American Institute of Mathematical Sciences

Advanced
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

[1]

Manuel Torrilhon. H-Theorem for nonlinear regularized 13-moment equations in kinetic gas theory. Kinetic & Related Models, 2012, 5 (1) : 185-201. doi: 10.3934/krm.2012.5.185
[2]

Jessy Mallet, Stéphane Brull, Bruno Dubroca. General moment system for plasma physics based on minimum entropy principle. Kinetic & Related Models, 2015, 8 (3) : 533-558. doi: 10.3934/krm.2015.8.533
[3]

Yi Zhang, Yuyun Zhao, Tao Xu, Xin Liu. $p$th Moment absolute exponential stability of stochastic control system with Markovian switching. Journal of Industrial & Management Optimization, 2016, 12 (2) : 471-486. doi: 10.3934/jimo.2016.12.471
[4]

Peter Bella, Arianna Giunti. Green's function for elliptic systems: Moment bounds. Networks & Heterogeneous Media, 2018, 13 (1) : 155-176. doi: 10.3934/nhm.2018007
[5]

Sho Matsumoto, Jonathan Novak. A moment method for invariant ensembles. Electronic Research Announcements, 2018, 25: 60-71. doi: 10.3934/era.2018.25.007
[6]

Paweł Lubowiecki, Henryk Żołądek. The Hess-Appelrot system. I. Invariant torus and its normal hyperbolicity. Journal of Geometric Mechanics, 2012, 4 (4) : 443-467. doi: 10.3934/jgm.2012.4.443
[7]

Gilberto M. Kremer, Wilson Marques Jr.. Fourteen moment theory for granular gases. Kinetic & Related Models, 2011, 4 (1) : 317-331. doi: 10.3934/krm.2011.4.317
[8]

Henning Struchtrup. Unique moment set from the order of magnitude method. Kinetic & Related Models, 2012, 5 (2) : 417-440. doi: 10.3934/krm.2012.5.417
[9]

Darryl D. Holm, Cesare Tronci. Geodesic Vlasov equations and their integrable moment closures. Journal of Geometric Mechanics, 2009, 1 (2) : 181-208. doi: 10.3934/jgm.2009.1.181
[10]

Miroslava Růžičková, Irada Dzhalladova, Jitka Laitochová, Josef Diblík. Solution to a stochastic pursuit model using moment equations. Discrete & Continuous Dynamical Systems - B, 2018, 23 (1) : 473-485. doi: 10.3934/dcdsb.2018032
[11]

Florian Méhats, Olivier Pinaud. A problem of moment realizability in quantum statistical physics. Kinetic & Related Models, 2011, 4 (4) : 1143-1158. doi: 10.3934/krm.2011.4.1143
[12]

Giacomo Dimarco. The moment guided Monte Carlo method for the Boltzmann equation. Kinetic & Related Models, 2013, 6 (2) : 291-315. doi: 10.3934/krm.2013.6.291
[13]

Laurent Baratchart, Sylvain Chevillard, Douglas Hardin, Juliette Leblond, Eduardo Andrade Lima, Jean-Paul Marmorat. Magnetic moment estimation and bounded extremal problems. Inverse Problems & Imaging, 2019, 13 (1) : 39-67. doi: 10.3934/ipi.2019003
[14]

Swann Marx, Tillmann Weisser, Didier Henrion, Jean Bernard Lasserre. A moment approach for entropy solutions to nonlinear hyperbolic PDEs. Mathematical Control & Related Fields, 2020, 10 (1) : 113-140. doi: 10.3934/mcrf.2019032
[15]

Radosław Kurek, Paweł Lubowiecki, Henryk Żołądek. The Hess-Appelrot system. Ⅲ. Splitting of separatrices and chaos. Discrete & Continuous Dynamical Systems - A, 2018, 38 (4) : 1955-1981. doi: 10.3934/dcds.2018079
[16]

Henri Schurz. Moment attractivity, stability and contractivity exponents of stochastic dynamical systems. Discrete & Continuous Dynamical Systems - A, 2001, 7 (3) : 487-515. doi: 10.3934/dcds.2001.7.487
[17]

YunKyong Hyon. Hysteretic behavior of a moment-closure approximation for FENE model. Kinetic & Related Models, 2014, 7 (3) : 493-507. doi: 10.3934/krm.2014.7.493
[18]

Youngae Lee. Topological solutions in the Maxwell-Chern-Simons model with anomalous magnetic moment. Discrete & Continuous Dynamical Systems - A, 2018, 38 (3) : 1293-1314. doi: 10.3934/dcds.2018053
[19]

Luciano Pandolfi. Riesz systems and moment method in the study of viscoelasticity in one space dimension. Discrete & Continuous Dynamical Systems - B, 2010, 14 (4) : 1487-1510. doi: 10.3934/dcdsb.2010.14.1487
[20]

T. Hillen. On the $L^2$-moment closure of transport equations: The general case. Discrete & Continuous Dynamical Systems - B, 2005, 5 (2) : 299-318. doi: 10.3934/dcdsb.2005.5.299

2018 Impact Factor: 1.38

Tools

Metrics

  • PDF downloads (31)
  • HTML views (0)
  • Cited by (6)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences