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An Entropic Scheme for an Angular Moment Model for the Classical Fokker-Planck-Landau Equation of Electrons

Published online by Cambridge University Press:  03 June 2015

Jessy Mallet ,
Stéphane Brull  and
Bruno Dubroca
Jessy Mallet*
Affiliation:
Univ. Bordeaux, CELIA, UMR 5107, F- 33400 Talence, France Univ. Bordeaux, IMB, UMR 5251, F- 33400 Talence, France
Stéphane Brull*
Affiliation:
Univ. Bordeaux, IMB, UMR 5251, F- 33400 Talence, France
Bruno Dubroca*
Affiliation:
Univ. Bordeaux, CELIA, UMR 5107, F- 33400 Talence, France Univ. Bordeaux, IMB, UMR 5251, F- 33400 Talence, France
*
Corresponding author.Email:Stephane.Brull@math.u-bordeauxl.fr
Email:mallet@celia.u-bordeauxl.fr
Email:dubroca@celia.u-bordeauxl.fr
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Abstract

In plasma physics domain, the electron transport is described with the Fokker-Planck-Landau equation. The direct numerical solution of the kinetic equation is usually intractable due to the large number of independent variables. That is why we propose in this paper a new model whose derivation is based on an angular closure in the phase space and retains only the energy of particles as kinetic dimension. To find a solution compatible with physics conditions, the closure of the moment system is obtained under a minimum entropy principle. This model is proved to satisfy the fundamental properties like a H theorem. Moreover an entropic discretization in the velocity variable is proposed on the semi-discrete model. Finally, we validate on numerical test cases the fundamental properties of the full discrete model.

Keywords

35Q8465L0476Y05Entropy minimizationLandau-Fokker-Planck equationmoment systemsentropic scheme
Type
Research Article
Information
Communications in Computational Physics , Volume 15 , Issue 2 , February 2014 , pp. 422 - 450
Copyright
Copyright © Global Science Press Limited 2014

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References

[1]Anile, A.M., Pennisi, S. and Sammartino, M., A thermodynamical approach to Eddington factors. J. Math. Phys., 32 (1991) 544.Google Scholar
[2]Bathnagar, P.L., Gross, E.P. and Krook, M., A model for collision processes in gases. Phys. Rev., 94 (1954) 511–525.Google Scholar
[3]Berthon, C., Frank, M., Sarazin, C. and Turpault, R., Numerical methods for balance laws with space dependent flux: application to radiotherapy dose calculation. Comm. Comp. Phys., 10 (2011) 1184–1210.Google Scholar
[4]Buet, C. and Cordier, S., Numerical analysis of conservative and entropy schemes for the Fokker-Planck-Landau equation. SIAM J. Numer. Anal. 36, No. 3 (1999) 953–973.Google Scholar
[5]Buet, C., Cordier, S., Degond, P. and Lemou, M., Fast algorithms fot numerical, Conservative, and entropy approximations of the Fokker-Planck-Landau equation. J. Comp. Phys., 133 (1997) 310–322.Google Scholar
[6]Buet, C. and Dellacherie, S., On the Chang and Cooper scheme applied to a linear Fokker-Planck equation. Comm. in Math. Sc., 8 (2010) 1079–1090.Google Scholar
[7]Buet, C., Dellacherie, S. and Sentis, R., Résolution numérique d’une équation de Fokker-Planck ionique avec température électronique. Acad, C. R.Sci. Paris Serie I Math., 327 (1998) 93–98.Google Scholar
[8]Chen, F., Introduction to Plasma Physics and Controlled Fusion. Plenum Press, New York, 1984.Google Scholar
[9]Crispel, P., Degond, P. and Vignal, M.-H., A plasma expansion model based on the full Euler-Poisson system. Math. Mod. Meth. Appl. Sci., 17 (2007) 1129–1158.Google Scholar
[10]Crispel, P., Degond, P. and Vignal, M.-H., Quasi-neutral fluid models for current-carrying plasmas. J. Comp. Phys., 205 (2005) 408–438.Google Scholar
[11]Crispel, P., Degond, P. and Vignal, M.-H., An asymptotic preserving scheme for the two-fluid Euler-Poisson model in the quasi-neutral limit. J. Comp. Phys., 223 (2007) 208–234.CrossRefGoogle Scholar
[12]Crouseilles, N. and Filbet, F., Numerical approximation of collisional plasmas by high order methods. J. Comp. Phys., 201 (2004) 546–572.Google Scholar
[13]Degond, P., Lucquin-Desreux, B., An entropy scheme for the fokker-planck collision operator of plasma kinetic theory. Nummer. Math, 68 (1994) 239–262.Google Scholar
[14]Delcroix, J.L. and Bers, A., Physique des Plasmas. InterEditions, Paris, V. 2 (1994).Google Scholar
[15]Dellacherie, S., Sur un schéma numerique semi-discret appliqué a un opérateur de Fokker-Planck isotrope. Acad, C. R.Sci. Paris Série I Math., 328 (1999) 1219–1224.Google Scholar
[16]Dellacherie, S., Numerical resolution of an ion-electron collision operator in axisymmetrical geometry. Transp. Theory and Stat. Phys., 31 (2002) 397–429.Google Scholar
[17]Dellacherie, S., Buet, C. and Sentis, R., Numerical solution of an ionic Fokker-Planck equation with electronic temperature. SIAM J. Numer. Anal., 39 (2001) 1219–1253.Google Scholar
[18]Dellacherie, S., Contribution a l’analyse et a la simulation numérique des équations cinétiques décrivant un plasma chaud. PhD thesis, University Denis Diderot Paris VII, 1998.Google Scholar
[19]Dubroca, B. and Feugeas, J.L., Entropic moment closure hierarchy for the radiative transfert equation. Acad, C. R.Sci. Paris Ser. I, 329 (1999) 915–920.Google Scholar
[20]Duclous, R., Modélisation et Simulation Numérique multi-échelle du transport cinétique électronique. PhD thesis, University Bordeaux 1, 2009.Google Scholar
[21]Duclous, R., Dubroca, B., Filbet, F. and Tikhonchuk, V., High order resolution of the Maxwell-Fokker-Planck-Landau model intended for ICF application. J. Comp. Phys., 228 (2009) 5072–5100.Google Scholar
[22]Frank, M., Dubroca, B. and Klar, A., Partial moment entropy approximation to radiative transfer. J. Comp. Phys., 218 (2006) 1–18.Google Scholar
[23]Grad, H., On Kinetic theory of the rarefied gases. Comm. Pure and Appl. Math., Vol.II (1949) 331–407.Google Scholar
[24]Harten, A., Lax, P. D. and Van Leer, B., On upstream differencing and Godunov-type schemes for hyperbolic conservation laws. Siam Review, 25 (1983) 35–61.Google Scholar
[25]Hauck, C. and McClarren, R., Positive PN closures. SIAM J. Sci. Comp., 32 (2010) 2603–2626.Google Scholar
[26]Junk, M., Maximum entropy for reduced moment problems. Math. Mod. Meth. in Appl. Sci., 10 (2000) 1001–1025.Google Scholar
[27]Kingham, R.J. and Bell, A.R., An implicit Vlasov-Fokker-Planck code to model non-local electron transport in 2-D with magnetic fields. J. Comp. Phys., 194 (2004) 1–34.Google Scholar
[28]Laval, G., La Fusion Thermonucléaire Inertielle par Laser. Eyrolles, P. 1, V. 1, editors Daufray, R. et Watteau, J.P., Paris, France, 1994.Google Scholar
[29]Levermore, D., Moment closure hierarchies for kinetic theories. J. Stat. Phys., 83 (1996).Google Scholar
[30]Minerbo, G.N., Maximum entropy Eddington factors. J. Quant. Spectrosc. Radiat. Transfer, 20 (1978) 541.Google Scholar
[31]Schneider, J., Entropic approximation in kinetic theory. ESAIM: M2AN, 38 (2004) 541–561.Google Scholar
[32]Sentoku, Y. and Kemp, A.J., Numerical method for particle simulations at extreme densities and temperatures: weighted particles, relativistic collisions and reduced currents. J. Comp. Phys., 227 (2008) 6846–6861.CrossRefGoogle Scholar