American Institute of Mathematical Sciences

Advanced
September  2011, 4(3): 735-766. doi: 10.3934/krm.2011.4.735

Non--local macroscopic models based on Gaussian closures for the Spizer-Härm regime

Thierry Goudon 1, and  Martin Parisot 1,

1. 

Project-Team SIMPAF–INRIA Lille Nord Europe, Park Plazza, 40 avenue Halley, F-59650 Villeneuve d’Ascq cedex, France, France

Received  March 2011 Revised  June 2011 Published  August 2011

The Spitzer-Härm regime arising in plasma physics leads asymptotically to a nonlinear diffusion equation for the electron temperature. In this work we propose a hierarchy of models intended to retain more features of the underlying modeling based on kinetic equations. These models are of non--local type. Nevertheless, owing to energy discretization they can lead to coupled systems of diffusion equations. We make the connection between the different models precise and bring out some mathematical properties of the models. A numerical scheme is designed for the approximate models, and simulations validate the proposed approach.
Keywords: Plasma physics, hydrodynamic limits..
Mathematics Subject Classification: 35Q99, 35B25, 82C7.
Citation: Thierry Goudon, Martin Parisot. Non--local macroscopic models based on Gaussian closures for the Spizer-Härm regime. Kinetic & Related Models, 2011, 4 (3) : 735-766. doi: 10.3934/krm.2011.4.735
References:
[1]

F. Alouani Bibi and J.-P. Matte, Nonlocal electron heat transport and electronion energy transfer in the presence of strong collisional heating,, Laser and Particle Beams, 22 (2004), 103.   Google Scholar

[2]

E. M. Epperlein and R. Short, A practical nonlocal model for electron heat transport in laser plasmas,, Phys. Fluids B, 3 (1991), 3092.  doi: 10.1063/1.859789.  Google Scholar

[3]

M. Frank, D. Levermore and M. Shäfer, Diffusive corrections to $\mathbb P_N$ approximations,, Multiscale Model. Simul., 9 (2011), 1.   Google Scholar

[4]

T. Goudon and M. Parisot, On the Spitzer-Härm regime and non-local approximations: modeling, analysis and numerical simulations,, SIAM Multiscale Model. Simul., (2011).   Google Scholar

[5]

B. Graille, "Modélisation de Mélanges Gazeux Réactifs Ionisés Dissipatifs,", Ph.D. thesis, (2004).   Google Scholar

[6]

Y. Guo, The Landau equation in a periodic box,, Comm. Math. Phys., 231 (2002), 391.  doi: 10.1007/s00220-002-0729-9.  Google Scholar

[7]

D. Levermore, "Boundary Conditions for Moment Closures,", IPAM KT 2009, (2009).   Google Scholar

[8]

D. Levermore, "Kinetic Theory, Gaussian Moment Closures, and Fluid Approximations,", IPAM KT 2009, (2009).   Google Scholar

[9]

T.-P.Liu and Y. Zeng, Large time behavior of solutions for general quasilinear hyperbolic-parabolic systems of conservation laws,, Mem. Amer. Math. Soc., 125 (1997).   Google Scholar

[10]

J.-F. Luciani and P. Mora, Resummation methods of the Chapman-Enskog expansion for a strongly inhomogeneous plasma,, J. Stat. Phys., 43 (1986), 281.  doi: 10.1007/BF01010582.  Google Scholar

[11]

J.-F. Luciani and P. Mora, Nonlocal electron transport in laser created plasmas,, Laser and Particle Beams, 12 (1994), 387.  doi: 10.1017/S0263034600008247.  Google Scholar

[12]

J.-F. Luciani, P. Mora and R. Pellat, Quasistatic heat front and delocalized heat flux,, Phys. Fluids, 28 (1985), 835.  doi: 10.1063/1.865052.  Google Scholar

[13]

P. Nicolaï, J.-L. Feugeas and G. Schurtz, A practical nonlocal model for heat transport in magnetized laser plasmas,, Phys. of Plasmas, 13 (2006), 032701.   Google Scholar

[14]

M. Parisot, Finite volume schemes on unstructured grids for generalized Spitzer-Härm model,, Tech. Rep., (2011).   Google Scholar

[15]

E. J. Routh, "A Treatise on the Stability of a Given State of Motion,", Macmillan and Co., (1877).   Google Scholar

[16]

G. P. Schurtz, P. Nicolaï and M. Busquet, A nonlocal electron conduction model for multidimensional radiation hydrodynamics codes,, Physics of Plasmas, 7 (2000), 4238.  doi: 10.1063/1.1289512.  Google Scholar

[17]

I. P. Shkarofsky, Cartesian tensor expansion of the Fokker-Planck equation,, Can. J. Phys., 41 (1963), 1753.  doi: 10.1139/p63-179.  Google Scholar

[18]

L. Spitzer and R. Härm, Transport phenomena in a completely ionized gas,, Phys. Rev., 89 (1953), 977.  doi: 10.1103/PhysRev.89.977.  Google Scholar

show all references

References:
[1]

F. Alouani Bibi and J.-P. Matte, Nonlocal electron heat transport and electronion energy transfer in the presence of strong collisional heating,, Laser and Particle Beams, 22 (2004), 103.   Google Scholar

[2]

E. M. Epperlein and R. Short, A practical nonlocal model for electron heat transport in laser plasmas,, Phys. Fluids B, 3 (1991), 3092.  doi: 10.1063/1.859789.  Google Scholar

[3]

M. Frank, D. Levermore and M. Shäfer, Diffusive corrections to $\mathbb P_N$ approximations,, Multiscale Model. Simul., 9 (2011), 1.   Google Scholar

[4]

T. Goudon and M. Parisot, On the Spitzer-Härm regime and non-local approximations: modeling, analysis and numerical simulations,, SIAM Multiscale Model. Simul., (2011).   Google Scholar

[5]

B. Graille, "Modélisation de Mélanges Gazeux Réactifs Ionisés Dissipatifs,", Ph.D. thesis, (2004).   Google Scholar

[6]

Y. Guo, The Landau equation in a periodic box,, Comm. Math. Phys., 231 (2002), 391.  doi: 10.1007/s00220-002-0729-9.  Google Scholar

[7]

D. Levermore, "Boundary Conditions for Moment Closures,", IPAM KT 2009, (2009).   Google Scholar

[8]

D. Levermore, "Kinetic Theory, Gaussian Moment Closures, and Fluid Approximations,", IPAM KT 2009, (2009).   Google Scholar

[9]

T.-P.Liu and Y. Zeng, Large time behavior of solutions for general quasilinear hyperbolic-parabolic systems of conservation laws,, Mem. Amer. Math. Soc., 125 (1997).   Google Scholar

[10]

J.-F. Luciani and P. Mora, Resummation methods of the Chapman-Enskog expansion for a strongly inhomogeneous plasma,, J. Stat. Phys., 43 (1986), 281.  doi: 10.1007/BF01010582.  Google Scholar

[11]

J.-F. Luciani and P. Mora, Nonlocal electron transport in laser created plasmas,, Laser and Particle Beams, 12 (1994), 387.  doi: 10.1017/S0263034600008247.  Google Scholar

[12]

J.-F. Luciani, P. Mora and R. Pellat, Quasistatic heat front and delocalized heat flux,, Phys. Fluids, 28 (1985), 835.  doi: 10.1063/1.865052.  Google Scholar

[13]

P. Nicolaï, J.-L. Feugeas and G. Schurtz, A practical nonlocal model for heat transport in magnetized laser plasmas,, Phys. of Plasmas, 13 (2006), 032701.   Google Scholar

[14]

M. Parisot, Finite volume schemes on unstructured grids for generalized Spitzer-Härm model,, Tech. Rep., (2011).   Google Scholar

[15]

E. J. Routh, "A Treatise on the Stability of a Given State of Motion,", Macmillan and Co., (1877).   Google Scholar

[16]

G. P. Schurtz, P. Nicolaï and M. Busquet, A nonlocal electron conduction model for multidimensional radiation hydrodynamics codes,, Physics of Plasmas, 7 (2000), 4238.  doi: 10.1063/1.1289512.  Google Scholar

[17]

I. P. Shkarofsky, Cartesian tensor expansion of the Fokker-Planck equation,, Can. J. Phys., 41 (1963), 1753.  doi: 10.1139/p63-179.  Google Scholar

[18]

L. Spitzer and R. Härm, Transport phenomena in a completely ionized gas,, Phys. Rev., 89 (1953), 977.  doi: 10.1103/PhysRev.89.977.  Google Scholar

[1]

Sebastian Bauer. A non-relativistic model of plasma physics containing a radiation reaction term. Kinetic & Related Models, 2018, 11 (1) : 25-42. doi: 10.3934/krm.2018002
[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]

Baptiste Fedele, Claudia Negulescu. Numerical study of an anisotropic Vlasov equation arising in plasma physics. Kinetic & Related Models, 2018, 11 (6) : 1395-1426. doi: 10.3934/krm.2018055
[4]

Martin Seehafer. A local existence result for a plasma physics model containing a fully coupled magnetic field. Kinetic & Related Models, 2009, 2 (3) : 503-520. doi: 10.3934/krm.2009.2.503
[5]

Masahiro Suzuki. Asymptotic stability of stationary solutions to the Euler-Poisson equations arising in plasma physics. Kinetic & Related Models, 2011, 4 (2) : 569-588. doi: 10.3934/krm.2011.4.569
[6]

Claudia Negulescu, Anne Nouri, Philippe Ghendrih, Yanick Sarazin. Existence and uniqueness of the electric potential profile in the edge of tokamak plasmas when constrained by the plasma-wall boundary physics. Kinetic & Related Models, 2008, 1 (4) : 619-639. doi: 10.3934/krm.2008.1.619
[7]

Shixin Xu, Xingye Yue, Changrong Zhang. Homogenization: In mathematics or physics?. Discrete & Continuous Dynamical Systems - S, 2016, 9 (5) : 1575-1590. doi: 10.3934/dcdss.2016064
[8]

Silvia Caprino, Carlo Marchioro. On the plasma-charge model. Kinetic & Related Models, 2010, 3 (2) : 241-254. doi: 10.3934/krm.2010.3.241
[9]

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
[10]

Oded Schramm. Hyperfinite graph limits. Electronic Research Announcements, 2008, 15: 17-23. doi: 10.3934/era.2008.15.17
[11]

Dmitry Jakobson. On quantum limits on flat tori. Electronic Research Announcements, 1995, 1: 80-86.

[12]

Tobias Wichtrey. Harmonic limits of dynamical systems. Conference Publications, 2011, 2011 (Special) : 1432-1439. doi: 10.3934/proc.2011.2011.1432
[13]

Silvia Caprino, Carlo Marchioro. On a charge interacting with a plasma of unbounded mass. Kinetic & Related Models, 2011, 4 (1) : 215-226. doi: 10.3934/krm.2011.4.215
[14]

Guillaume Bal, Tomasz Komorowski, Lenya Ryzhik. Kinetic limits for waves in a random medium. Kinetic & Related Models, 2010, 3 (4) : 529-644. doi: 10.3934/krm.2010.3.529
[15]

Agnese Di Castro, Mayte Pérez-Llanos, José Miguel Urbano. Limits of anisotropic and degenerate elliptic problems. Communications on Pure & Applied Analysis, 2012, 11 (3) : 1217-1229. doi: 10.3934/cpaa.2012.11.1217
[16]

Martin D. Buhmann, Slawomir Dinew. Limits of radial basis function interpolants. Communications on Pure & Applied Analysis, 2007, 6 (3) : 569-585. doi: 10.3934/cpaa.2007.6.569
[17]

Menglu Feng, Mei Choi Chiu, Hoi Ying Wong. Pairs trading with illiquidity and position limits. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-19. doi: 10.3934/jimo.2019090
[18]

Adrian Constantin, Joachim Escher. Introduction to the special issue on hydrodynamic model equations. Communications on Pure & Applied Analysis, 2012, 11 (4) : i-iii. doi: 10.3934/cpaa.2012.11.4i
[19]

Yachun Li, Shengguo Zhu. Existence results for compressible radiation hydrodynamic equations with vacuum. Communications on Pure & Applied Analysis, 2015, 14 (3) : 1023-1052. doi: 10.3934/cpaa.2015.14.1023
[20]

Boling Guo, Guangwu Wang. Existence of the solution for the viscous bipolar quantum hydrodynamic model. Discrete & Continuous Dynamical Systems - A, 2017, 37 (6) : 3183-3210. doi: 10.3934/dcds.2017136

2018 Impact Factor: 1.38

Tools

Metrics

  • PDF downloads (8)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences