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Boltzmann-type equations for multi-agent systems with label switching
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 |
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.
References:
[1] |
A. Aimi, M. Diligenti, M. 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. Andries, K. 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. Bhatnagar, E. 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. Bisi, A. V. Bobylev, M. 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. Bisi, M. 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. Bisi, M. 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. Bisi, M. 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. Bobylev, M. Bisi, M. Groppi, G. 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. Boscarino, S.-Y. Cho, G. 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. Cho, S. Boscarino, G. 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. Cho, S. Boscarino, G. 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. Cravero, G. Puppo, M. 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. Groppi, G. 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. Groppi, G. 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. Haack, C. 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. Kestin, K. Knierim, E. A. Mason, B. Najafi, S. 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. Klingenberg, M. 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. Levy, G. 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. Russo, P. 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. Simić, M. 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. Aimi, M. Diligenti, M. 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. Andries, K. 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. Bhatnagar, E. 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. Bisi, A. V. Bobylev, M. 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. Bisi, M. 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. Bisi, M. 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. Bisi, M. 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. Bobylev, M. Bisi, M. Groppi, G. 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. Boscarino, S.-Y. Cho, G. 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. Cho, S. Boscarino, G. 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. Cho, S. Boscarino, G. 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. Cravero, G. Puppo, M. 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. Groppi, G. 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. Groppi, G. 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. Haack, C. 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. Kestin, K. Knierim, E. A. Mason, B. Najafi, S. 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. Klingenberg, M. 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. Levy, G. 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. Russo, P. 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. Simić, M. 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. |











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