October  2021, 14(5): 895-928. doi: 10.3934/krm.2021029

BGK models for inert mixtures: Comparison and applications

1. 

Department of Mathematics and Computer Science, University of Catania, Viale Andrea Doria 6, 95125 Catania, Italy

2. 

Department of Mathematics and Research Institute of Natural Science, Gyeongsang National University, Jinju-daero 501, 52828 Jinju, Republic of Korea

3. 

Department of Mathematical, Physical and Computer Sciences, University of Parma Parco Area delle Scienze 53/A, I–43124 Parma, Italy

* Corresponding author: Seung Yeon Cho

Received  February 2021 Revised  July 2021 Published  October 2021 Early access  September 2021

Consistent BGK models for inert mixtures are compared, first in their kinetic behavior and then versus the hydrodynamic limits that can be derived in different collision-dominated regimes. The comparison is carried out both analytically and numerically, for the latter using an asymptotic preserving semi-Lagrangian scheme for the BGK models. Application to the plane shock wave in a binary mixture of noble gases is also presented.

Citation: Sebastiano Boscarino, Seung Yeon Cho, Maria Groppi, Giovanni Russo. BGK models for inert mixtures: Comparison and applications. Kinetic and Related Models, 2021, 14 (5) : 895-928. doi: 10.3934/krm.2021029
References:
[1]

A. AimiM. DiligentiM. Groppi and C. Guardasoni, On the numerical solution of a BGK-type model for chemical reactions, Eur. J. Mech. B Fluids, 26 (2007), 455-472.  doi: 10.1016/j.euromechflu.2006.10.001.

[2]

P. AndriesK. Aoki and B. Perthame, A consistent BGK-type model for gas mixtures, J. Stat. Phys., 106 (2002), 993-1018.  doi: 10.1023/A:1014033703134.

[3]

P. L. BhatnagarE. P. Gross and K. Krook, A model for collision processes in gases, Phys. Rev., 94 (1954), 511-524.  doi: 10.1103/PhysRev.94.511.

[4]

M. BisiA. V. BobylevM. Groppi and G. Spiga, Hydrodynamic equations from a BGK model for inert gas mixtures, In: AIP Conference Proceedings, AIP Publishing LLC, 2132 (2019), 130010.  doi: 10.1063/1.5119630.

[5]

M. BisiM. Groppi and G. Martalò, Macroscopic equations for inert gas mixtures in different hydrodynamic regimes, J. Phys. A: Math. and Theor., 54 (2021), 085201.  doi: 10.1088/1751-8121/abbd1b.

[6]

M. BisiM. Groppi and G. Martalò, The evaporation-condensation problem for a binary mixture of rarefied gases, Contin. Mech. Thermodyn., 32 (2020), 1109-1126.  doi: 10.1007/s00161-019-00814-x.

[7]

M. BisiM. Groppi and G. Spiga, Kinetic Bhatnagar-Gross-Krook model for fast reactive mixtures and its hydrodynamic limit, Phys. Rev. E, 81 (2010), 036327.  doi: 10.1103/PhysRevE.81.036327.

[8]

M. Bisi and G. Spiga, Navier–Stokes hydrodynamic limit of BGK kinetic equations for an inert mixture of polyatomic gases, In: "From Particle Systems to Partial Differential Equations V" (eds. P. Goncalves and A. J. Soares), Springer Proceedings in Mathematics and Statistics, 258 (2018), 13-31.  doi: 10.1007/978-3-319-99689-9_1.

[9]

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

[10]

S. Boscarino, S. Y. Cho and G. Russo, A local velocity grid conservative semi-Lagrangian schemes for BGK model, preprint, arXiv: 2107.08626.

[11]

S. BoscarinoS.-Y. ChoG. Russo and S.-B. Yun, High order conservative semi-Lagrangian scheme for the BGK model of the Boltzmann equation, Commun. Comput. Phys., 29 (2021), 1-56.  doi: 10.4208/cicp.OA-2020-0050.

[12]

S. Boscarino, S.-Y. Cho, G. Russo and S.-B. Yun, Convergence estimates of a semi-Lagrangian scheme for the ellipsoidal BGK model for polyatomic molecules, preprint, arXiv: 2003.00215.

[13]

S. Brull and C. Prigent, Local discrete velocity grids for multi-species rarefied flow simulations, Commun. Comput. Phys., 28 (2020), 1274-1304.  doi: 10.4208/cicp.OA-2019-0089.

[14]

C. Cercignani, The Boltzmann Equation and its Applications, Springer, New York, 1988. doi: 10.1007/978-1-4612-1039-9.

[15]

S. Y. Cho, S. Boscarino, M. Groppi and G. Russo, Conservative semi-Lagrangian schemes for a general consistent BGK model for inert gas mixtures, preprint, arXiv: 2012.02497.

[16]

S. Y. ChoS. BoscarinoG. Russo and S.-B. Yun, Conservative semi-Lagrangian schemes for kinetic equations Part I: Reconstruction, J. Comput. Phys., 432 (2021), 110159.  doi: 10.1016/j.jcp.2021.110159.

[17]

S. Y. ChoS. BoscarinoG. Russo and S.-B. Yun, Conservative semi-Lagrangian schemes for kinetic equations Part II: Applications, J. Comput. Phys., 436 (2021), 110281.  doi: 10.1016/j.jcp.2021.110281.

[18]

C. K. Chu, Kinetic-theoretic description of the formation of a shock wave, Phys. Fluids, 8 (1965), 12-22.  doi: 10.1063/1.1761077.

[19]

I. CraveroG. PuppoM. Semplice and G. Visconti, CWENO: Uniformly accurate reconstructions for balance laws, Math. Comp., 87 (2018), 1689-1719.  doi: 10.1090/mcom/3273.

[20]

V. S. Galkin and N. K. Makashev, Kinetic derivation of the gas-dynamic equation for multicomponent mixtures of light and heavy particles, Fluid Dyn., 29 (1994), 140-155.  doi: 10.1007/BF02330636.

[21]

M. Groppi, S. Rjasanow and G. Spiga, A kinetic relaxation approach to fast reactive mixtures: Shock wave structure, J. Stat. Mech. Theory Exp., 2009 (2009), P10010. doi: 10.1088/1742-5468/2009/10/P10010.

[22]

M. GroppiG. Russo and G. Stracquadanio, High order semi-Lagrangian methods for the BGK equation, Commun. Math. Sci., 14 (2016), 389-414.  doi: 10.4310/CMS.2016.v14.n2.a4.

[23]

M. GroppiG. Russo and G. Stracquadanio, Boundary conditions for semi-Lagrangian methods for the BGK model, Commun. Appl. Ind. Math., 7 (2016), 138-164.  doi: 10.1515/caim-2016-0025.

[24]

M. Groppi, G. Russo and G. Stracquadanio, Semi-Lagrangian approximation of BGK models for inert and reactive gas mixtures, In: "From Particle Systems to Partial Differential Equations V" ((Eds.) P. Gonçalves and A. Soares), Springer Proceedings in Mathematics and Statistics, 258 (2018), 53–80. doi: 10.1007/978-3-319-99689-9_5.

[25]

M. Groppi and G. Spiga, A Bhatnagar-Gross-Krook-type approach for chemically reacting gas mixtures, Phys. Fluids, 16 (2004), 4273-4284.  doi: 10.1063/1.1808651.

[26]

J. R. HaackC. D. Hauck and M. S. Murillo, A conservative, entropic multispecies BGK model, J. Stat. Phys., 168 (2017), 826-856.  doi: 10.1007/s10955-017-1824-9.

[27]

J. KestinK. KnierimE. A. MasonB. NajafiS. T. Ro and M. Waldman, Equilibrium and transport properties of the noble gases and their mixtures at low density, J. Phys. Chem. Ref. Data, 13 (1984), 229-303.  doi: 10.1063/1.555703.

[28]

C. Klingenberg and M. Pirner, Existence, uniqueness and positivity of solutions for BGK models for mixtures, J. Differential Equations, 264 (2018), 702-727.  doi: 10.1016/j.jde.2017.09.019.

[29]

C. KlingenbergM. Pirner and G. Puppo, A consistent kinetic model for a two-component mixture with an application to plasma, Kinet. Relat. Models, 10 (2017), 445-465.  doi: 10.3934/krm.2017017.

[30] M. N. Kogan, Rarefied Gas Dynamics, Plenum Press, New York, 1969.  doi: 10.1007/978-1-4899-6381-9.
[31]

D. LevyG. Puppo and G. Russo, Central WENO schemes for hyperbolic systems of conservation laws, ESAIM: Math. Model. Numer. Anal., 33 (1999), 547-571.  doi: 10.1051/m2an:1999152.

[32]

D. Madjarević and S. Simić, Shock structure in helium-argon mixture-a comparison of hyperbolic multi-temperature model with experiment, EPL, 102 (2013), 44002.  doi: 10.1209/0295-5075/102/44002.

[33]

T. Ruggeri and S. Simić, On the hyperbolic system of a mixture of Eulerian fluids: A comparison between single- and multi-temperature models, Math. Methods Appl. Sci., 30 (2007), 827-849.  doi: 10.1002/mma.813.

[34]

G. Russo and F. Filbet, Semilagrangian schemes applied to moving boundary problems for the BGK model of rarefied gas dynamics, Kinet. Relat. Models, 2 (2009), 231-250.  doi: 10.3934/krm.2009.2.231.

[35]

G. RussoP. Santagati and S.-B. Yun, Convergence of a semi-Lagrangian scheme for the BGK model of the Boltzmann equation, SIAM J. Numer. Anal., 50 (2012), 1111-1135.  doi: 10.1137/100800348.

[36]

G. Russo and S.-B. Yun, Convergence of a semi-Lagrangian scheme for the ellipsoidal BGK model of the Boltzmann equation, SIAM J. Numer. Anal., 56 (2018), 3580-3610.  doi: 10.1137/17M1163360.

[37]

P. Santagati and G. Russo, A new class of large time step methods for the BGK models of the Boltzmann equation, preprint, arXiv: 1103.5247.

[38]

S. SimićM. Pavic-Colic and D. Madjarević, Non-equilibrium mixtures of gases: Modelling and computation, Riv. di Mat. della Univ. di Parma, 6 (2015), 135-214. 

[39]

J. Vranjes and P. S. Krstic, Collisions, magnetization, and transport coefficients in the lower solar atmosphere, Astron. Astrophys., 554 (2013), A22. doi: 10.1051/0004-6361/201220738.

show all references

References:
[1]

A. AimiM. DiligentiM. Groppi and C. Guardasoni, On the numerical solution of a BGK-type model for chemical reactions, Eur. J. Mech. B Fluids, 26 (2007), 455-472.  doi: 10.1016/j.euromechflu.2006.10.001.

[2]

P. AndriesK. Aoki and B. Perthame, A consistent BGK-type model for gas mixtures, J. Stat. Phys., 106 (2002), 993-1018.  doi: 10.1023/A:1014033703134.

[3]

P. L. BhatnagarE. P. Gross and K. Krook, A model for collision processes in gases, Phys. Rev., 94 (1954), 511-524.  doi: 10.1103/PhysRev.94.511.

[4]

M. BisiA. V. BobylevM. Groppi and G. Spiga, Hydrodynamic equations from a BGK model for inert gas mixtures, In: AIP Conference Proceedings, AIP Publishing LLC, 2132 (2019), 130010.  doi: 10.1063/1.5119630.

[5]

M. BisiM. Groppi and G. Martalò, Macroscopic equations for inert gas mixtures in different hydrodynamic regimes, J. Phys. A: Math. and Theor., 54 (2021), 085201.  doi: 10.1088/1751-8121/abbd1b.

[6]

M. BisiM. Groppi and G. Martalò, The evaporation-condensation problem for a binary mixture of rarefied gases, Contin. Mech. Thermodyn., 32 (2020), 1109-1126.  doi: 10.1007/s00161-019-00814-x.

[7]

M. BisiM. Groppi and G. Spiga, Kinetic Bhatnagar-Gross-Krook model for fast reactive mixtures and its hydrodynamic limit, Phys. Rev. E, 81 (2010), 036327.  doi: 10.1103/PhysRevE.81.036327.

[8]

M. Bisi and G. Spiga, Navier–Stokes hydrodynamic limit of BGK kinetic equations for an inert mixture of polyatomic gases, In: "From Particle Systems to Partial Differential Equations V" (eds. P. Goncalves and A. J. Soares), Springer Proceedings in Mathematics and Statistics, 258 (2018), 13-31.  doi: 10.1007/978-3-319-99689-9_1.

[9]

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

[10]

S. Boscarino, S. Y. Cho and G. Russo, A local velocity grid conservative semi-Lagrangian schemes for BGK model, preprint, arXiv: 2107.08626.

[11]

S. BoscarinoS.-Y. ChoG. Russo and S.-B. Yun, High order conservative semi-Lagrangian scheme for the BGK model of the Boltzmann equation, Commun. Comput. Phys., 29 (2021), 1-56.  doi: 10.4208/cicp.OA-2020-0050.

[12]

S. Boscarino, S.-Y. Cho, G. Russo and S.-B. Yun, Convergence estimates of a semi-Lagrangian scheme for the ellipsoidal BGK model for polyatomic molecules, preprint, arXiv: 2003.00215.

[13]

S. Brull and C. Prigent, Local discrete velocity grids for multi-species rarefied flow simulations, Commun. Comput. Phys., 28 (2020), 1274-1304.  doi: 10.4208/cicp.OA-2019-0089.

[14]

C. Cercignani, The Boltzmann Equation and its Applications, Springer, New York, 1988. doi: 10.1007/978-1-4612-1039-9.

[15]

S. Y. Cho, S. Boscarino, M. Groppi and G. Russo, Conservative semi-Lagrangian schemes for a general consistent BGK model for inert gas mixtures, preprint, arXiv: 2012.02497.

[16]

S. Y. ChoS. BoscarinoG. Russo and S.-B. Yun, Conservative semi-Lagrangian schemes for kinetic equations Part I: Reconstruction, J. Comput. Phys., 432 (2021), 110159.  doi: 10.1016/j.jcp.2021.110159.

[17]

S. Y. ChoS. BoscarinoG. Russo and S.-B. Yun, Conservative semi-Lagrangian schemes for kinetic equations Part II: Applications, J. Comput. Phys., 436 (2021), 110281.  doi: 10.1016/j.jcp.2021.110281.

[18]

C. K. Chu, Kinetic-theoretic description of the formation of a shock wave, Phys. Fluids, 8 (1965), 12-22.  doi: 10.1063/1.1761077.

[19]

I. CraveroG. PuppoM. Semplice and G. Visconti, CWENO: Uniformly accurate reconstructions for balance laws, Math. Comp., 87 (2018), 1689-1719.  doi: 10.1090/mcom/3273.

[20]

V. S. Galkin and N. K. Makashev, Kinetic derivation of the gas-dynamic equation for multicomponent mixtures of light and heavy particles, Fluid Dyn., 29 (1994), 140-155.  doi: 10.1007/BF02330636.

[21]

M. Groppi, S. Rjasanow and G. Spiga, A kinetic relaxation approach to fast reactive mixtures: Shock wave structure, J. Stat. Mech. Theory Exp., 2009 (2009), P10010. doi: 10.1088/1742-5468/2009/10/P10010.

[22]

M. GroppiG. Russo and G. Stracquadanio, High order semi-Lagrangian methods for the BGK equation, Commun. Math. Sci., 14 (2016), 389-414.  doi: 10.4310/CMS.2016.v14.n2.a4.

[23]

M. GroppiG. Russo and G. Stracquadanio, Boundary conditions for semi-Lagrangian methods for the BGK model, Commun. Appl. Ind. Math., 7 (2016), 138-164.  doi: 10.1515/caim-2016-0025.

[24]

M. Groppi, G. Russo and G. Stracquadanio, Semi-Lagrangian approximation of BGK models for inert and reactive gas mixtures, In: "From Particle Systems to Partial Differential Equations V" ((Eds.) P. Gonçalves and A. Soares), Springer Proceedings in Mathematics and Statistics, 258 (2018), 53–80. doi: 10.1007/978-3-319-99689-9_5.

[25]

M. Groppi and G. Spiga, A Bhatnagar-Gross-Krook-type approach for chemically reacting gas mixtures, Phys. Fluids, 16 (2004), 4273-4284.  doi: 10.1063/1.1808651.

[26]

J. R. HaackC. D. Hauck and M. S. Murillo, A conservative, entropic multispecies BGK model, J. Stat. Phys., 168 (2017), 826-856.  doi: 10.1007/s10955-017-1824-9.

[27]

J. KestinK. KnierimE. A. MasonB. NajafiS. T. Ro and M. Waldman, Equilibrium and transport properties of the noble gases and their mixtures at low density, J. Phys. Chem. Ref. Data, 13 (1984), 229-303.  doi: 10.1063/1.555703.

[28]

C. Klingenberg and M. Pirner, Existence, uniqueness and positivity of solutions for BGK models for mixtures, J. Differential Equations, 264 (2018), 702-727.  doi: 10.1016/j.jde.2017.09.019.

[29]

C. KlingenbergM. Pirner and G. Puppo, A consistent kinetic model for a two-component mixture with an application to plasma, Kinet. Relat. Models, 10 (2017), 445-465.  doi: 10.3934/krm.2017017.

[30] M. N. Kogan, Rarefied Gas Dynamics, Plenum Press, New York, 1969.  doi: 10.1007/978-1-4899-6381-9.
[31]

D. LevyG. Puppo and G. Russo, Central WENO schemes for hyperbolic systems of conservation laws, ESAIM: Math. Model. Numer. Anal., 33 (1999), 547-571.  doi: 10.1051/m2an:1999152.

[32]

D. Madjarević and S. Simić, Shock structure in helium-argon mixture-a comparison of hyperbolic multi-temperature model with experiment, EPL, 102 (2013), 44002.  doi: 10.1209/0295-5075/102/44002.

[33]

T. Ruggeri and S. Simić, On the hyperbolic system of a mixture of Eulerian fluids: A comparison between single- and multi-temperature models, Math. Methods Appl. Sci., 30 (2007), 827-849.  doi: 10.1002/mma.813.

[34]

G. Russo and F. Filbet, Semilagrangian schemes applied to moving boundary problems for the BGK model of rarefied gas dynamics, Kinet. Relat. Models, 2 (2009), 231-250.  doi: 10.3934/krm.2009.2.231.

[35]

G. RussoP. Santagati and S.-B. Yun, Convergence of a semi-Lagrangian scheme for the BGK model of the Boltzmann equation, SIAM J. Numer. Anal., 50 (2012), 1111-1135.  doi: 10.1137/100800348.

[36]

G. Russo and S.-B. Yun, Convergence of a semi-Lagrangian scheme for the ellipsoidal BGK model of the Boltzmann equation, SIAM J. Numer. Anal., 56 (2018), 3580-3610.  doi: 10.1137/17M1163360.

[37]

P. Santagati and G. Russo, A new class of large time step methods for the BGK models of the Boltzmann equation, preprint, arXiv: 1103.5247.

[38]

S. SimićM. Pavic-Colic and D. Madjarević, Non-equilibrium mixtures of gases: Modelling and computation, Riv. di Mat. della Univ. di Parma, 6 (2015), 135-214. 

[39]

J. Vranjes and P. S. Krstic, Collisions, magnetization, and transport coefficients in the lower solar atmosphere, Astron. Astrophys., 554 (2013), A22. doi: 10.1051/0004-6361/201220738.

Figure 1.  Time evolution of relative $ L^1 $-norm of the differences in the distribution functions $ g_1 $ between BGK models for various values of $ \varepsilon $. In (a) and (b), the $ x $-axes stand for time and the $ y $-axes are the values obtained by (30)
Figure 2.  Comparison of the three BGK models for $ \varepsilon = 10^{-2} $ (Left) and $ \varepsilon = 10^{-3} $ (Right) with initial data in (29)
Figure 3.  Comparison of the three BGK models for $ \varepsilon = 10^{-4} $ with initial data in (29)
Figure 4.  Comparison of BGK model (10) and NS equations (15) for $ \varepsilon = 10^{-2} $ with initial data in (31)
Figure 5.  Comparison of BGK model (10) and NS equations (15) for $ \varepsilon = 10^{-3} $ (Left) and $ \varepsilon = 10^{-4} $ (Right) with initial data in (31)
Figure 6.  Comparison of the scaled BBGSP model (23) and Navier-Stokes equations (24) with $ \kappa = 1 $ for $ \varepsilon = 10^{-2} $ with initial data in (31)
Figure 7.  Comparison of the scaled BBGSP model (23) and Navier-Stokes equations (24) with $ \kappa = 1 $ for $ \varepsilon = 10^{-3} $ (Left) and $ \varepsilon = 10^{-4} $ (Right) with initial data in (31)
Figure 8.  Comparison of the numerical solution of the scaled BBGSP model (23) for $ \varepsilon = \kappa = 10^{-3} $ with: (left) global velocity and temperature Euler system (15) for $ \varepsilon = 0 $ and (right) multi-velocity and multi-temperature Euler system (24) for $ \varepsilon = 0 $, $ \kappa = 10^{-3} $. We use the initial data in (6.3)
Figure 9.  Comparison of the scaled BBGSP model (23) for $ \varepsilon = \kappa = 10^{-4} $ with: (left) global velocity and temperature Euler system (15) for $ \varepsilon = 0 $ and (right) multi-velocity and multi-temperature Euler system (24) for $ \varepsilon = 0 $, $ \kappa = 10^{-4} $
Figure 10.  Comparison of the scaled BBGSP model (23) for $ \varepsilon = \kappa = 10^{-5} $ with multi-velocity and multi-temperature Euler system (24) for $ \varepsilon = 0 $, $ \kappa = 10^{-5} $
Figure 11.  BDF3-QCWENO35 for $ \varepsilon = 10^{-0} $. Neon and Argon with $ n_1 = 0.1m_1,\quad n_2 = 0.9m_2 $. Black dashed lines are reference NS solutions and solid lines are BGK solutions
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