied gases,, Comm. Pure Appl. Math., 2 (1949), 331.  doi: 10.1002/cpa.3160020403.  Google Scholar

[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.  doi: 10.1007/s102550200003.  Google Scholar

[6]

S. Jin and M. Slemrod, Regularization of the Burnett equations via relaxation,, J. Stat. Phys., 103 (2001), 1009.  doi: 10.1023/A:1010365123288.  Google Scholar

[7]

G. Karniadakis, A. Beskok and N. Aluru, Microflows and Nanoflows: Fundamentals and Simulation,, Springer, (2005).   Google Scholar

[8]

I. Müller and T. Ruggeri, Rational Extended Thermodynamics,, $2^{nd}$ edition, (1998).  doi: 10.1007/978-1-4612-2210-1.  Google Scholar

[9]

L. Mieussens and H. Struchtrup, Numerical comparison of Bhatnagar-Gross-Krook models with proper Prandtl number,, Phys. Fluids, 16 (2004), 2797.  doi: 10.1063/1.1758217.  Google Scholar

[10]

H. Struchtrup and M. Torrilhon, Regularization of Grad's 13 moment equations: Derivation and linear analysis,, Phys. Fluids, 15 (2003), 2668.  doi: 10.1063/1.1597472.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[13]

M. Torrilhon, Hyperbolic moment equations in kinetic gas theory based on multi-variate Pearson-IV-distributions,, Commun. Comput. Phys., 7 (2010), 639.  doi: 10.4208/cicp.2009.09.049.  Google Scholar

[14]

Wolfram Research, Mathematica 9,, , ().   Google Scholar

show all references

References:
[1]

G. A. Bird, Molecular Gas Dynamics and the Direct Simulation of Gas Flows,, Clarendon Press, (1995).   Google Scholar

[2]

S. Chapman and T. G. Cowling, The Mathematical Theory of Non-uniform Gases,, $3^{rd}$ edition, (1990).   Google Scholar

[3]

H. Grad, Note on $N$-dimensional Hermite polynomials,, Comm. Pure Appl. Math., 2 (1949), 325.  doi: 10.1002/cpa.3160020402.  Google Scholar

[4]

H. Grad, On the kinetic theory of rarefied gases,, Comm. Pure Appl. Math., 2 (1949), 331.  doi: 10.1002/cpa.3160020403.  Google Scholar

[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.  doi: 10.1007/s102550200003.  Google Scholar

[6]

S. Jin and M. Slemrod, Regularization of the Burnett equations via relaxation,, J. Stat. Phys., 103 (2001), 1009.  doi: 10.1023/A:1010365123288.  Google Scholar

[7]

G. Karniadakis, A. Beskok and N. Aluru, Microflows and Nanoflows: Fundamentals and Simulation,, Springer, (2005).   Google Scholar

[8]

I. Müller and T. Ruggeri, Rational Extended Thermodynamics,, $2^{nd}$ edition, (1998).  doi: 10.1007/978-1-4612-2210-1.  Google Scholar

[9]

L. Mieussens and H. Struchtrup, Numerical comparison of Bhatnagar-Gross-Krook models with proper Prandtl number,, Phys. Fluids, 16 (2004), 2797.  doi: 10.1063/1.1758217.  Google Scholar

[10]

H. Struchtrup and M. Torrilhon, Regularization of Grad's 13 moment equations: Derivation and linear analysis,, Phys. Fluids, 15 (2003), 2668.  doi: 10.1063/1.1597472.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[13]

M. Torrilhon, Hyperbolic moment equations in kinetic gas theory based on multi-variate Pearson-IV-distributions,, Commun. Comput. Phys., 7 (2010), 639.  doi: 10.4208/cicp.2009.09.049.  Google Scholar

[14]

Wolfram Research, Mathematica 9,, , ().   Google Scholar

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