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

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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

Abstract
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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 and 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), 103-108.
[2]	E. M. Epperlein and R. Short, A practical nonlocal model for electron heat transport in laser plasmas, Phys. Fluids B, 3 (1991), 3092-3098. doi: 10.1063/1.859789.
[3]	M. Frank, D. Levermore and M. Shäfer, Diffusive corrections to $\mathbb P_N$ approximations, Multiscale Model. Simul., 9 (2011), 1-28.
[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), To appear.
[5]	B. Graille, "Modélisation de Mélanges Gazeux Réactifs Ionisés Dissipatifs," Ph.D. thesis, Ecole Polytechnique, 2004.
[6]	Y. Guo, The Landau equation in a periodic box, Comm. Math. Phys., 231 (2002), 391-434. doi: 10.1007/s00220-002-0729-9.
[7]	D. Levermore, "Boundary Conditions for Moment Closures," IPAM KT 2009, UCLA, CA, 2009.
[8]	D. Levermore, "Kinetic Theory, Gaussian Moment Closures, and Fluid Approximations," IPAM KT 2009, Culminating Retreat, Lake Arrowhead, CA, 2009.
[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), viii+120.
[10]	J.-F. Luciani and P. Mora, Resummation methods of the Chapman-Enskog expansion for a strongly inhomogeneous plasma, J. Stat. Phys., 43 (1986), 281-302. doi: 10.1007/BF01010582.
[11]	J.-F. Luciani and P. Mora, Nonlocal electron transport in laser created plasmas, Laser and Particle Beams, 12 (1994), 387-400. doi: 10.1017/S0263034600008247.
[12]	J.-F. Luciani, P. Mora and R. Pellat, Quasistatic heat front and delocalized heat flux, Phys. Fluids, 28 (1985), 835-845. doi: 10.1063/1.865052.
[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), 032701-1/032701-13.
[14]	M. Parisot, Finite volume schemes on unstructured grids for generalized Spitzer-Härm model, Tech. Rep., INRIA, 2011, In preparation.
[15]	E. J. Routh, "A Treatise on the Stability of a Given State of Motion," Macmillan and Co., 1877.
[16]	G. P. Schurtz, P. Nicolaï and M. Busquet, A nonlocal electron conduction model for multidimensional radiation hydrodynamics codes, Physics of Plasmas, 7 (2000), 4238-4250. doi: 10.1063/1.1289512.
[17]	I. P. Shkarofsky, Cartesian tensor expansion of the Fokker-Planck equation, Can. J. Phys., 41 (1963), 1753-1775. doi: 10.1139/p63-179.
[18]	L. Spitzer and R. Härm, Transport phenomena in a completely ionized gas, Phys. Rev., 89 (1953), 977-981. doi: 10.1103/PhysRev.89.977.

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), 103-108.
[2]	E. M. Epperlein and R. Short, A practical nonlocal model for electron heat transport in laser plasmas, Phys. Fluids B, 3 (1991), 3092-3098. doi: 10.1063/1.859789.
[3]	M. Frank, D. Levermore and M. Shäfer, Diffusive corrections to $\mathbb P_N$ approximations, Multiscale Model. Simul., 9 (2011), 1-28.
[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), To appear.
[5]	B. Graille, "Modélisation de Mélanges Gazeux Réactifs Ionisés Dissipatifs," Ph.D. thesis, Ecole Polytechnique, 2004.
[6]	Y. Guo, The Landau equation in a periodic box, Comm. Math. Phys., 231 (2002), 391-434. doi: 10.1007/s00220-002-0729-9.
[7]	D. Levermore, "Boundary Conditions for Moment Closures," IPAM KT 2009, UCLA, CA, 2009.
[8]	D. Levermore, "Kinetic Theory, Gaussian Moment Closures, and Fluid Approximations," IPAM KT 2009, Culminating Retreat, Lake Arrowhead, CA, 2009.
[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), viii+120.
[10]	J.-F. Luciani and P. Mora, Resummation methods of the Chapman-Enskog expansion for a strongly inhomogeneous plasma, J. Stat. Phys., 43 (1986), 281-302. doi: 10.1007/BF01010582.
[11]	J.-F. Luciani and P. Mora, Nonlocal electron transport in laser created plasmas, Laser and Particle Beams, 12 (1994), 387-400. doi: 10.1017/S0263034600008247.
[12]	J.-F. Luciani, P. Mora and R. Pellat, Quasistatic heat front and delocalized heat flux, Phys. Fluids, 28 (1985), 835-845. doi: 10.1063/1.865052.
[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), 032701-1/032701-13.
[14]	M. Parisot, Finite volume schemes on unstructured grids for generalized Spitzer-Härm model, Tech. Rep., INRIA, 2011, In preparation.
[15]	E. J. Routh, "A Treatise on the Stability of a Given State of Motion," Macmillan and Co., 1877.
[16]	G. P. Schurtz, P. Nicolaï and M. Busquet, A nonlocal electron conduction model for multidimensional radiation hydrodynamics codes, Physics of Plasmas, 7 (2000), 4238-4250. doi: 10.1063/1.1289512.
[17]	I. P. Shkarofsky, Cartesian tensor expansion of the Fokker-Planck equation, Can. J. Phys., 41 (1963), 1753-1775. doi: 10.1139/p63-179.
[18]	L. Spitzer and R. Härm, Transport phenomena in a completely ionized gas, Phys. Rev., 89 (1953), 977-981. doi: 10.1103/PhysRev.89.977.

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