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September  2015, 8(3): 533-558. doi: 10.3934/krm.2015.8.533

General moment system for plasma physics based on minimum entropy principle

1. 

Univ. Bordeaux, CELIA, UMR 5107, F- 33400 TALENCE, France, France

2. 

Université de Bordeaux 1, 351, cours de la Libération, 33405 TALENCE Cedex

Received  June 2014 Revised  April 2015 Published  June 2015

In plasma physics domain, the electrons transport can be described from kinetic and hydrodynamical models. Both methods present disadvantages and thus cannot be considered in practical computations for Inertial Confinement Fusion (ICF). That is why we propose in this paper a new model which is intermediate between these two descriptions. More precisely, the derivation of such models is based on an angular closure in the phase space and retains only the energy of particles as a kinetic variable. The closure of the moment system is obtained from a minimum entropy principle. The resulting continuous model is proved to satisfy fundamental properties. Moreover the model is discretized w.r.t the energy variable and the semi-discretized scheme is shown to satisfy conservation properties and entropy decay.
Citation: Jessy Mallet, Stéphane Brull, Bruno Dubroca. General moment system for plasma physics based on minimum entropy principle. Kinetic and Related Models, 2015, 8 (3) : 533-558. doi: 10.3934/krm.2015.8.533
References:
[1]

G. Alldredge, C. Hauck and A. L. Tits, High-order entropy-based closures for linear transport in slab geometry II: A computational study of the optimization problem, SIAM J. Sci. Comput., 34 (2012), B361-B391. doi: 10.1137/11084772X.

[2]

P. L. Bathnagar, E. P. Gross and M. Krook, A model for collision processes in gases, Phys. Rev., 94 (1954), 511-525.

[3]

Yu. A. Berezin, V. N. Khudick and M. S. Pekker, Conservative finite-difference schemes for the Fokker-Planck equation not violating the law of an increasing entropy, J. Comput. Phys., 69 (1987), 163-174. doi: 10.1016/0021-9991(87)90160-4.

[4]

S. Brull, Existence of solutions to discrete coagulation-fragmentation equations with transport and diffusion, Annales de la Faculté de Toulouse, 18 (2008), 1-22.

[5]

S. Brull, P. Degond, F. Deluzet and A. Mouton, An asymptotic preserving scheme for a bi-fluid Euler-Lorentz system, Kinetic and Related Models, 4 (2011), 991-1023.

[6]

C. Buet and S. Cordier, Conservative and entropy decaying numerical scheme for the isotropic Fokker-Planck-Landau equation, J. Comput. Phys., 145 (1998), 228-245. doi: 10.1006/jcph.1998.6015.

[7]

C. Buet and S. Cordier, Numerical analysis of conservative and entropy schemes for the Fokker-Planck-Landau equation, SIAM J. Numer Anal., 36 (1999), 953-973. doi: 10.1137/S0036142997322102.

[8]

C. Buet, S. Cordier, P. Degond and M. Lemou, Fast algorithms fot numerical, conservative, and entropy approximations of the Fokker-Planck-Landau equation, J. Comput. Phys., 133 (1997), 310-322. doi: 10.1006/jcph.1997.5669.

[9]

K. M. Case and P. F. Zweifel, Linear Transport Theory, Addison-Wesley Publishing Company, Reading, Massachusetts, 1967.

[10]

F. Chen, Introduction to Plasma Physics and Controlled Fusion, Plenum Press, New York, 1984. doi: 10.1007/978-1-4757-5595-4.

[11]

P. Crispel, P. Degond and M. H. Vignal, A plasma expansion model based on the full Euler-Poisson system, Math. Models Methods Appl. Sci., 17 (2007), 1129-1158. doi: 10.1142/S0218202507002224.

[12]

P. Crispel, P. Degond and M. H. Vignal, Quasi-neutral fluid models for current-carrying plasmas, J. Comput. Phys., 205 (2005), 408-438. doi: 10.1016/j.jcp.2004.11.011.

[13]

P. Crispel, P. Degond and M.-H. Vignal, An asymptotic preserving scheme for the two-fluid Euler-Poisson model in the quasi-neutral limit, J. Comput. Phys., 223 (2007), 208-234. doi: 10.1016/j.jcp.2006.09.004.

[14]

N. Crouseilles and F. Filbet, Numerical approximation of collisional plasmas by high order methods, J. Comput. Phys., 201 (2004), 546-572. doi: 10.1016/j.jcp.2004.06.007.

[15]

J. L. Delcroix and A. Bers, Physique des Plasmas, InterEditions, Paris, 1994.

[16]

S. Dellacherie, Contribution à L'analyse et à la Simulation Numérique des Équations Cinétiques Décrivant un Plasma Chaud, PhD thesis, Université Denis Diderot Paris VII, 1998.

[17]

S. Dellacherie, Sur un schéma numerique semi-discret appliqué à un opérateur de Fokker-Planck isotrope. C. R. Acad. Sci. Paris Série I Math., 328 (1999), 1219-1224. doi: 10.1016/S0764-4442(99)80443-1.

[18]

S. Dellacherie, C. Buet and R. Sentis, Numerical solution of an ionic Fokker-Planck equation with electronic temperature, SIAM J. Numer. Anal., 39 (2001), 1219-1253. doi: 10.1137/S0036142999359669.

[19]

B. Dubroca and J. L. Feugeas, Entropic moment closure hierarchy for the radiative transfert equation, C. R. Acad. Sci. Paris Séries I, 329 (1999), 915-920. doi: 10.1016/S0764-4442(00)87499-6.

[20]

R. Duclous, B. Dubroca, F. Filbet and V. Tikhonchuk, High order resolution of the Maxwell-Fokker-Planck-Landau model intended for ICF application, J. Comput. Phys., 228 (2009), 5072-5100. doi: 10.1016/j.jcp.2009.04.005.

[21]

M. Frank, B. Dubroca and A. Klar, Partial moment entropy approximation to radiative transfer, J. Comp. Phys., 218 (2006), 1-18. doi: 10.1016/j.jcp.2006.01.038.

[22]

H. Grad, On kinetic theory of the rarefied gases, Comm. Pure and Appl. Math., 2 (1949), 331-407. doi: 10.1002/cpa.3160020403.

[23]

S. Guisset, S. Brull, E. d'Humière, B. Dubroca, S. Karpov and I. Potapenko, Asymptotic-preserving scheme for the $M_1$-Maxwell system in the quasi-neutral regime, to appear in Comm. Comput. Phys.

[24]

C. Hauck and R. McClarren, Positive $P_N$ closure, SIAM J. Sci. Comput., 32 (2010), 2603-2626. doi: 10.1137/090764918.

[25]

C. Hauck, High-order entropy-based closures for linear transport in slab geometries, Commun. Math. Sci., 9 (2011), 187-205. doi: 10.4310/CMS.2011.v9.n1.a9.

[26]

M. Junk, Domain of definition of Levermore's five-moment system, J. Stat. Phys., 93 (1998), 1143-1167. doi: 10.1023/B:JOSS.0000033155.07331.d9.

[27]

M. Junk, Maximum entropy for reduced moment problems, Math. Models Methods Appl. Sci., 10 (2000), 1001-1025. doi: 10.1142/S0218202500000513.

[28]

E. W. Larsen and G. C. Pomraning, The $P_N$ theory as an asymptotic limit of transport theory in planar geometry - I: Analysis, Nucl. Sci. Eng., {109} (1991), 49-75.

[29]

E. W. Larsen and G. C. Pomraning, The $P_N$ theory as an asymptotic limit of transport theory in planar geometry - II: Numerical results, Nucl. Sci. Eng., 109 (1991), 76-85.

[30]

D. Levermore, Moment closure hierarchies for kinetic theories, J. Stat. Phys., 83 (1996), 1021-1065. doi: 10.1007/BF02179552.

[31]

R. G. McClarren, J. P. Holloway and T. A. Brunner, On solutions to the $P_n$ equations for thermal radiative transfert, J. Comput. Phys., 227 (2008), 2864-2885. doi: 10.1016/j.jcp.2007.11.027.

[32]

R. G. McClarren, J. P. Holloway and T. A. Brunner, Analytic $P_1$ solutions for time dependent, thermal radiative transfert in several geometries, J. Quant. Spec. Rad. Transfer, 109 (2008), 389-403.

[33]

J. Mallet, S. Brull and B. Dubroca, An entropic scheme for an angular moment model for the classical Maxwell-Fokker-Planck equation of electrons, Comm. Comput. Phys., 15 (2014), 422-450.

[34]

G. N. Minerbo, Maximum entropy Eddington factors, J. Quant. Spectrosc. Radiat. Transfer, 20 (1978), 541-545. doi: 10.1016/0022-4073(78)90024-9.

[35]

J. Schneider, Entropic approximation in kinetic theory, ESAIM: M2AN, 38 (2004), 541-561. doi: 10.1051/m2an:2004025.

[36]

Y. Sentoku and A. J. Kemp, Numerical method for particle simulations at extreme densities and temperatures: Weighted particles, relativistic collisions and reduced currents, J. Comput. Phys., 227 (2008), 6846-6861. doi: 10.1016/j.jcp.2008.03.043.

[37]

M. Tzoufras, A. R. Bell, P. A. Norreys and F. S. Tsung, A Vlasov-Fokker-Planck code for high energy density physics, J. Comput. Phys., 230 (2011), 6475-6494. doi: 10.1016/j.jcp.2011.04.034.

show all references

References:
[1]

G. Alldredge, C. Hauck and A. L. Tits, High-order entropy-based closures for linear transport in slab geometry II: A computational study of the optimization problem, SIAM J. Sci. Comput., 34 (2012), B361-B391. doi: 10.1137/11084772X.

[2]

P. L. Bathnagar, E. P. Gross and M. Krook, A model for collision processes in gases, Phys. Rev., 94 (1954), 511-525.

[3]

Yu. A. Berezin, V. N. Khudick and M. S. Pekker, Conservative finite-difference schemes for the Fokker-Planck equation not violating the law of an increasing entropy, J. Comput. Phys., 69 (1987), 163-174. doi: 10.1016/0021-9991(87)90160-4.

[4]

S. Brull, Existence of solutions to discrete coagulation-fragmentation equations with transport and diffusion, Annales de la Faculté de Toulouse, 18 (2008), 1-22.

[5]

S. Brull, P. Degond, F. Deluzet and A. Mouton, An asymptotic preserving scheme for a bi-fluid Euler-Lorentz system, Kinetic and Related Models, 4 (2011), 991-1023.

[6]

C. Buet and S. Cordier, Conservative and entropy decaying numerical scheme for the isotropic Fokker-Planck-Landau equation, J. Comput. Phys., 145 (1998), 228-245. doi: 10.1006/jcph.1998.6015.

[7]

C. Buet and S. Cordier, Numerical analysis of conservative and entropy schemes for the Fokker-Planck-Landau equation, SIAM J. Numer Anal., 36 (1999), 953-973. doi: 10.1137/S0036142997322102.

[8]

C. Buet, S. Cordier, P. Degond and M. Lemou, Fast algorithms fot numerical, conservative, and entropy approximations of the Fokker-Planck-Landau equation, J. Comput. Phys., 133 (1997), 310-322. doi: 10.1006/jcph.1997.5669.

[9]

K. M. Case and P. F. Zweifel, Linear Transport Theory, Addison-Wesley Publishing Company, Reading, Massachusetts, 1967.

[10]

F. Chen, Introduction to Plasma Physics and Controlled Fusion, Plenum Press, New York, 1984. doi: 10.1007/978-1-4757-5595-4.

[11]

P. Crispel, P. Degond and M. H. Vignal, A plasma expansion model based on the full Euler-Poisson system, Math. Models Methods Appl. Sci., 17 (2007), 1129-1158. doi: 10.1142/S0218202507002224.

[12]

P. Crispel, P. Degond and M. H. Vignal, Quasi-neutral fluid models for current-carrying plasmas, J. Comput. Phys., 205 (2005), 408-438. doi: 10.1016/j.jcp.2004.11.011.

[13]

P. Crispel, P. Degond and M.-H. Vignal, An asymptotic preserving scheme for the two-fluid Euler-Poisson model in the quasi-neutral limit, J. Comput. Phys., 223 (2007), 208-234. doi: 10.1016/j.jcp.2006.09.004.

[14]

N. Crouseilles and F. Filbet, Numerical approximation of collisional plasmas by high order methods, J. Comput. Phys., 201 (2004), 546-572. doi: 10.1016/j.jcp.2004.06.007.

[15]

J. L. Delcroix and A. Bers, Physique des Plasmas, InterEditions, Paris, 1994.

[16]

S. Dellacherie, Contribution à L'analyse et à la Simulation Numérique des Équations Cinétiques Décrivant un Plasma Chaud, PhD thesis, Université Denis Diderot Paris VII, 1998.

[17]

S. Dellacherie, Sur un schéma numerique semi-discret appliqué à un opérateur de Fokker-Planck isotrope. C. R. Acad. Sci. Paris Série I Math., 328 (1999), 1219-1224. doi: 10.1016/S0764-4442(99)80443-1.

[18]

S. Dellacherie, C. Buet and R. Sentis, Numerical solution of an ionic Fokker-Planck equation with electronic temperature, SIAM J. Numer. Anal., 39 (2001), 1219-1253. doi: 10.1137/S0036142999359669.

[19]

B. Dubroca and J. L. Feugeas, Entropic moment closure hierarchy for the radiative transfert equation, C. R. Acad. Sci. Paris Séries I, 329 (1999), 915-920. doi: 10.1016/S0764-4442(00)87499-6.

[20]

R. Duclous, B. Dubroca, F. Filbet and V. Tikhonchuk, High order resolution of the Maxwell-Fokker-Planck-Landau model intended for ICF application, J. Comput. Phys., 228 (2009), 5072-5100. doi: 10.1016/j.jcp.2009.04.005.

[21]

M. Frank, B. Dubroca and A. Klar, Partial moment entropy approximation to radiative transfer, J. Comp. Phys., 218 (2006), 1-18. doi: 10.1016/j.jcp.2006.01.038.

[22]

H. Grad, On kinetic theory of the rarefied gases, Comm. Pure and Appl. Math., 2 (1949), 331-407. doi: 10.1002/cpa.3160020403.

[23]

S. Guisset, S. Brull, E. d'Humière, B. Dubroca, S. Karpov and I. Potapenko, Asymptotic-preserving scheme for the $M_1$-Maxwell system in the quasi-neutral regime, to appear in Comm. Comput. Phys.

[24]

C. Hauck and R. McClarren, Positive $P_N$ closure, SIAM J. Sci. Comput., 32 (2010), 2603-2626. doi: 10.1137/090764918.

[25]

C. Hauck, High-order entropy-based closures for linear transport in slab geometries, Commun. Math. Sci., 9 (2011), 187-205. doi: 10.4310/CMS.2011.v9.n1.a9.

[26]

M. Junk, Domain of definition of Levermore's five-moment system, J. Stat. Phys., 93 (1998), 1143-1167. doi: 10.1023/B:JOSS.0000033155.07331.d9.

[27]

M. Junk, Maximum entropy for reduced moment problems, Math. Models Methods Appl. Sci., 10 (2000), 1001-1025. doi: 10.1142/S0218202500000513.

[28]

E. W. Larsen and G. C. Pomraning, The $P_N$ theory as an asymptotic limit of transport theory in planar geometry - I: Analysis, Nucl. Sci. Eng., {109} (1991), 49-75.

[29]

E. W. Larsen and G. C. Pomraning, The $P_N$ theory as an asymptotic limit of transport theory in planar geometry - II: Numerical results, Nucl. Sci. Eng., 109 (1991), 76-85.

[30]

D. Levermore, Moment closure hierarchies for kinetic theories, J. Stat. Phys., 83 (1996), 1021-1065. doi: 10.1007/BF02179552.

[31]

R. G. McClarren, J. P. Holloway and T. A. Brunner, On solutions to the $P_n$ equations for thermal radiative transfert, J. Comput. Phys., 227 (2008), 2864-2885. doi: 10.1016/j.jcp.2007.11.027.

[32]

R. G. McClarren, J. P. Holloway and T. A. Brunner, Analytic $P_1$ solutions for time dependent, thermal radiative transfert in several geometries, J. Quant. Spec. Rad. Transfer, 109 (2008), 389-403.

[33]

J. Mallet, S. Brull and B. Dubroca, An entropic scheme for an angular moment model for the classical Maxwell-Fokker-Planck equation of electrons, Comm. Comput. Phys., 15 (2014), 422-450.

[34]

G. N. Minerbo, Maximum entropy Eddington factors, J. Quant. Spectrosc. Radiat. Transfer, 20 (1978), 541-545. doi: 10.1016/0022-4073(78)90024-9.

[35]

J. Schneider, Entropic approximation in kinetic theory, ESAIM: M2AN, 38 (2004), 541-561. doi: 10.1051/m2an:2004025.

[36]

Y. Sentoku and A. J. Kemp, Numerical method for particle simulations at extreme densities and temperatures: Weighted particles, relativistic collisions and reduced currents, J. Comput. Phys., 227 (2008), 6846-6861. doi: 10.1016/j.jcp.2008.03.043.

[37]

M. Tzoufras, A. R. Bell, P. A. Norreys and F. S. Tsung, A Vlasov-Fokker-Planck code for high energy density physics, J. Comput. Phys., 230 (2011), 6475-6494. doi: 10.1016/j.jcp.2011.04.034.

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