doi: 10.3934/nhm.2022018
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Modelling and numerical study of the polyatomic bitemperature Euler system

Univ. Bordeaux, CNRS, Bordeaux INP, IMB, UMR 5251, F-33400 Talence, France

* Corresponding author: Stéphane Brull

Received  April 2021 Revised  January 2022 Early access April 2022

This paper is devoted to the study of the bitemperature Euler system in a polyatomic setting. Physically, this model describes a mixture of one species of ions and one species of electrons in the quasi-neutral regime. We firstly derive the model starting from a kinetic polyatomic model and performing next a fluid limit. This kinetic model is shown to satisfy fundamental properties. Some exact solutions are presented. Finally, a numerical scheme is derived and proved to coincide with an approximation designed in [3] and extended to second order and two space dimensions in [6]. Some numerical tests are presented.

Citation: Denise Aregba-Driollet, Stéphane Brull. Modelling and numerical study of the polyatomic bitemperature Euler system. Networks and Heterogeneous Media, doi: 10.3934/nhm.2022018
References:
[1]

R. Abgrall and S. Karni, A comment on the computation of non-conservative products, J. Comput. Phys., 229 (2010), 2759-2763.  doi: 10.1016/j.jcp.2009.12.015.

[2]

P. AndriesP. Le TallecJ.-P. Perlat and B. Perthame, Entropy condition for the ES BGK model of Boltzmann equation for mono and polyatomic gases, Eur. J. Mech. B Fluids, 19 (2000), 813-830.  doi: 10.1016/S0997-7546(00)01103-1.

[3]

D. Aregba-DriolletJ. BreilS. BrullB. Dubroca and E. Estibals, Modelling and numerical approximation for the nonconservative bitemperature Euler model, ESAIM Math. Model. Numer. Anal., 52 (2018), 1353-1383.  doi: 10.1051/m2an/2017007.

[4]

D. Aregba-Driollet and S. Brull, A viscous approximation of the bitemperature Euler system, Comm. Math. Sci., 17 (2019), 1135-1147.  doi: 10.4310/CMS.2019.v17.n4.a14.

[5]

D. Aregba-DriolletS. Brull and Y.-J. Peng, Global existence of smooth solutions for a non-conservative bitemperature Euler model, SIAM J. Math. Anal., 53 (2021), 1886-1907.  doi: 10.1137/20M1353812.

[6]

D. Aregba-DriolletS. Brull and C. Prigent, A discrete velocity numerical scheme for the two-dimensional bitemperature Euler system, SIAM J. Numer. Anal., 60 (2022), 28-51.  doi: 10.1137/21M1407185.

[7]

D. Aregba-Driollet and R. Natalini, Discrete kinetic schemes for multidimensional systems of conservation laws, SIAM J. Numer. Anal., 37 (2000), 1973-2004.  doi: 10.1137/S0036142998343075.

[8]

T. ArimaS. TaniguchiT. Ruggeri and M. Sugiyama, Monoatomic rarefied gas as a singular limit of polyatomic gas in extended thermodynamics, Physics letter A, 337 (2013), 2136-2140.  doi: 10.1016/j.physleta.2013.06.035.

[9]

C. BarangerM. BisiS. Brull and L. Desvillettes, On the Chapman-Enskog asymptotics of mixture of monoatomic and polyatomic rarefied gases, Kin. Rel. Mod., 11 (2018), 821-858.  doi: 10.3934/krm.2018033.

[10]

F. BernardA. Iollo and G. Puppo, BGK polyatomic model for rarefied flows, J. Sci. Comput., 78 (2019), 1893-1916.  doi: 10.1007/s10915-018-0864-x.

[11]

M. Bisi, R. Monaco and A. J. Soares, A BGK model for reactive mixtures of polyatomic gases with continuous internal energy, J. Phys. A, 51 (2018), 125501, 29 pp. doi: 10.1088/1751-8121/aaac8e.

[12]

M. BisiT. Ruggeri and G. Spiga, Dynamical pressure in a polyatomic gas: Interplay between kinetic theory and Extended Thermodynamics, Kinet. Relat. Models, 11 (2018), 71-95.  doi: 10.3934/krm.2018004.

[13]

M. Bisi and G. Spiga, On a kinetic BGK model for slow chemical reactions, Kinet. Relat. Models, 4 (2011), 153-167.  doi: 10.3934/krm.2011.4.153.

[14]

M. Bisi and R. Travaglini, A polyatomic model for mixtures for monoatomic and polyatomic gases, Phys. A, 547 (2020), 124441, 18 pp. doi: 10.1016/j.physa.2020.124441.

[15]

A. V. BobylevM. BisiM. GroppiG. Spiga and I. Potapenko, A general consistent BGK model for gas mixtures, Kinet. Relat. Models, 11 (2018), 1377-1393.  doi: 10.3934/krm.2018054.

[16]

C. Borgnakke and P. S. Larsen, Statistical collision model for Monte-Carlo simulation of polyatomic mixtures, Journ. Comput. Phys., 18 (1975), 405-420.  doi: 10.1016/0021-9991(75)90094-7.

[17]

J.-F BourgatL. DesvillettesP. Le Tallec and B. Perthame, Microreversible collisions for polyatomic gases and Boltzmann's theorem, Eur. Journ. Fluid Mech., 13 (1994), 237-254. 

[18] A. Bressan, Hyperbolic Systems of Conservation Laws, Oxford Lecture Series in Mathematics and its Applications, 20. Oxford University Press, Oxford, 2000. 
[19]

S. Brull, An Ellipsoidal Statistical Model for a monoatomic and polyatomic gas mixture, Comm. Math. Sci., 19 (2021), 2177-2194.  doi: 10.4310/CMS.2021.v19.n8.a5.

[20]

S. BrullB. Dubroca and C. Prigent, A kinetic approach of the bi-temperature Euler model, Kinet. Relat. Models, 13 (2020), 33-61.  doi: 10.3934/krm.2020002.

[21]

S. Brull and J. Schneider, On the Ellipsoidal Statistical Model for polyatomic gases, Cont. Mech. Thermodyn, 20 (2009), 489-508.  doi: 10.1007/s00161-009-0095-3.

[22]

C. Chalons and F. Coquel, Navier-Stokes equations with several independent pressure laws and explicit predictor-corrector schemes, Numer. Math., 101 (2005), 451-478.  doi: 10.1007/s00211-005-0612-7.

[23]

F. Coquel and C. Marmignon, Numerical methods for weakly ionized gas, Astrophysics and Space Science, 260 (1998), 15-27.  doi: 10.1023/A:1001870802972.

[24]

G. Dal MasoP. G. Le Floch and F. Murat, Definition and weak stability of nonconservative products, J. Math. Pures et Appl., 74 (1995), 483-548. 

[25]

L. Desvillettes, Sur un modèle de type Borgnakke-Larsen conduisant à des lois d'énergie non-linéaires en température pour les gas parfaits polyatomiques, Ann. Fac. Sci. Toulouse Math., 6 (1997), 257-262.  doi: 10.5802/afst.864.

[26]

L. DesvillettesR. Monaco and F. Salvarani, A kinetic model allowing to obtain the energy law of polytropic gases in the presence of chemical reactions, Eur. J. Mech. B Fluids, 24 (2005), 219-236.  doi: 10.1016/j.euromechflu.2004.07.004.

[27]

A. Ern and V. Giovangigli, The kinetic equilibrium regime, Physica A, 260 (1998), 49-72. 

[28]

E. Estibals, H. Guillard and A. Sangam, Derivation and numerical approximation of two-temperature Euler plasma model, J. Comput. Phys., 444 (2021), 110565, 48 pp. doi: 10.1016/j.jcp.2021.110565.

[29]

H. FunaganeS. TakataK. Aoki and K. Kugimoto, Poiseuille flow and thermal transpiration of a rarefied polyatomic gas through a circular tube with applications to microflows, Boll. Unione Mat. Ital., 4 (2011), 19-46. 

[30]

V. Giovangigli, Multicomponent Flow Modeling. Modeling and Simulation in Science, Engineering and Technology, Birkhäuser Boston, Inc., Boston, MA, 1999. doi: 10.1007/978-1-4612-1580-6.

[31]

B. GrailleT. Magin and M. Massot, Kinetic theory of plasmas, Maths models and methods in the Appl. Sci., 19 (2009), 527-599.  doi: 10.1142/S021820250900353X.

[32]

J. M. Greene, Improved Bhatnagar-Gross-Krook model of electron-ion collisions, Phys.Fluids, 16 (1973), 2022-2023. 

[33]

J. D. Huba, NRL Plasma Formulary, Revised 2013 version, NRL.

[34]

C. KlingenbergM. Pirner and G. Puppo, A consistent kinetic model for a two-component mixture of polyatomic molecules, Comm. in Math. Sci., 17 (2019), 149-173.  doi: 10.4310/CMS.2019.v17.n1.a6.

[35]

S. KosugeK. Aoki and T. Goto, Shock wave structure in polyatomic gases: Numerical analysis using a model Boltzmann equation, AIP Conf. Proc., 1786 (2016), 180004.  doi: 10.1063/1.4967673.

[36]

C. Pares, Numerical methods for nonconservative hyperbolic systems: A theoretical framework, SIAM J. Numer. Anal., 44 (2006), 300-321.  doi: 10.1137/050628052.

[37]

B. Perthame and C. Simeoni, A kinetic scheme for the Saint-Venant system with a source term, Calcolo, 38 (2001), 201-231.  doi: 10.1007/s10092-001-8181-3.

[38]

S. TakataH. Funagane and K. Aoki, Fluid modeling for the Knudsen compressor: Case of polyatomic gases, Kinet. Relat. Models, 3 (2010), 353-372.  doi: 10.3934/krm.2010.3.353.

[39]

Q. Wargnier, S. Faure, S. B. Graille, T. Magin and M. Massot, Numerical treatment of the nonconservative product in a multiscale fluid model for plasmas in thermal nonequilibrium: Application to solar physics, SIAM J. Sci. Comput., 42 (2020), B492–B519. doi: 10.1137/18M1194225.

[40] B. Zel'dovich and P. Raizer, Physics of Shock Waves and High-Temperature Hydrodynamic Phenomena, Academic Press, 1966. 

show all references

References:
[1]

R. Abgrall and S. Karni, A comment on the computation of non-conservative products, J. Comput. Phys., 229 (2010), 2759-2763.  doi: 10.1016/j.jcp.2009.12.015.

[2]

P. AndriesP. Le TallecJ.-P. Perlat and B. Perthame, Entropy condition for the ES BGK model of Boltzmann equation for mono and polyatomic gases, Eur. J. Mech. B Fluids, 19 (2000), 813-830.  doi: 10.1016/S0997-7546(00)01103-1.

[3]

D. Aregba-DriolletJ. BreilS. BrullB. Dubroca and E. Estibals, Modelling and numerical approximation for the nonconservative bitemperature Euler model, ESAIM Math. Model. Numer. Anal., 52 (2018), 1353-1383.  doi: 10.1051/m2an/2017007.

[4]

D. Aregba-Driollet and S. Brull, A viscous approximation of the bitemperature Euler system, Comm. Math. Sci., 17 (2019), 1135-1147.  doi: 10.4310/CMS.2019.v17.n4.a14.

[5]

D. Aregba-DriolletS. Brull and Y.-J. Peng, Global existence of smooth solutions for a non-conservative bitemperature Euler model, SIAM J. Math. Anal., 53 (2021), 1886-1907.  doi: 10.1137/20M1353812.

[6]

D. Aregba-DriolletS. Brull and C. Prigent, A discrete velocity numerical scheme for the two-dimensional bitemperature Euler system, SIAM J. Numer. Anal., 60 (2022), 28-51.  doi: 10.1137/21M1407185.

[7]

D. Aregba-Driollet and R. Natalini, Discrete kinetic schemes for multidimensional systems of conservation laws, SIAM J. Numer. Anal., 37 (2000), 1973-2004.  doi: 10.1137/S0036142998343075.

[8]

T. ArimaS. TaniguchiT. Ruggeri and M. Sugiyama, Monoatomic rarefied gas as a singular limit of polyatomic gas in extended thermodynamics, Physics letter A, 337 (2013), 2136-2140.  doi: 10.1016/j.physleta.2013.06.035.

[9]

C. BarangerM. BisiS. Brull and L. Desvillettes, On the Chapman-Enskog asymptotics of mixture of monoatomic and polyatomic rarefied gases, Kin. Rel. Mod., 11 (2018), 821-858.  doi: 10.3934/krm.2018033.

[10]

F. BernardA. Iollo and G. Puppo, BGK polyatomic model for rarefied flows, J. Sci. Comput., 78 (2019), 1893-1916.  doi: 10.1007/s10915-018-0864-x.

[11]

M. Bisi, R. Monaco and A. J. Soares, A BGK model for reactive mixtures of polyatomic gases with continuous internal energy, J. Phys. A, 51 (2018), 125501, 29 pp. doi: 10.1088/1751-8121/aaac8e.

[12]

M. BisiT. Ruggeri and G. Spiga, Dynamical pressure in a polyatomic gas: Interplay between kinetic theory and Extended Thermodynamics, Kinet. Relat. Models, 11 (2018), 71-95.  doi: 10.3934/krm.2018004.

[13]

M. Bisi and G. Spiga, On a kinetic BGK model for slow chemical reactions, Kinet. Relat. Models, 4 (2011), 153-167.  doi: 10.3934/krm.2011.4.153.

[14]

M. Bisi and R. Travaglini, A polyatomic model for mixtures for monoatomic and polyatomic gases, Phys. A, 547 (2020), 124441, 18 pp. doi: 10.1016/j.physa.2020.124441.

[15]

A. V. BobylevM. BisiM. GroppiG. Spiga and I. Potapenko, A general consistent BGK model for gas mixtures, Kinet. Relat. Models, 11 (2018), 1377-1393.  doi: 10.3934/krm.2018054.

[16]

C. Borgnakke and P. S. Larsen, Statistical collision model for Monte-Carlo simulation of polyatomic mixtures, Journ. Comput. Phys., 18 (1975), 405-420.  doi: 10.1016/0021-9991(75)90094-7.

[17]

J.-F BourgatL. DesvillettesP. Le Tallec and B. Perthame, Microreversible collisions for polyatomic gases and Boltzmann's theorem, Eur. Journ. Fluid Mech., 13 (1994), 237-254. 

[18] A. Bressan, Hyperbolic Systems of Conservation Laws, Oxford Lecture Series in Mathematics and its Applications, 20. Oxford University Press, Oxford, 2000. 
[19]

S. Brull, An Ellipsoidal Statistical Model for a monoatomic and polyatomic gas mixture, Comm. Math. Sci., 19 (2021), 2177-2194.  doi: 10.4310/CMS.2021.v19.n8.a5.

[20]

S. BrullB. Dubroca and C. Prigent, A kinetic approach of the bi-temperature Euler model, Kinet. Relat. Models, 13 (2020), 33-61.  doi: 10.3934/krm.2020002.

[21]

S. Brull and J. Schneider, On the Ellipsoidal Statistical Model for polyatomic gases, Cont. Mech. Thermodyn, 20 (2009), 489-508.  doi: 10.1007/s00161-009-0095-3.

[22]

C. Chalons and F. Coquel, Navier-Stokes equations with several independent pressure laws and explicit predictor-corrector schemes, Numer. Math., 101 (2005), 451-478.  doi: 10.1007/s00211-005-0612-7.

[23]

F. Coquel and C. Marmignon, Numerical methods for weakly ionized gas, Astrophysics and Space Science, 260 (1998), 15-27.  doi: 10.1023/A:1001870802972.

[24]

G. Dal MasoP. G. Le Floch and F. Murat, Definition and weak stability of nonconservative products, J. Math. Pures et Appl., 74 (1995), 483-548. 

[25]

L. Desvillettes, Sur un modèle de type Borgnakke-Larsen conduisant à des lois d'énergie non-linéaires en température pour les gas parfaits polyatomiques, Ann. Fac. Sci. Toulouse Math., 6 (1997), 257-262.  doi: 10.5802/afst.864.

[26]

L. DesvillettesR. Monaco and F. Salvarani, A kinetic model allowing to obtain the energy law of polytropic gases in the presence of chemical reactions, Eur. J. Mech. B Fluids, 24 (2005), 219-236.  doi: 10.1016/j.euromechflu.2004.07.004.

[27]

A. Ern and V. Giovangigli, The kinetic equilibrium regime, Physica A, 260 (1998), 49-72. 

[28]

E. Estibals, H. Guillard and A. Sangam, Derivation and numerical approximation of two-temperature Euler plasma model, J. Comput. Phys., 444 (2021), 110565, 48 pp. doi: 10.1016/j.jcp.2021.110565.

[29]

H. FunaganeS. TakataK. Aoki and K. Kugimoto, Poiseuille flow and thermal transpiration of a rarefied polyatomic gas through a circular tube with applications to microflows, Boll. Unione Mat. Ital., 4 (2011), 19-46. 

[30]

V. Giovangigli, Multicomponent Flow Modeling. Modeling and Simulation in Science, Engineering and Technology, Birkhäuser Boston, Inc., Boston, MA, 1999. doi: 10.1007/978-1-4612-1580-6.

[31]

B. GrailleT. Magin and M. Massot, Kinetic theory of plasmas, Maths models and methods in the Appl. Sci., 19 (2009), 527-599.  doi: 10.1142/S021820250900353X.

[32]

J. M. Greene, Improved Bhatnagar-Gross-Krook model of electron-ion collisions, Phys.Fluids, 16 (1973), 2022-2023. 

[33]

J. D. Huba, NRL Plasma Formulary, Revised 2013 version, NRL.

[34]

C. KlingenbergM. Pirner and G. Puppo, A consistent kinetic model for a two-component mixture of polyatomic molecules, Comm. in Math. Sci., 17 (2019), 149-173.  doi: 10.4310/CMS.2019.v17.n1.a6.

[35]

S. KosugeK. Aoki and T. Goto, Shock wave structure in polyatomic gases: Numerical analysis using a model Boltzmann equation, AIP Conf. Proc., 1786 (2016), 180004.  doi: 10.1063/1.4967673.

[36]

C. Pares, Numerical methods for nonconservative hyperbolic systems: A theoretical framework, SIAM J. Numer. Anal., 44 (2006), 300-321.  doi: 10.1137/050628052.

[37]

B. Perthame and C. Simeoni, A kinetic scheme for the Saint-Venant system with a source term, Calcolo, 38 (2001), 201-231.  doi: 10.1007/s10092-001-8181-3.

[38]

S. TakataH. Funagane and K. Aoki, Fluid modeling for the Knudsen compressor: Case of polyatomic gases, Kinet. Relat. Models, 3 (2010), 353-372.  doi: 10.3934/krm.2010.3.353.

[39]

Q. Wargnier, S. Faure, S. B. Graille, T. Magin and M. Massot, Numerical treatment of the nonconservative product in a multiscale fluid model for plasmas in thermal nonequilibrium: Application to solar physics, SIAM J. Sci. Comput., 42 (2020), B492–B519. doi: 10.1137/18M1194225.

[40] B. Zel'dovich and P. Raizer, Physics of Shock Waves and High-Temperature Hydrodynamic Phenomena, Academic Press, 1966. 
Figure 1.  Double rarefaction. Left: density. Right: velocity
Figure 2.  Double rarefaction. Left: electronic temperature. Right : ionic temperature
Figure 3.  Total density (left) and electronic temperature (right) at time $ t = 4.0901\times 10^{-7} $s for an implosion test case with $ \nu_{ei} $ given by the NRL formulary with a grid of 500 by 500 points
Figure 4.  Implosion test case with $ \nu_{ei} $ given by the NRL formulary with a grid of 500 by 500 points. Density along the first bisector at 4 different times: the peak occurs for $ t = 9.2 \times 10^{-7} $ sec
Figure 5.  Implosion test case with $ \nu_{ei} $ given by the NRL formulary with a grid of 500 by 500 points. Left: isovalues of the density when the peak occurs. Right: isovalues of the electronic and ionic temperatures when the peak occurs
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