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February  2020, 13(1): 33-61. doi: 10.3934/krm.2020002

A kinetic approach of the bi-temperature Euler model

Stéphane Brull 1, , Bruno Dubroca 2, and  Corentin Prigent 1,

1. 

Univ. Bordeaux, CNRS, Bordeaux INP, IMB, UMR 5251, F-33400 Talence, France
2. 

Univ. Bordeaux, Laboratoire des Composites ThermoStructuraux (LCTS), UMR 5801: CNRS-Herakles(Safran)-CEA-UBx, 3, Allée de La Boétie, 33600 Pessac, France

Received  November 2018 Revised  September 2019 Published  December 2019

We are interested in the numerical approximation of the bi-temperature Euler equations, which is a non conservative hyperbolic system introduced in [4]. We consider a conservative underlying kinetic model, the Vlasov-BGK-Poisson system. We perform a scaling on this system in order to obtain its hydrodynamic limit. We present a deterministic numerical method to approximate this kinetic system. The method is shown to be Asymptotic-Preserving in the hydrodynamic limit, which means that any stability condition of the method is independant of any parameter $ \varepsilon $, with $ \varepsilon \rightarrow 0 $. We prove that the method is, under appropriate choices, consistant with the solution for bi-temperature Euler. Finally, our method is compared to methods for the fluid model (HLL, Suliciu).

Keywords: Bgk model, asympotic preserving scheme, nonconservative hyperbolic system, plasma physics, kinetic scheme.
Mathematics Subject Classification: 82B40, 82D10, 34K28, 76N99.
Citation: Stéphane Brull, Bruno Dubroca, Corentin Prigent. A kinetic approach of the bi-temperature Euler model. Kinetic & Related Models, 2020, 13 (1) : 33-61. doi: 10.3934/krm.2020002
References:
[1]

R. Abgrall and S. Karni, A comment on the computation of non-conservative products, Journal of Computational Physics, 229 (2010), 2759-2763.  doi: 10.1016/j.jcp.2009.12.015.  Google Scholar

[2]

P. Andries, P. Le Tallec, J.-P. Perlat and B. Perthame, The Gaussian-BGK model of Boltzmann equation with small Prandtl number, European Journal of Mechanics-B/Fluids, 19 (2000), 813-830.  doi: 10.1016/S0997-7546(00)01103-1.  Google Scholar

[3]

D. Aregba and S. Brull, About viscous approximations of the bitemperature Euler system, Communications in Math Sciences, 17 (2019), 1135-1147.  doi: 10.4310/CMS.2019.v17.n4.a14.  Google Scholar

[4]

D. Aregba-Driollet, J. Breil, S. Brull, B. Dubroca and E. Estibals, Modelling and numerical approximation for the nonconservative bitemperature Euler model, ESAIM: Mathematical Modelling and Numerical Analysis, 52 (2018), 1353-1383.  doi: 10.1051/m2an/2017007.  Google Scholar

[5]

R. Belaouar, N. Crouseilles, P. Degond and E. Sonnendrücker, An asymptotically stable semi-Lagrangian scheme in the quasi-neutral limit, J. Sci. Comput., 41 (2009), 341-365.  doi: 10.1007/s10915-009-9302-4.  Google Scholar

[6]

A. V. Bobylev, M. Bisi, M. Groppi, G. Spiga and I. F. Potapenko, A general consistent BGK model for gas mixtures, Kinetic and Related Models, 11 (2018), 1377-1393.  doi: 10.3934/krm.2018054.  Google Scholar

[7]

F. Bouchut, Nonlinear Stability of Finite Volume Methods for Hyperbolic Conservation Laws and Well-Balanced Schemes for Sources, Birkauser Verlag, Basel, 2004. doi: 10.1007/b93802.  Google Scholar

[8]

S. Brull, P. Degond, F. Deluzet and A. Mouton, Asymptotic-preserving scheme for a bi-fluid Euler-Lorentz model, Kinetic and Related Models, 4 (2011), 991-1023.   Google Scholar

[9]

S. Brull, X. Lhebrard and B. Dubroca, Modelling and entropy satisfying relaxation scheme for the nonconservative bitemperature Euler system with transverse magnetic field, preprint. Google Scholar

[10]

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.  Google Scholar

[11]

C. Chalons and F. Coquel, A new comment on the computation of non-conservative products using Roe-type path conservative schemes, J. Comput. Phys., 335 (2017), 592-604.  doi: 10.1016/j.jcp.2017.01.016.  Google Scholar

[12]

C. Chalons, M. Girardin and S. Kokh, Large time step and asymptotic preserving numerical schemes for the gas dynamics equations with source terms, SIAM Journal on Scientific Computing, 35 (2013), A2874-A2902.  doi: 10.1137/130908671.  Google Scholar

[13]

F. Chen, Introduction to Plasma Physics and Controlled Fusion, Number vol. 1 in Introduction to Plasma Physics and Controlled Fusion. Springer, 1984. Google Scholar

[14]

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

[15]

P. Crispel, P. Degond and M.-H. Vignal, An asymptotically stable discretization for the Euler-Poisson system in the quasi-neutral limit, C. R. Math. Acad. Sci. Paris, 341 (2005), 323-328.  doi: 10.1016/j.crma.2005.07.008.  Google Scholar

[16]

P. Crispel, P. Degond and M.-H. Vignal, An asymptotic preserving scheme for the two-fluid Euler-Poisson model in the quasineutral limit, J. Comput. Phys., 223 (2007), 208-234.  doi: 10.1016/j.jcp.2006.09.004.  Google Scholar

[17]

G. Dal Maso, P. Le Floch and F. Murat., Definition and weak stability of nonconservative products, Journal de Mathématiques Pures et Appliquées, 74 (1995), 483-548.   Google Scholar

[18]

P. Degond and F. Deluzet, Asymptotic-preserving methods and multiscale models for plasma physics, J. Comput. Phys., 336 (2017), 429-457.  doi: 10.1016/j.jcp.2017.02.009.  Google Scholar

[19]

P. Degond, F. Deluzet and D. Savelief, Numerical approximation of the Euler-Maxwell model in the quasineutral limit, J. Comput. Phys., 231 (2012), 1917-1946.  doi: 10.1016/j.jcp.2011.11.011.  Google Scholar

[20]

P. Degond, H. Liu, D. Savelief and M.-H. Vignal, Numerical approximation of the Euler-Poisson-Boltzmann model in the quasineutral limit, J. Sci. Comput., 51 (2012), 59-86.  doi: 10.1007/s10915-011-9495-1.  Google Scholar

[21]

M. Dumbser and D. S. Balsara, A new efficient formulation of the HLLEM Riemann solver for general conservative and non-conservative hyperbolic systems, Journal of Computational Physics, 304 (2016), 275-319.  doi: 10.1016/j.jcp.2015.10.014.  Google Scholar

[22]

S. Guisset, S. Brull, E. D'Humières, B. Dubroca, S. Karpov and I. Potapenko, Asymptotic-preserving scheme for the M1-Maxwell system in the quasi-neutral regime, Communications in Computational Physics, 19 (2016), 301-328.  doi: 10.4208/cicp.131014.030615a.  Google Scholar

[23]

S. Jin, Efficient asymptotic-preserving (AP) schemes for some multiscale kinetic equations, SIAM J. Sci. Comput., 21 (1999), 441-454.  doi: 10.1137/S1064827598334599.  Google Scholar

[24]

C. Klingenberg, M. Pirner and G. Puppo, A consistent kinetic model for a two-component mixture with an application to plasma, Kinetic and Related Models, 10 (2017), 445-465.  doi: 10.3934/krm.2017017.  Google Scholar

[25]

C. Klingenberg, M. Pirner and G. Puppo, A consistent kinetic model for a two-component mixture of polyatomic molecules, Communication in Mathematical Sciences, 17 (2019), 149-173.  doi: 10.4310/CMS.2019.v17.n1.a6.  Google Scholar

[26]

P. G. LeFloch and S. Mishra, Numerical methods with controlled dissipation for small-scale dependent shocks, Acta Numerica, 23 (2014), 743-816.  doi: 10.1017/S0962492914000099.  Google Scholar

[27]

C. D. Levermore, Moment closure hierarchies for kinetic theories, Journal of Statistical Physics, 83 (1996), 1021-1065.  doi: 10.1007/BF02179552.  Google Scholar

[28]

C. Pars, Path-conservative numerical methods for nonconservative hyperbolic systems, In Numerical Methods for Balance Laws, volume 24 of Quad. Mat., pages 67–121. Dept. Math., Seconda Univ. Napoli, Caserta, 2009.  Google Scholar

[29]

K. Xu, X. He and C. Cai, Multiple temperature kinetic model and gas-kinetic method for hypersonic non-equilibrium flow computations, J. Comp. Phys., 227 (2008), 6779-6794.  doi: 10.1016/j.jcp.2008.03.035.  Google Scholar

show all references

References:
[1]

R. Abgrall and S. Karni, A comment on the computation of non-conservative products, Journal of Computational Physics, 229 (2010), 2759-2763.  doi: 10.1016/j.jcp.2009.12.015.  Google Scholar

[2]

P. Andries, P. Le Tallec, J.-P. Perlat and B. Perthame, The Gaussian-BGK model of Boltzmann equation with small Prandtl number, European Journal of Mechanics-B/Fluids, 19 (2000), 813-830.  doi: 10.1016/S0997-7546(00)01103-1.  Google Scholar

[3]

D. Aregba and S. Brull, About viscous approximations of the bitemperature Euler system, Communications in Math Sciences, 17 (2019), 1135-1147.  doi: 10.4310/CMS.2019.v17.n4.a14.  Google Scholar

[4]

D. Aregba-Driollet, J. Breil, S. Brull, B. Dubroca and E. Estibals, Modelling and numerical approximation for the nonconservative bitemperature Euler model, ESAIM: Mathematical Modelling and Numerical Analysis, 52 (2018), 1353-1383.  doi: 10.1051/m2an/2017007.  Google Scholar

[5]

R. Belaouar, N. Crouseilles, P. Degond and E. Sonnendrücker, An asymptotically stable semi-Lagrangian scheme in the quasi-neutral limit, J. Sci. Comput., 41 (2009), 341-365.  doi: 10.1007/s10915-009-9302-4.  Google Scholar

[6]

A. V. Bobylev, M. Bisi, M. Groppi, G. Spiga and I. F. Potapenko, A general consistent BGK model for gas mixtures, Kinetic and Related Models, 11 (2018), 1377-1393.  doi: 10.3934/krm.2018054.  Google Scholar

[7]

F. Bouchut, Nonlinear Stability of Finite Volume Methods for Hyperbolic Conservation Laws and Well-Balanced Schemes for Sources, Birkauser Verlag, Basel, 2004. doi: 10.1007/b93802.  Google Scholar

[8]

S. Brull, P. Degond, F. Deluzet and A. Mouton, Asymptotic-preserving scheme for a bi-fluid Euler-Lorentz model, Kinetic and Related Models, 4 (2011), 991-1023.   Google Scholar

[9]

S. Brull, X. Lhebrard and B. Dubroca, Modelling and entropy satisfying relaxation scheme for the nonconservative bitemperature Euler system with transverse magnetic field, preprint. Google Scholar

[10]

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.  Google Scholar

[11]

C. Chalons and F. Coquel, A new comment on the computation of non-conservative products using Roe-type path conservative schemes, J. Comput. Phys., 335 (2017), 592-604.  doi: 10.1016/j.jcp.2017.01.016.  Google Scholar

[12]

C. Chalons, M. Girardin and S. Kokh, Large time step and asymptotic preserving numerical schemes for the gas dynamics equations with source terms, SIAM Journal on Scientific Computing, 35 (2013), A2874-A2902.  doi: 10.1137/130908671.  Google Scholar

[13]

F. Chen, Introduction to Plasma Physics and Controlled Fusion, Number vol. 1 in Introduction to Plasma Physics and Controlled Fusion. Springer, 1984. Google Scholar

[14]

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

[15]

P. Crispel, P. Degond and M.-H. Vignal, An asymptotically stable discretization for the Euler-Poisson system in the quasi-neutral limit, C. R. Math. Acad. Sci. Paris, 341 (2005), 323-328.  doi: 10.1016/j.crma.2005.07.008.  Google Scholar

[16]

P. Crispel, P. Degond and M.-H. Vignal, An asymptotic preserving scheme for the two-fluid Euler-Poisson model in the quasineutral limit, J. Comput. Phys., 223 (2007), 208-234.  doi: 10.1016/j.jcp.2006.09.004.  Google Scholar

[17]

G. Dal Maso, P. Le Floch and F. Murat., Definition and weak stability of nonconservative products, Journal de Mathématiques Pures et Appliquées, 74 (1995), 483-548.   Google Scholar

[18]

P. Degond and F. Deluzet, Asymptotic-preserving methods and multiscale models for plasma physics, J. Comput. Phys., 336 (2017), 429-457.  doi: 10.1016/j.jcp.2017.02.009.  Google Scholar

[19]

P. Degond, F. Deluzet and D. Savelief, Numerical approximation of the Euler-Maxwell model in the quasineutral limit, J. Comput. Phys., 231 (2012), 1917-1946.  doi: 10.1016/j.jcp.2011.11.011.  Google Scholar

[20]

P. Degond, H. Liu, D. Savelief and M.-H. Vignal, Numerical approximation of the Euler-Poisson-Boltzmann model in the quasineutral limit, J. Sci. Comput., 51 (2012), 59-86.  doi: 10.1007/s10915-011-9495-1.  Google Scholar

[21]

M. Dumbser and D. S. Balsara, A new efficient formulation of the HLLEM Riemann solver for general conservative and non-conservative hyperbolic systems, Journal of Computational Physics, 304 (2016), 275-319.  doi: 10.1016/j.jcp.2015.10.014.  Google Scholar

[22]

S. Guisset, S. Brull, E. D'Humières, B. Dubroca, S. Karpov and I. Potapenko, Asymptotic-preserving scheme for the M1-Maxwell system in the quasi-neutral regime, Communications in Computational Physics, 19 (2016), 301-328.  doi: 10.4208/cicp.131014.030615a.  Google Scholar

[23]

S. Jin, Efficient asymptotic-preserving (AP) schemes for some multiscale kinetic equations, SIAM J. Sci. Comput., 21 (1999), 441-454.  doi: 10.1137/S1064827598334599.  Google Scholar

[24]

C. Klingenberg, M. Pirner and G. Puppo, A consistent kinetic model for a two-component mixture with an application to plasma, Kinetic and Related Models, 10 (2017), 445-465.  doi: 10.3934/krm.2017017.  Google Scholar

[25]

C. Klingenberg, M. Pirner and G. Puppo, A consistent kinetic model for a two-component mixture of polyatomic molecules, Communication in Mathematical Sciences, 17 (2019), 149-173.  doi: 10.4310/CMS.2019.v17.n1.a6.  Google Scholar

[26]

P. G. LeFloch and S. Mishra, Numerical methods with controlled dissipation for small-scale dependent shocks, Acta Numerica, 23 (2014), 743-816.  doi: 10.1017/S0962492914000099.  Google Scholar

[27]

C. D. Levermore, Moment closure hierarchies for kinetic theories, Journal of Statistical Physics, 83 (1996), 1021-1065.  doi: 10.1007/BF02179552.  Google Scholar

[28]

C. Pars, Path-conservative numerical methods for nonconservative hyperbolic systems, In Numerical Methods for Balance Laws, volume 24 of Quad. Mat., pages 67–121. Dept. Math., Seconda Univ. Napoli, Caserta, 2009.  Google Scholar

[29]

K. Xu, X. He and C. Cai, Multiple temperature kinetic model and gas-kinetic method for hypersonic non-equilibrium flow computations, J. Comp. Phys., 227 (2008), 6779-6794.  doi: 10.1016/j.jcp.2008.03.035.  Google Scholar

Figure 1.  Density and velocity solutions of shock tube test case with a mass ratio of 10 with 120000 space points, 40 velocity points and a domain length of 8
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Figure 2.  Electronic and ionic temperatures of a shock tube test case with a mass ratio of 10 with 120000 space points, 40 velocity points and a domain length of 8
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Figure 3.  Different jump relations through shocks of electronic and ionic temperatures for a shock tube test case with a mass ratio of 10 with 120000 space points, 40 velocity points and a domain length of 8
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Figure 4.  Electric field obtained by the kinetic scheme and by Ohm's law for the Sod tube test case with identical initial temperatures
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Figure 5.  Density and velocity solutions of shock tube test case with a mass ratio of 10 with 120000 space points, 40 velocity points and a domain length of 8
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Figure 6.  Electronic and ionic temperatures of a shock tube test case with a mass ratio of 10 with 120000 space points, 40 velocity points and a domain length of 8
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Figure 7.  Different jump relations of electronic and ionic temperatures for a shock tube test case with a mass ratio of 10 with 120000 space points, 40 velocity points and a domain length of 8
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Figure 8.  Electric field obtained by the kinetic scheme and by Ohm's law for the Sod tube test case with different initial temperatures
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Figure 9.  Density and velocity solution of a rarefaction wave test case with a mass ratio of 10 with 120000 space points, 40 velocity points and a domain length of 8
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Figure 10.  Temperature solutions of a rarefaction wave test case with a mass ratio of 10 with 120000 space points, 40 velocity points and a domain length of 8
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Figure 11.  Zoom on temperature solutions of a rarefaction wave test case with a mass ratio of 10 with 120000 space points, 40 velocity points and a domain length of 8
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Figure 12.  Electric field obtained by the kinetic scheme and by Ohm's law for the double rarefaction test case
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Figure 13.  Density and veloctiy solutions of a shock wave test case with a mass ratio of 10 with 120000 space points, 40 velocity points and a domain length of 8
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Figure 14.  Electronic and ionic temperature of a shock wave test case with a mass ratio of 10 with 120000 space points, 40 velocity points and a domain length of 8
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Figure 15.  Electronic and ionic temperature of a shock wave test case with a mass ratio of 10 with 120000 space points, 40 velocity points and a domain length of 8
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Figure 16.  Electric field obtained by the kinetic scheme and by Ohm's law for the double shock test case
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Figure 17.  Density and velocity solutions of shock tube test case with a mass ratio of 1000 with 100000 space points, 40 velocity points and a domain length of 8
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Figure 18.  Electronic and ionic temperatures of a shock tube test case with a mass ratio of 1000 with 100000 space points, 40 velocity points and a domain length of 8
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Figure 19.  Different jump relations of electronic and ionic temperatures for a shock tube test case with a mass ratio of 1000 with 100000 space points, 40 velocity points and a domain length of 8
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Figure 20.  Electric field obtained by the kinetic scheme and by Ohm's law for a Sod tube test with different initial temperatures
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