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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 & 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). doi: 10.1137/11084772X. Google Scholar

[2]

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

[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. doi: 10.1016/0021-9991(87)90160-4. Google Scholar

[4]

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

[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. Google Scholar

[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. doi: 10.1006/jcph.1998.6015. Google Scholar

[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. doi: 10.1137/S0036142997322102. Google Scholar

[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. doi: 10.1006/jcph.1997.5669. Google Scholar

[9]

K. M. Case and P. F. Zweifel, Linear Transport Theory,, Addison-Wesley Publishing Company, (1967). Google Scholar

[10]

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

[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. doi: 10.1142/S0218202507002224. Google Scholar

[12]

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

[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. doi: 10.1016/j.jcp.2006.09.004. Google Scholar

[14]

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

[15]

J. L. Delcroix and A. Bers, Physique des Plasmas,, InterEditions, (1994). Google Scholar

[16]

S. Dellacherie, Contribution à L'analyse et à la Simulation Numérique des Équations Cinétiques Décrivant un Plasma Chaud,, PhD thesis, (1998). Google Scholar

[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. doi: 10.1016/S0764-4442(99)80443-1. Google Scholar

[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. doi: 10.1137/S0036142999359669. Google Scholar

[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. doi: 10.1016/S0764-4442(00)87499-6. Google Scholar

[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. doi: 10.1016/j.jcp.2009.04.005. Google Scholar

[21]

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

[22]

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

[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., (). Google Scholar

[24]

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

[25]

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

[26]

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

[27]

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

[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. Google Scholar

[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. Google Scholar

[30]

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

[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. doi: 10.1016/j.jcp.2007.11.027. Google Scholar

[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. Google Scholar

[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. Google Scholar

[34]

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

[35]

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

[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. doi: 10.1016/j.jcp.2008.03.043. Google Scholar

[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. doi: 10.1016/j.jcp.2011.04.034. Google Scholar

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). doi: 10.1137/11084772X. Google Scholar

[2]

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

[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. doi: 10.1016/0021-9991(87)90160-4. Google Scholar

[4]

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

[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. Google Scholar

[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. doi: 10.1006/jcph.1998.6015. Google Scholar

[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. doi: 10.1137/S0036142997322102. Google Scholar

[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. doi: 10.1006/jcph.1997.5669. Google Scholar

[9]

K. M. Case and P. F. Zweifel, Linear Transport Theory,, Addison-Wesley Publishing Company, (1967). Google Scholar

[10]

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

[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. doi: 10.1142/S0218202507002224. Google Scholar

[12]

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

[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. doi: 10.1016/j.jcp.2006.09.004. Google Scholar

[14]

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

[15]

J. L. Delcroix and A. Bers, Physique des Plasmas,, InterEditions, (1994). Google Scholar

[16]

S. Dellacherie, Contribution à L'analyse et à la Simulation Numérique des Équations Cinétiques Décrivant un Plasma Chaud,, PhD thesis, (1998). Google Scholar

[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. doi: 10.1016/S0764-4442(99)80443-1. Google Scholar

[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. doi: 10.1137/S0036142999359669. Google Scholar

[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. doi: 10.1016/S0764-4442(00)87499-6. Google Scholar

[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. doi: 10.1016/j.jcp.2009.04.005. Google Scholar

[21]

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

[22]

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

[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., (). Google Scholar

[24]

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

[25]

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

[26]

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

[27]

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

[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. Google Scholar

[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. Google Scholar

[30]

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

[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. doi: 10.1016/j.jcp.2007.11.027. Google Scholar

[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. Google Scholar

[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. Google Scholar

[34]

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

[35]

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

[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. doi: 10.1016/j.jcp.2008.03.043. Google Scholar

[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. doi: 10.1016/j.jcp.2011.04.034. Google Scholar

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