# American Institute of Mathematical Sciences

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

 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.
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

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##### 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
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