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

A kinetic approach of the bi-temperature Euler model

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).

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. AndriesP. Le TallecJ.-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-DriolletJ. BreilS. BrullB. 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. BelaouarN. CrouseillesP. 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. BobylevM. BisiM. GroppiG. 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. BrullP. DegondF. 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. ChalonsM. 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. CrispelP. 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. CrispelP. 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 MasoP. 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. DegondF. 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. DegondH. LiuD. 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. GuissetS. BrullE. D'HumièresB. DubrocaS. 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. KlingenbergM. 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. KlingenbergM. 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. XuX. 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. AndriesP. Le TallecJ.-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-DriolletJ. BreilS. BrullB. 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. BelaouarN. CrouseillesP. 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. BobylevM. BisiM. GroppiG. 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. BrullP. DegondF. 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. ChalonsM. 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. CrispelP. 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. CrispelP. 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 MasoP. 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. DegondF. 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. DegondH. LiuD. 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. GuissetS. BrullE. D'HumièresB. DubrocaS. 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. KlingenbergM. 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. KlingenbergM. 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. XuX. 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
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
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
Figure 4.  Electric field obtained by the kinetic scheme and by Ohm's law for the Sod tube test case with identical initial temperatures
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
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
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
Figure 8.  Electric field obtained by the kinetic scheme and by Ohm's law for the Sod tube test case with different initial temperatures
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
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
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
Figure 12.  Electric field obtained by the kinetic scheme and by Ohm's law for the double rarefaction test case
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
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
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
Figure 16.  Electric field obtained by the kinetic scheme and by Ohm's law for the double shock test case
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
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
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
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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