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The Neumann numerical boundary condition for transport equations
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 |
We are interested in the numerical approximation of the bi-temperature Euler equations, which is a non conservative hyperbolic system introduced in [
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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.
|
[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. |
[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. |
[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. |
[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. |
[13] |
F. Chen, Introduction to Plasma Physics and Controlled Fusion, Number vol. 1 in Introduction to Plasma Physics and Controlled Fusion. Springer, 1984. |
[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. |
[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. |
[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. |
[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.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[27] |
C. D. Levermore,
Moment closure hierarchies for kinetic theories, Journal of Statistical Physics, 83 (1996), 1021-1065.
doi: 10.1007/BF02179552. |
[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. |
[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. |
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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.
|
[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. |
[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. |
[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. |
[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. |
[13] |
F. Chen, Introduction to Plasma Physics and Controlled Fusion, Number vol. 1 in Introduction to Plasma Physics and Controlled Fusion. Springer, 1984. |
[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. |
[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. |
[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. |
[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.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[27] |
C. D. Levermore,
Moment closure hierarchies for kinetic theories, Journal of Statistical Physics, 83 (1996), 1021-1065.
doi: 10.1007/BF02179552. |
[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. |
[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. |




















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