This work is devoted to the derivation of an asymptotic-preserving scheme for the electronic model in the diffusive regime. The case without electric field and the homogeneous case are studied. The derivation of the scheme is based on an approximate Riemann solver where the intermediate states are chosen consistent with the integral form of the approximate Riemann solver. This choice can be modified to enable the derivation of a numerical scheme which also satisfies the admissible conditions and is well-suited for capturing steady states. Moreover, it enjoys asymptotic-preserving properties and handles the diffusive limit recovering the correct diffusion equation. Numerical tests cases are presented, in each case, the asymptotic-preserving scheme is compared to the classical [A. Harten, P.D. Lax and B. Van Leer, SIAM Rev. 25 (1983) 35–61.] scheme usually used for the electronic model. It is shown that the new scheme gives comparable results with respect to the scheme in the classical regime. On the contrary, in the diffusive regime, the asymptotic-preserving scheme coincides with the expected diffusion equation, while the scheme suffers from a severe lack of accuracy because of its unphysical numerical viscosity.
Mots-clés : Electronic M1moment model, approximate Riemann solvers, Godunov type schemes, asymptotic preserving schemes, diffusive limit, plasma physics
@article{M2AN_2017__51_5_1805_0,
author = {Guisset, S\'ebastien and Brull, St\'ephane and D{\textquoteright}Humi\`eres, Emmanuel and Dubroca, Bruno},
title = {Asymptotic-preserving well-balanced scheme for the electronic $M_{1}$ model in the diffusive limit: {Particular} cases},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {1805--1826},
publisher = {EDP-Sciences},
volume = {51},
number = {5},
year = {2017},
doi = {10.1051/m2an/2016079},
mrnumber = {3731550},
zbl = {1406.78021},
language = {en},
url = {http://archive.numdam.org/articles/10.1051/m2an/2016079/}
}
TY - JOUR
AU - Guisset, Sébastien
AU - Brull, Stéphane
AU - D’Humières, Emmanuel
AU - Dubroca, Bruno
TI - Asymptotic-preserving well-balanced scheme for the electronic $M_{1}$ model in the diffusive limit: Particular cases
JO - ESAIM: Mathematical Modelling and Numerical Analysis
PY - 2017
SP - 1805
EP - 1826
VL - 51
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PB - EDP-Sciences
UR - http://archive.numdam.org/articles/10.1051/m2an/2016079/
DO - 10.1051/m2an/2016079
LA - en
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Guisset, Sébastien; Brull, Stéphane; D’Humières, Emmanuel; Dubroca, Bruno. Asymptotic-preserving well-balanced scheme for the electronic $M_{1}$ model in the diffusive limit: Particular cases. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 51 (2017) no. 5, pp. 1805-1826. doi : 10.1051/m2an/2016079. http://archive.numdam.org/articles/10.1051/m2an/2016079/
and , Upwind methods for hyperbolic conservation laws with source terms. Comput. Fluids 23 (1994) 1049–1071. | DOI | MR | Zbl
, and , 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 | MR | Zbl
E. Audit, P. Charrier, J.-P. Chièze and B. Dubroca, A radiation hydrodynamics scheme valid from the transport to the diffusion limit. Preprint (2002). | arXiv
R. Balescu. Transport Processes in Plasma, Vol. 1. Elsevier, Amsterdam (1988).
, and , Uniformly stable numerical schemes for the Boltzmann equation preserving the compressible Navier Stokes asymptotics. J. Comput. Phys. 227 (2008) 3781–3803. | DOI | MR | Zbl
, and , An asymptotic preserving relaxation scheme for a moment model of radiative transfer. C.R. Acad. Sci. Paris, Ser. I 344 (2007) 467–472. | DOI | MR | Zbl
, and , An HLLC Scheme to Solve The M1 Model of Radiative Transfer in Two Space Dimensions. J. Scient. Comput. 31 (2007) 347–389. | DOI | MR | Zbl
and , Asymptotic preserving HLL schemes. Numer. Methods Partial Differ. Equ. 27 (2011) 1396–1422. | DOI | MR | Zbl
F. Bouchut, Nonlinear Stability of Finite Volume Methods for Hyperbolic Conservation Laws, and Well-Balanced Schemes for sources. Frontiers in Mathematics series. Birkhauser (2004). | MR | Zbl
and , A subsonic-well-balanced reconstruction scheme for shallow water flows. SIAM J. Numer. Anal. 48 (2010) 1733–1758. | DOI | MR | Zbl
S.I. Braginskii, Reviews of Plasma Physics. In vol. 1. Edited by M.A Leontovich. Consultants Bureau New York (1965) 205.
, , and , Nonlocal heat wave propagation due to skin layer plasma heating by short laser pulses. Comput. Phys. Commun. 164 (2004) 67. | DOI
and , Asymptotic Preserving Scheme and Numerical Methods for Radiative Hydrodynamic Models. C.R. Acad. Sci. Paris, Tome, Série I 338 (2004) 951–956. | DOI | MR | Zbl
, , and , Diffusion limit of the Lorentz model: asymptotic preserving schemes. ESAIM: M2AN 36 (2002) 631–655. | DOI | Numdam | MR | Zbl
and , Asymptotic analysis of fluid models for the coupling of radiation and hydrodynamics. J. Quant. Spectrosc. Radiat. Transfer 85 (2004) 385–418. | DOI
and , Asymptotic preserving and positive schemes for radiation hydrodynamics. J. Comptut. Phys. 215 (2006) 717–740. | DOI | MR | Zbl
, and , Uniformly accurate schemes for hyperbolic systems with relaxation. SIAM J. Numer. Anal. 34 (1997) 246–281. | DOI | MR | Zbl
and , Un schéma équilibre adapté au modèle d’atmosphère avec termes de gravité. C.R. Acad. Sci. , Ser. I 318 (1994) 73–76. | MR | Zbl
, , and , Numerical schemes of diffusion asymptotics and moment closures for kinetic equations. J. Sci. Comput. 36 (2008) 113–149. | DOI | MR | Zbl
, Numerical approximations of the 10-moment Gaussian closure. Math. Comput. 75 (2006) 1809–1831. | DOI | MR | Zbl
, , , and , Godunov-type schemes for hyperbolic systems with parameter-dependent source. The case of Euler system with friction. Math. Models Methods Appl. Sci. 20 (2010) 2109–2166. MR 2740716 (2011m:65179). | DOI | MR | Zbl
, and , Well-balanced time implicit formulation of relaxation schemes for the euler equations. SIAM J. Sci. Comput. 30 (2008) 394–415. | DOI | MR | Zbl
S. Chapman and T.G. Cowling, The Mathematical Theory of Non-Uniform Gases. Cambridge University Press, Cambridge, England (1995). | Zbl
P. Charrier, B. Dubroca, G. Duffa and R. Turpault, Multigroup model for radiating flows during atmospheric hypersonic re-entry. Proceedings of International Workshop on Radiation of High Temperature Gases in Atmospheric Entry. Lisbonne, Portugal (2003) 103–110.
and , Numerical passage from kinetic to fluid equations. SIAM J. Numer. Anal. 28 (1991) 26–42. | DOI | MR | Zbl
, and , Quasi-neutral fluid models for current-carrying plasmas. J. Comput. Phys. 205 (2005) 408–438. | DOI | MR | Zbl
, and , An asymptotic preserving scheme for the two-fluid Euler-Poisson model in the quasi-neutral limit. J. Comput. Phys. 223 (2007) 208–234. | DOI | MR | Zbl
, , , and , Asymptotic-Preserving Particle-In-Cell method for the Vlasov-Poisson system near quasineutrality. J. Comput. Phys. 229 (2010) 5630–5652. | DOI | MR | Zbl
, , and , Numerical approximation of the Euler-Poisson-Boltzmann model in the quasineutral limit. C.R. Acad. Sci. Paris, Ser. I 341 (2005) 323–328. | Zbl
, and , Numerical approximation of the Euler-Maxwell model in the quasineutral limit. J. Comput. Phys. 231 (2012) 1917–1946. | DOI | MR | Zbl
, , and , Well-balanced schemes to capture non-explicit steady states. Part 1: Ripa model. Math. Comput. 85 (2016) 1571–1602. | DOI | MR | Zbl
, , , , and , Parametric instabilities of electromagnetic waves in plasmas. Phys. Fluids 17 (1974) 778. | DOI
, and , Angular moment model for the Fokker-Planck equation. Europ. Phys. J. D 60 (2010) 301. | DOI
and , étude théorique et numérique d’une hiéarchie de modèles aux moments pour le transfert radiatif. C.R. Acad. Sci. Paris, Ser. I 329 (1999) 915–920. | DOI | MR | Zbl
and , Entropic moment closure hierarchy for the radiative transfert equation. C.R. Acad. Sci. Paris Ser. I 329 (1999) 915. | MR | Zbl
and , Phys. Fluids B 4 (1992) 2211. | DOI
, Positive and entropy stable Godunov-type schemes for gas dynamics and MHD equations in Lagrangian or Eulerian coordinates. Numer. Math. 94 (2003) 673–713. MR 1990589 (2004e:65094). | DOI | MR | Zbl
and , An asymptotic-preserving well-balanced scheme for the hyperbolic heat equations. C.R. Math. Acad. Sci. Paris 334 (2002) 337–342. | DOI | MR | Zbl
and , Space localization and well-balanced schemes for discrete kinetic models in diffusive regimes. SIAM J. Numer. Anal. 41 (2003) 641–658. | DOI | MR | Zbl
. On the kinetic theory of rarefied gases. Commun. Pure Appl. Math. 2 (1949) 331–407. | DOI | MR | Zbl
and , A well-balanced scheme for the numerical processing of source terms in hyperbolic equations. SIAM J. Numer. Anal. 33 (1996) 1–16. | DOI | MR | Zbl
and , Towards physically-realizable and hyperbolic moment closures for kinetic theory. Continuum Mech. Thermodyn. 21 (2009) 467–493. | DOI | MR | Zbl
, , , , and , Asymptotic-preserving scheme for the Fokker-Planck-Landau-Maxwell system in the quasi-neutral regime. Commun. Comput. Phys. 19 (2016) 301–328. | DOI | MR | Zbl
, and , On upstream differencing and Godunov-type schemes for hyperbolic conservation laws. SIAM Rev. 25 (1983) 35–61. | DOI | MR | Zbl
and , Simulation of fluid and particles flows: asymptotic preserving schemes for bubbling and flowing regimes. J. Comput. Phys. 227 (2008) 7929–7951. | DOI | MR | Zbl
, Efficient Asymptotic-Preserving (AP) Schemes for Some Multiscale Kinetic Equations. SIAM J. Sci. Comput. 21 (1999) 441–454. | DOI | MR | Zbl
and , Fully discrete numerical transfer in diffusive regimes. Trans. Theory Stat. Phys. 22 (1993) 739–791. | DOI | MR | Zbl
and , The discrete-ordinate method in diffusive regimes. Trans. Theory Stat. Phys. 20 (1991) 413–439. | DOI | MR | Zbl
and , Numerical Schemes for Hyperbolic Conservation Laws with Stiff Relaxation Terms. J. Comput. Phys. 126 (1996) 449-467. | DOI | MR | Zbl
and , Discretization of the multiscale semiconductor Boltzmann equation by diffusive relaxation scheme. J. Comput. Phys. 161 (2000) 312–330. | DOI | MR | Zbl
, and , Uniformly accurate diffusive relaxation schemes for multiscale transport equations. SIAM J. Numer. Anal. 38 (2000) 913–936. | DOI | MR | Zbl
and , The relaxation scheme for systems of conservation laws in arbitrary space dimension. Commun. Pure Appl. Math. 45 (1995) 235–276. | DOI | MR | Zbl
, An asymptotic-induced scheme for nonstationary transport equations in the diffusive limit. SIAM J. Numer. Anal. 35 (1998) 1073–1094. | DOI | MR | Zbl
, An asymptotic preserving numerical scheme for kinetic equations in the low Mach number limit. SIAM J. Numer. Anal. 36 (1999) 1507–1527. | DOI | MR | Zbl
and , Numerical passage from radiative heat transfer to nonlinear diffusion models. Math. Models Methods Appl. Sci. 11 (2001) 749–767. | DOI | MR | Zbl
and , Uniform stability of a finite difference scheme for transport equations in the diffusion limit. SIAM J. Numer. Anal. 40 (2002) 891–913. | DOI | MR | Zbl
, On the vibration of the electronic plasma. J. Phys. (USSR) 10 (1946) 25–34. | MR | Zbl
and , Asymptotic solutions of numerical transport problems in optically thick, diffusive regimes II. J. Comput. Phys. 83 (1989) 212–236. | DOI | MR | Zbl
, and , Asymptotic solutions of numerical transport problems in optically thick, diffusive regimes. J. Comput. Phys. 69 (1987) 283–324. | DOI | Zbl
and , A new asymptotic preserving scheme based on micro-macro formulation for linear kinetic equations in the diffusion limit. SIAM J. Sci. Comput. 31 (2008) 334–368. | DOI | Zbl
, Moment closure hierarchies for kinetic theories. J. Stat. Phys. 83 (1996) 1021–1065. | DOI | Zbl
and , Analysis of an asymptotic preserving scheme for linear kinetic equations in the diffusion limit. SIAM J. Numer. Anal. 48 (2010) 7561–7586. | Zbl
, and , An entropic scheme for an angular moment model for the classical Fokker-Planck-Landeau equation of electrons. Commun. Comput. Phys. 15 (2015) 422–450. | DOI | Zbl
, and , General moment system for plasma physics based on minimum entropy principle. Kinetic Relat Mod. 8 (2015) 533–558. | DOI | Zbl
, , , , , , and , Nonlocal heat wave propagation due to skin layer plasma heating by short laser pulses. Phys. Plasmas 20 (2013) 022702.
and , Towards realizable hyperbolic moment closures for viscous heat-conducting gas flows based on a maximum-entropy distribution. Contin. Mech. Thermodyn. 25 (2012) 573–603. | DOI | Zbl
, , , , , , G. Slark and K. Oades, Phys. Plasmas 11 (2004) 5573.
, Maximum entropy Eddigton Factors. J. Quant. Spectrosc. Radiat. Transfer 20 (1978) 541. | DOI
I. Muller and T. Ruggeri, Rational Extended Thermodynamics. Springer, New York (1998). | Zbl
, , and . Phys. Plasmas 7 (2000) 4250. | DOI
. An averaged formulation of the M1 radiation model with presumed probability density function for turbulent flows. J. Quant. Spectrosc. Radiat. Trans. 83 (2004) 493–517. | DOI
, and , A factored operator method for solving coupled radiation-hydrodynamics models. Trans. Theory. Stat. Phys. 31 (2002) 531–557. | DOI | Zbl
, and , A model of ultrashort laser pulse absorption in solid targets. Phys. Plasmas 3 (1996) 360. | DOI
, , , , , , H. Takabe and K. Mima, Phys. Rev. Lett. 78 (1997) 250.
I.P. Shkarofsky and T.W. Johnston, and The Particle Kinetics of Plasmas M.P. Bachynski, Addison-Wesley Reading, Massachusetts (1966).
and , Phys. Rev. 89 (1953) 977. | DOI | Zbl
H. Struchtrup, Macroscopic Transport Equations for Rarefied Gas Flows. Springer, Berlin (2005). | Zbl
E.F. Toro, Riemann Solvers and Numerical Methods for Fluids dynamics. Springer, Berlin (1999). | Zbl
, A consistent multigroup model for radiative transfer and its underlying mean opacity. J. Quant. Spectrosc. Radiat. Transfer 94 (2005) 357–371. | DOI
, , and , Multigroup half space moment appproximations to the radiative heat transfer equations. J. Comput. Phys. 198 (2004) 363. | DOI | Zbl
, , and , Phys. Plasmas 5 (1998) 1491. | DOI
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