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Strong solutions for the Alber equation and stability of unidirectional wave spectra
Angular moments models for rarefied gas dynamics. Numerical comparisons with kinetic and Navier-Stokes equations
CEA-DAM Ile-de-France, France |
Angular moments models based on a minimum entropy problem have been largely used to describe the transport of photons [
References:
[1] |
G. W. Alldredge, C. D. 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] |
C. Berthon, P. Charrier and B. Dubroca,
An HLLC Scheme to Solve The M1 Model of Radiative Transfer in Two Space Dimensions, J. Sci. Comput., 31 (2007), 347-389.
doi: 10.1007/s10915-006-9108-6. |
[3] |
F. Bouchut, Nonlinear Stability of Finite Volume Methods for Hyperbolic Conservation Laws, and Well-Balanced Schemes for Sources, Birkhäuser Verlag, Basel, (2004).
doi: 10.1007/b93802. |
[4] |
S. Brull and L. Mieussens,
Local discrete velocity grids for deterministic rarefied flow simulations, J. Comput. Phys., 266 (2014), 22-46.
doi: 10.1016/j.jcp.2014.01.050. |
[5] |
S. Chapman,
On the law of distribution of molecular velocities, and on the theory of viscosity and thermal conduction, in a non-uniform simple monatomic gas, Phil. Trans. Roy. Soc. London Ser. A, 216 (1916), 538-548.
doi: 10.1098/rsta.1916.0006. |
[6] |
S. Chapman and T. G. Cowling, The Mathematical Theory of Non-Uniform Gases, Cambridge University Press, Cambridge, 1939.
![]() ![]() |
[7] |
P. Charrier, B. Dubroca, G. Duffa, and R. Turpault, Multigroup model for radiating flows during atmospheric hypersonic re-entry, in Proceedings of International Workshop on Radiation of High Temperature Gases in Atmospheric Entry, Lisbonne, Portugal, 2003,103–110. |
[8] | |
[9] |
B. Dubroca, J.-L. Feugeas and M. Frank, Angular moment model for the Fokker-Planck equation, Eur. Phys. J. D, 60, (2010) 302–307.
doi: 10.1140/epjd/e2010-00190-8. |
[10] |
B. Dubroca and J. L. Feugeas,
Étude théorique et numérique d'une hiéarchie de modèles aux moments pour le transfert radiatif, C. R. Acad. Sci. Paris Ser. I Math., 329 (1999), 915-920.
doi: 10.1016/S0764-4442(00)87499-6. |
[11] |
R. Duclous, B. Dubroca and M. Frank,
Deterministic partial differential equation model for dose calculation in electron radiotherapy, Phys. Med. Biol., 55 (2010), 3843-3857.
doi: 10.1088/0031-9155/55/13/018. |
[12] |
D. Enskog, Kinetische Theorie der Vorgänge in Mässig Verdünnten Gasen, Uppsala, 1917. |
[13] |
F. Filbet and T. Rey, A hierarchy of hybrid numerical methods for multi-scale kinetic equation, SIAM J. Sci. Computing, 37 (2015), A1218–A1247.
doi: 10.1137/140958773. |
[14] |
M. González, E. Audit and P. Huynh,
Heracles: A three-dimensional radiation hydrodynamics code, A and A, 464 (2007), 429-435.
|
[15] |
H. Grad,
On the kinetic theory of rarefied gases, Commun. Pure Appl. Math., 2 (1949), 331-407.
doi: 10.1002/cpa.3160020403. |
[16] |
E. P. Gross, P. L. Bathnagar and M. Krook, A model for collision processes in gases. Ⅰ. Small amplitude processes in charged and neutral one-component systems, Phys. Rev., 94 (1954), 511.
doi: 10.1103/PhysRev.94.511. |
[17] |
S. Guisset, D. Aregba, S. Brull and B. Dubroca,
The M1 angular moments model in a velocity-adaptive frame for rarefied gas dynamics applications, Multiscale Model. Simul., 15 (2017), 1719-1747.
doi: 10.1137/16M1099327. |
[18] |
S. Guisset, S. Brull, E. d'Humières, B. Dubroca and V. Tikhonchuk,
Classical transport theory for the collisional electronic M1 model, Phys. A, 446 (2016), 182-194.
doi: 10.1016/j.physa.2015.12.001. |
[19] |
S. Guisset, S. Brull, B. Dubroca, E. d'Humières, S. Karpov and I. Potapenko,
Asymptotic-preserving scheme for the M1-Maxwell system in the quasi-neutral regime, Commun. Comput. Phys., 19 (2016), 301-328.
doi: 10.4208/cicp.131014.030615a. |
[20] |
S. Guisset, J. G. Moreau, R. Nuter, S. Brull, E. d'Humieres, B. Dubroca and V. T. Tikhonchuk, Limits of the M1 and M2 angular moments models for kinetic plasma physics studies, J. Phys. A, 48 (2015), 335501.
doi: 10.1088/1751-8113/48/33/335501. |
[21] |
A. Harten, P. D. Lax and B. Van Leer,
On upstream differencing and Godunov-type schemes for hyperbolic conservation laws, SIAM Rev., 25 (1983), 35-61.
doi: 10.1137/1025002. |
[22] |
P. Helluy, M. Massaro, L. Navoret, N. Pham and T. Strub, Reduced Vlasov-Maxwell modeling, PIERS Proceedings, (2014), 2622–2627. |
[23] |
F. Hermeline, A finite volume method for the approximation of the PN Boltzmann and similar systems of equations on general meshes, CEA Report, (2015). |
[24] |
M. Junk and A. Unterreiter,
Maximum entropy moment systems and Galilean invariance, Contin. Mech. Thermodyn., 14 (2002), 563-576.
doi: 10.1007/s00161-002-0096-y. |
[25] |
B. Van Leer,
Towards the ultimate conservative difference scheme Ⅲ. Upstream-centered finite-difference schemes for ideal compressible flow, J. Comput. Phys., 23 (1977), 263-275.
doi: 10.1016/0021-9991(77)90094-8. |
[26] |
C. D. Levermore,
Moment closure hierarchies for kinetic theories, J. Stat. Phys., 83 (1996), 1021-1065.
doi: 10.1007/BF02179552. |
[27] |
J. Mallet, S. Brull and B. Dubroca,
General moment system for plasma physics based on minimum entropy principle, Kinet. Relat. Models, 8 (2015), 533-558.
doi: 10.3934/krm.2015.8.533. |
[28] |
J. McDonald and M. Torrilhon,
An affordable robust moment closures for CFD based on the maximum-entropy hierarchy, J. Comput. Phys., 251 (2013), 500-523.
doi: 10.1016/j.jcp.2013.05.046. |
[29] |
J. G. McDonald and C. P. T. Groth,
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: 10.1007/s00161-012-0252-y. |
[30] |
L. Mieussens,
Discrete velocity model and implicit scheme for the BGK equation of rarefied gas dynamics, Math. Models Methods Appl. Sci., 10 (2000), 1121-1149.
doi: 10.1142/S0218202500000562. |
[31] |
G. N. Minerbo,
Maximum entropy eddington factors, J. Quant. Spectrosc. Radiat. Trans., 20 (1978), 541-545.
doi: 10.1016/0022-4073(78)90024-9. |
[32] |
I. Muller and T. Ruggeri, Rational Extended Thermodynamics, Springer-Verlag, New York, NY, 1998.
doi: 10.1007/978-1-4612-2210-1. |
[33] |
G. C. Pomraning,
Maximum entropy Eddington factors and flux limited diffusion theory, J. Quant. Spectrosc. Radiat. Trans., 26 (1981), 385-388.
doi: 10.1016/0022-4073(81)90101-1. |
[34] |
J.-F. Ripoll,
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.
|
[35] |
J.-F. Ripoll, B. Dubroca and E. Audit,
A factored operator method for solving coupled radiation-hydrodynamics models, Transport Theory Statist. Phys., 31 (2002), 531-557.
doi: 10.1081/TT-120015513. |
[36] |
J. Schneider,
Entropic approximation in kinetic theory, M2AN Math. Model. Numer. Anal., 38 (2004), 541-561.
doi: 10.1051/m2an:2004025. |
[37] |
H. Struchtrup, Macroscopic Transport Equations for Rarefied Gas Flows, Springer, Berlin, (2005). |
[38] |
B. Su,
More on boundary conditions for differential approximations, J. Quant. Spectrosc. Radiat. Trans., 64 (2000), 409-419.
doi: 10.1016/S0022-4073(99)00128-4. |
[39] | |
[40] |
M. Torrilhon,
Modeling nonequilibrium gas flow based on moment equations, Annu. Rev. Fluid Mech., 48 (2016), 429-458.
|
[41] |
R. Turpault,
A consistent multigroup model for radiative transfer and its underlying mean opacity, J. Quant. Spectrosc. Radiat. Trans., 94 (2005), 357-371.
doi: 10.1016/j.jqsrt.2004.09.042. |
[42] |
R. Turpault, M. Frank, B. Dubroca and A. Klar,
Multigroup half space moment approximations to the radiative heat transfer equations, J. Comput. Phys., 198 (2004), 363-371.
|
show all references
References:
[1] |
G. W. Alldredge, C. D. 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] |
C. Berthon, P. Charrier and B. Dubroca,
An HLLC Scheme to Solve The M1 Model of Radiative Transfer in Two Space Dimensions, J. Sci. Comput., 31 (2007), 347-389.
doi: 10.1007/s10915-006-9108-6. |
[3] |
F. Bouchut, Nonlinear Stability of Finite Volume Methods for Hyperbolic Conservation Laws, and Well-Balanced Schemes for Sources, Birkhäuser Verlag, Basel, (2004).
doi: 10.1007/b93802. |
[4] |
S. Brull and L. Mieussens,
Local discrete velocity grids for deterministic rarefied flow simulations, J. Comput. Phys., 266 (2014), 22-46.
doi: 10.1016/j.jcp.2014.01.050. |
[5] |
S. Chapman,
On the law of distribution of molecular velocities, and on the theory of viscosity and thermal conduction, in a non-uniform simple monatomic gas, Phil. Trans. Roy. Soc. London Ser. A, 216 (1916), 538-548.
doi: 10.1098/rsta.1916.0006. |
[6] |
S. Chapman and T. G. Cowling, The Mathematical Theory of Non-Uniform Gases, Cambridge University Press, Cambridge, 1939.
![]() ![]() |
[7] |
P. Charrier, B. Dubroca, G. Duffa, and R. Turpault, Multigroup model for radiating flows during atmospheric hypersonic re-entry, in Proceedings of International Workshop on Radiation of High Temperature Gases in Atmospheric Entry, Lisbonne, Portugal, 2003,103–110. |
[8] | |
[9] |
B. Dubroca, J.-L. Feugeas and M. Frank, Angular moment model for the Fokker-Planck equation, Eur. Phys. J. D, 60, (2010) 302–307.
doi: 10.1140/epjd/e2010-00190-8. |
[10] |
B. Dubroca and J. L. Feugeas,
Étude théorique et numérique d'une hiéarchie de modèles aux moments pour le transfert radiatif, C. R. Acad. Sci. Paris Ser. I Math., 329 (1999), 915-920.
doi: 10.1016/S0764-4442(00)87499-6. |
[11] |
R. Duclous, B. Dubroca and M. Frank,
Deterministic partial differential equation model for dose calculation in electron radiotherapy, Phys. Med. Biol., 55 (2010), 3843-3857.
doi: 10.1088/0031-9155/55/13/018. |
[12] |
D. Enskog, Kinetische Theorie der Vorgänge in Mässig Verdünnten Gasen, Uppsala, 1917. |
[13] |
F. Filbet and T. Rey, A hierarchy of hybrid numerical methods for multi-scale kinetic equation, SIAM J. Sci. Computing, 37 (2015), A1218–A1247.
doi: 10.1137/140958773. |
[14] |
M. González, E. Audit and P. Huynh,
Heracles: A three-dimensional radiation hydrodynamics code, A and A, 464 (2007), 429-435.
|
[15] |
H. Grad,
On the kinetic theory of rarefied gases, Commun. Pure Appl. Math., 2 (1949), 331-407.
doi: 10.1002/cpa.3160020403. |
[16] |
E. P. Gross, P. L. Bathnagar and M. Krook, A model for collision processes in gases. Ⅰ. Small amplitude processes in charged and neutral one-component systems, Phys. Rev., 94 (1954), 511.
doi: 10.1103/PhysRev.94.511. |
[17] |
S. Guisset, D. Aregba, S. Brull and B. Dubroca,
The M1 angular moments model in a velocity-adaptive frame for rarefied gas dynamics applications, Multiscale Model. Simul., 15 (2017), 1719-1747.
doi: 10.1137/16M1099327. |
[18] |
S. Guisset, S. Brull, E. d'Humières, B. Dubroca and V. Tikhonchuk,
Classical transport theory for the collisional electronic M1 model, Phys. A, 446 (2016), 182-194.
doi: 10.1016/j.physa.2015.12.001. |
[19] |
S. Guisset, S. Brull, B. Dubroca, E. d'Humières, S. Karpov and I. Potapenko,
Asymptotic-preserving scheme for the M1-Maxwell system in the quasi-neutral regime, Commun. Comput. Phys., 19 (2016), 301-328.
doi: 10.4208/cicp.131014.030615a. |
[20] |
S. Guisset, J. G. Moreau, R. Nuter, S. Brull, E. d'Humieres, B. Dubroca and V. T. Tikhonchuk, Limits of the M1 and M2 angular moments models for kinetic plasma physics studies, J. Phys. A, 48 (2015), 335501.
doi: 10.1088/1751-8113/48/33/335501. |
[21] |
A. Harten, P. D. Lax and B. Van Leer,
On upstream differencing and Godunov-type schemes for hyperbolic conservation laws, SIAM Rev., 25 (1983), 35-61.
doi: 10.1137/1025002. |
[22] |
P. Helluy, M. Massaro, L. Navoret, N. Pham and T. Strub, Reduced Vlasov-Maxwell modeling, PIERS Proceedings, (2014), 2622–2627. |
[23] |
F. Hermeline, A finite volume method for the approximation of the PN Boltzmann and similar systems of equations on general meshes, CEA Report, (2015). |
[24] |
M. Junk and A. Unterreiter,
Maximum entropy moment systems and Galilean invariance, Contin. Mech. Thermodyn., 14 (2002), 563-576.
doi: 10.1007/s00161-002-0096-y. |
[25] |
B. Van Leer,
Towards the ultimate conservative difference scheme Ⅲ. Upstream-centered finite-difference schemes for ideal compressible flow, J. Comput. Phys., 23 (1977), 263-275.
doi: 10.1016/0021-9991(77)90094-8. |
[26] |
C. D. Levermore,
Moment closure hierarchies for kinetic theories, J. Stat. Phys., 83 (1996), 1021-1065.
doi: 10.1007/BF02179552. |
[27] |
J. Mallet, S. Brull and B. Dubroca,
General moment system for plasma physics based on minimum entropy principle, Kinet. Relat. Models, 8 (2015), 533-558.
doi: 10.3934/krm.2015.8.533. |
[28] |
J. McDonald and M. Torrilhon,
An affordable robust moment closures for CFD based on the maximum-entropy hierarchy, J. Comput. Phys., 251 (2013), 500-523.
doi: 10.1016/j.jcp.2013.05.046. |
[29] |
J. G. McDonald and C. P. T. Groth,
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: 10.1007/s00161-012-0252-y. |
[30] |
L. Mieussens,
Discrete velocity model and implicit scheme for the BGK equation of rarefied gas dynamics, Math. Models Methods Appl. Sci., 10 (2000), 1121-1149.
doi: 10.1142/S0218202500000562. |
[31] |
G. N. Minerbo,
Maximum entropy eddington factors, J. Quant. Spectrosc. Radiat. Trans., 20 (1978), 541-545.
doi: 10.1016/0022-4073(78)90024-9. |
[32] |
I. Muller and T. Ruggeri, Rational Extended Thermodynamics, Springer-Verlag, New York, NY, 1998.
doi: 10.1007/978-1-4612-2210-1. |
[33] |
G. C. Pomraning,
Maximum entropy Eddington factors and flux limited diffusion theory, J. Quant. Spectrosc. Radiat. Trans., 26 (1981), 385-388.
doi: 10.1016/0022-4073(81)90101-1. |
[34] |
J.-F. Ripoll,
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.
|
[35] |
J.-F. Ripoll, B. Dubroca and E. Audit,
A factored operator method for solving coupled radiation-hydrodynamics models, Transport Theory Statist. Phys., 31 (2002), 531-557.
doi: 10.1081/TT-120015513. |
[36] |
J. Schneider,
Entropic approximation in kinetic theory, M2AN Math. Model. Numer. Anal., 38 (2004), 541-561.
doi: 10.1051/m2an:2004025. |
[37] |
H. Struchtrup, Macroscopic Transport Equations for Rarefied Gas Flows, Springer, Berlin, (2005). |
[38] |
B. Su,
More on boundary conditions for differential approximations, J. Quant. Spectrosc. Radiat. Trans., 64 (2000), 409-419.
doi: 10.1016/S0022-4073(99)00128-4. |
[39] | |
[40] |
M. Torrilhon,
Modeling nonequilibrium gas flow based on moment equations, Annu. Rev. Fluid Mech., 48 (2016), 429-458.
|
[41] |
R. Turpault,
A consistent multigroup model for radiative transfer and its underlying mean opacity, J. Quant. Spectrosc. Radiat. Trans., 94 (2005), 357-371.
doi: 10.1016/j.jqsrt.2004.09.042. |
[42] |
R. Turpault, M. Frank, B. Dubroca and A. Klar,
Multigroup half space moment approximations to the radiative heat transfer equations, J. Comput. Phys., 198 (2004), 363-371.
|












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