F. Alexander, H. Chen, S. Chen, and G. Doolen, Lattice Boltzmann model for compressible fluids, Physical Review A, vol.46, issue.4, pp.1967-1970, 1992.
DOI : 10.1103/PhysRevA.46.1967

D. Aregba-driollet and R. Natalini, Discrete Kinetic Schemes for Multidimensional Systems of Conservation Laws, SIAM Journal on Numerical Analysis, vol.37, issue.6, pp.1973-2004, 2000.
DOI : 10.1137/S0036142998343075
URL : https://hal.archives-ouvertes.fr/hal-00959531

D. Aregba-driollet, R. Natalini, and S. Tang, Explicit diffusive kinetic schemes for nonlinear degenerate parabolic systems, Mathematics of Computation, vol.73, issue.245, pp.63-94, 2004.
DOI : 10.1090/S0025-5718-03-01549-7
URL : https://hal.archives-ouvertes.fr/hal-00387865

B. F. Armaly, F. Durst, J. C. Pereira, and B. Schonung, Experimental and theoretical investigation of backward-facing step flow, Journal of Fluid Mechanics, vol.86, issue.-1, pp.473-496, 1983.
DOI : 10.1017/S0022112067001892

P. Asinari, T. Ohwada, E. Chiavazzo, and A. F. Di-rienzo, Link-wise artificial compressibility method, Journal of Computational Physics, vol.231, issue.15, pp.5109-5143, 2012.
DOI : 10.1016/j.jcp.2012.04.027
URL : https://hal.archives-ouvertes.fr/hal-01287495

E. Audusse, M. Bristeau, B. Perthame, and J. Sainte-marie, A multilayer Saint-Venant system with mass exchanges for shallow water flows. Derivation and numerical validation, ESAIM: Mathematical Modelling and Numerical Analysis, vol.45, issue.1, pp.45-169, 2011.
DOI : 10.1051/m2an/2010036
URL : https://hal.archives-ouvertes.fr/inria-00551428

C. Bardos, F. Golse, and C. D. Levermore, Fluid dynamic limits of kinetic equations. I. Formal derivations, Journal of Statistical Physics, vol.16, issue.1-2, pp.323-344, 1991.
DOI : 10.1007/BF01026608

C. Bardos, F. Golse, and C. D. Levermore, Fluid dynamic limits of kinetic equations II convergence proofs for the boltzmann equation, Communications on Pure and Applied Mathematics, vol.82, issue.162, pp.667-753, 1993.
DOI : 10.1002/cpa.3160460503

C. Bardos, F. Golse, and C. D. Levermore, The Acoustic Limit for the Boltzmann Equation, Archive for Rational Mechanics and Analysis, vol.153, issue.3, pp.177-204, 2000.
DOI : 10.1007/s002050000080

F. Berthelin and F. Bouchut, Kinetic invariant domains and relaxation limit from a BGK model to isentropic gas dynamics, Asymptotic analysis, pp.31153-176, 2002.

F. Berthelin and F. Bouchut, Relaxation to isentropic gas dynamics for a BGK system with single kinetic entropy, Methods and Applications of Analysis, vol.9, issue.2, pp.313-327, 2002.
DOI : 10.4310/MAA.2002.v9.n2.a7

G. Biswas, M. Breuer, and F. Durst, Backward-Facing Step Flows for Various Expansion Ratios at Low and Moderate Reynolds Numbers, Journal of Fluids Engineering, vol.126, issue.3, pp.362-374, 2004.
DOI : 10.1115/1.1760532

J. Bernsdorf, Simulation of complex flows and multi-physics with the lattice-boltzmann method, pp.114-76, 2008.

B. Boghosian and C. Levermore, A cellular automaton for Burgers's equation, Complex Systems, vol.1, pp.17-30, 1987.

F. Bouchut, Construction of BGK models with a family of kinetic entropies for a given system of conservation laws, J. Stat. Phys, pp.95-113, 1999.

F. Bouchut, Entropy satisfying flux vector splittings and kinetic BGK models, Numerische Mathematik, vol.94, issue.4, pp.623-672, 2003.
DOI : 10.1007/s00211-002-0426-9

F. Bouchut, A REDUCED STABILITY CONDITION FOR NONLINEAR RELAXATION TO CONSERVATION LAWS, Journal of Hyperbolic Differential Equations, vol.01, issue.01, pp.149-170, 2004.
DOI : 10.1142/S0219891604000020

F. Bouchut, Nonlinear stability of finite volume methods for hyperbolic conservation laws, and well-balanced schemes for sources, Frontiers in Mathematics series, 2004.

F. Bouchut and H. Frid, FINITE DIFFERENCE SCHEMES WITH CROSS DERIVATIVES CORRECTORS FOR MULTIDIMENSIONAL PARABOLIC SYSTEMS, Journal of Hyperbolic Differential Equations, vol.03, issue.01, pp.27-52, 2006.
DOI : 10.1142/S0219891606000719

F. Bouchut, F. R. Guarguaglini, and R. Natalini, Diffusive BGK approximations for nonlinear multidimensional parabolic equations, Indiana University Mathematics Journal, vol.49, issue.2, pp.723-749, 2000.
DOI : 10.1512/iumj.2000.49.1811

F. Bouchut, C. Klingenberg, and K. Waagan, A multiwave approximate Riemann solver for ideal MHD based on relaxation. I: theoretical framework, Numerische Mathematik, vol.153, issue.2, pp.7-42, 2007.
DOI : 10.1007/s00211-007-0108-8

Y. Brenier, Résolution d'´ equations d'´ evolution quasilinéaires en dimension N d'espacè a l'aide d'´ equations linéaires en dimension N + 1, J

Y. Brenier, Averaged Multivalued Solutions for Scalar Conservation Laws, SIAM Journal on Numerical Analysis, vol.21, issue.6, pp.1013-1037, 1984.
DOI : 10.1137/0721063

D. Bresch, B. Desjardins, E. Grenier, and C. Lin, Low Mach Number Limit of Viscous Polytropic Flows: Formal Asymptotics in the Periodic Case, Studies in Applied Mathematics, vol.109, issue.2, pp.125-149, 2002.
DOI : 10.1111/1467-9590.01440

D. Bresch and G. Métivier, Anelastic Limits for Euler-Type Systems, Applied Mathematics Research eXpress, vol.2010, issue.2, pp.119-141
DOI : 10.1093/amrx/abq012
URL : https://hal.archives-ouvertes.fr/hal-00777129

M. Breuer, J. Bernsdorf, T. Zeiser, and F. Durst, Accurate computations of the laminar flow past a square cylinder based on two different methods: lattice-Boltzmann and finite-volume, International Journal of Heat and Fluid Flow, vol.21, issue.2, pp.186-196, 2000.
DOI : 10.1016/S0142-727X(99)00081-8

M. Carfora and R. Natalini, A discrete kinetic approximation for the incompressible Navier-Stokes equations, ESAIM: Mathematical Modelling and Numerical Analysis, vol.42, issue.1, pp.93-112, 2008.
DOI : 10.1051/m2an:2007055

C. Chalons, M. Girardin, and S. Kokh, Abstract, Communications in Computational Physics, vol.35, issue.01, 2016.
DOI : 10.1016/j.compfluid.2013.07.019

G. Q. Chen, C. D. Levermore, and T. Liu, Hyperbolic conservation laws with stiff relaxation terms and entropy, Communications on Pure and Applied Mathematics, vol.44, issue.6, pp.47-787, 1994.
DOI : 10.1002/cpa.3160470602
URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.50.312

S. Chikatamarla and I. Karlin, Lattices for the lattice Boltzmann method, Physical Review E, vol.79, issue.4, p.46701, 2009.
DOI : 10.1103/PhysRevE.79.046701

D. S. Clague, B. D. Kandhai, R. Zhang, and P. M. Sloot, Hydraulic permeability of (un)bounded fibrous media using the lattice Boltzmann method, Physical Review E, vol.61, issue.1, pp.61-62, 2000.
DOI : 10.1103/PhysRevE.61.616

F. Cordier, P. Degond, and A. Kumbaro, An Asymptotic-Preserving all-speed scheme for the Euler and Navier???Stokes equations, Journal of Computational Physics, vol.231, issue.17, pp.5685-5704, 2012.
DOI : 10.1016/j.jcp.2012.04.025
URL : https://hal.archives-ouvertes.fr/hal-00614662

H. Darcy, Les fontaines publiques de la ville de Dijon, p.Dalmont, 1856.

P. Degond and M. Tang, Abstract, Communications in Computational Physics, vol.141, issue.01, pp.1-31, 2011.
DOI : 10.4208/cicp.210709.210610a

S. Dellacherie, Analysis of Godunov type schemes applied to the compressible Euler system at low Mach number, Journal of Computational Physics, vol.229, issue.4, pp.978-1016, 2010.
DOI : 10.1016/j.jcp.2009.09.044

S. Dellacherie, Construction and Analysis of Lattice Boltzmann Methods Applied to a 1D Convection-Diffusion Equation, Acta Applicandae Mathematicae, vol.11, issue.1, pp.69-140, 2014.
DOI : 10.1007/s10440-013-9850-3
URL : https://hal.archives-ouvertes.fr/hal-00717873

S. Dellacherie and P. Omnes, On the Godunov scheme applied to the variable cross-section linear wave equation, Finite volumes for complex applications vi: problems & perspectives, pp.313-321, 2011.

D. Humieres, I. Ginzburg, M. Krafczyk, P. Lallemand, and L. S. Luo, Multiple-relaxation-time lattice Boltzmann models in three dimensions, Phil. Trans. Royal Soc. London series A-Math. Phys. Eng. Sci, pp.360-437, 2002.

F. Dubois, D??composition de flux et discontinuit?? de contact stationnaire, Comptes Rendus de l'Acad??mie des Sciences - Series I - Mathematics, vol.330, issue.9, pp.847-850, 2000.
DOI : 10.1016/S0764-4442(00)00255-X

B. Elton, C. Levermore, and G. Rodrigue, Convergence of Convective???Diffusive Lattice Boltzmann Methods, SIAM Journal on Numerical Analysis, vol.32, issue.5, pp.1327-1354, 1995.
DOI : 10.1137/0732062

E. Erturk, Numerical solutions of 2-D steady incompressible flow over a backward-facing step, Part I: High Reynolds number solutions, Computers & Fluids, vol.37, issue.6, pp.633-655, 2008.
DOI : 10.1016/j.compfluid.2007.09.003

E. Florez, Y. Carranza, and Y. Ortiz, Numerical flow solutions on a backward-facing step using the lattice Boltzmann equation method, pp.31-74, 2011.

U. Ghia, K. N. Ghia, and C. T. Shin, High-Re solutions for incompressible flow using the Navier-Stokes equations and a multigrid method, Journal of Computational Physics, vol.48, issue.3, pp.48-387, 1982.
DOI : 10.1016/0021-9991(82)90058-4

J. Gressier, P. Villedieu, and J. Moschetta, Positivity of Flux Vector Splitting Schemes, Journal of Computational Physics, vol.155, issue.1, pp.199-220, 1999.
DOI : 10.1006/jcph.1999.6337

M. M. Gupta and J. C. Kalita, A new paradigm for solving Navier???Stokes equations: streamfunction???velocity formulation, Journal of Computational Physics, vol.207, issue.1, pp.52-68, 2005.
DOI : 10.1016/j.jcp.2005.01.002

F. J. Higuera and J. Jimenez, Boltzmann Approach to Lattice Gas Simulations, Europhysics Letters (EPL), vol.9, issue.7, pp.663-668, 1989.
DOI : 10.1209/0295-5075/9/7/009

T. Inamuro, M. Yoshino, and F. Ogino, Accuracy of the lattice Boltzmann method for small Knudsen number with finite Reynolds number, Physics of Fluids, vol.9, issue.11, p.3535, 1997.
DOI : 10.1063/1.869426

S. Jin, L. Pareschi, and G. Toscani, Diffusive Relaxation Schemes for Multiscale Discrete-Velocity Kinetic Equations, SIAM Journal on Numerical Analysis, vol.35, issue.6, pp.2405-2439, 1998.
DOI : 10.1137/S0036142997315962

S. Jin, L. Pareschi, and G. Toscani, Uniformly Accurate Diffusive Relaxation Schemes for Multiscale Transport Equations, SIAM Journal on Numerical Analysis, vol.38, issue.3, pp.913-936, 2000.
DOI : 10.1137/S0036142998347978

I. V. Karlin, A. N. Gorban, S. Succi, and V. Boffi, Maximum Entropy Principle for Lattice Kinetic Equations, Physical Review Letters, vol.81, issue.1, pp.6-9, 1998.
DOI : 10.1103/PhysRevLett.81.6

S. Klainerman and A. Majda, Compressible and incompressible fluids, Communications on Pure and Applied Mathematics, vol.33, issue.5, pp.629-651, 1982.
DOI : 10.1002/cpa.3160350503

P. Kumar and F. Topin, Micro-structural Impact of Different Strut Shapes and Porosity on Hydraulic Properties of Kelvin-Like Metal Foams, Transport in Porous Media 105, pp.57-81, 2014.
DOI : 10.1007/s11242-014-0358-8
URL : https://hal.archives-ouvertes.fr/hal-01459310

P. Lallemand and L. Luo, Theory of the lattice Boltzmann method: Dispersion, dissipation, isotropy, Galilean invariance, and stability, Physical Review E, vol.61, issue.6, pp.61-6546, 2000.
DOI : 10.1103/PhysRevE.61.6546

X. Li and C. Gu, An All-Speed Roe-type scheme and its asymptotic analysis of low Mach number behaviour, Journal of Computational Physics, vol.227, issue.10, pp.5144-5159, 2008.
DOI : 10.1016/j.jcp.2008.01.037

P. Lions and N. Masmoudi, Incompressible limit for a viscous compressible fluid, Journal de Math??matiques Pures et Appliqu??es, vol.77, issue.6, pp.585-627, 1998.
DOI : 10.1016/S0021-7824(98)80139-6

G. R. Mcnamara and G. Zanetti, Use of the Boltzmann Equation to Simulate Lattice-Gas Automata, Physical Review Letters, vol.61, issue.20, pp.2332-2335, 1988.
DOI : 10.1103/PhysRevLett.61.2332

J. C. Mandal and S. M. Deshpande, Higher Order Accurate Kinetic Flux Vector Splitting Method for Euler Equations, Nonlinear Hyperbolic Equations ? Theory, Computation Methods, and Applications, Notes on Numerical Fluid Mechanics Series, pp.384-392, 1989.
DOI : 10.1007/978-3-322-87869-4_39

B. Maury, The respiratory system in equations, Simulation and Applications, vol.7
DOI : 10.1007/978-88-470-5214-7
URL : https://hal.archives-ouvertes.fr/hal-00929739

R. Mei, L. S. Luo, P. Lallemand, and D. , Consistent initial conditions for lattice Boltzmann simulations, Computers & Fluids, vol.35, issue.8-9, pp.855-862, 2006.
DOI : 10.1016/j.compfluid.2005.08.008

X. Nicolas, M. Medale, S. Glockner, and S. Gounand, Benchmark Solution for a Three-Dimensional Mixed-Convection Flow, Part 1: Reference Solutions, Numerical Heat Transfer, Part B: Fundamentals, vol.37, issue.5, pp.325-345, 2011.
DOI : 10.1080/10407790.2011.616758
URL : https://hal.archives-ouvertes.fr/hal-00692092

S. Noelle, G. Bispen, K. R. Arun, M. Lukacova-medvidova, and C. Munz, A Weakly Asymptotic Preserving Low Mach Number Scheme for the Euler Equations of Gas Dynamics, SIAM Journal on Scientific Computing, vol.36, issue.6, pp.36-989, 2014.
DOI : 10.1137/120895627

M. A. Ol-'shanskii and V. M. Staroverov, On simulation of outflow boundary conditions in finite difference calculations for incompressible fluid, International Journal for Numerical Methods in Fluids, vol.127, issue.4, pp.499-534, 2000.
DOI : 10.1002/1097-0363(20000630)33:4<499::AID-FLD19>3.0.CO;2-7

H. Paillére, C. Viozat, A. Kumbaro, and I. Toumi, Comparison of low Mach number models for natural convection problems, Heat and Mass Transfer, vol.36, issue.6, pp.567-573, 2000.
DOI : 10.1007/s002310000116

T. C. Papanastasiou, N. Malamataris, and K. Ellwood, A new outflow boundary condition, International Journal for Numerical Methods in Fluids, vol.30, issue.5, pp.587-608, 1992.
DOI : 10.1002/fld.1650140506

B. Perthame, Boltzmann Type Schemes for Gas Dynamics and the Entropy Property, SIAM Journal on Numerical Analysis, vol.27, issue.6, pp.1405-1421, 1990.
DOI : 10.1137/0727081

B. Perthame and E. Tadmor, A kinetic equation with kinetic entropy functions for scalar conservation laws, Communications in Mathematical Physics, vol.81, issue.3, pp.501-517, 1991.
DOI : 10.1007/BF02099071

D. A. Perumal and A. K. Dass, Application of lattice Boltzmann method for incompressible viscous flows, Applied Mathematical Modelling, vol.37, issue.6, pp.4075-4092, 2013.
DOI : 10.1016/j.apm.2012.09.028

A. Poux, S. Glockner, E. Ahusborde, and M. Azaiez, Open boundary conditions for the velocity-correction scheme of the Navier???Stokes equations, Computers & Fluids, vol.70, pp.29-43, 2012.
DOI : 10.1016/j.compfluid.2012.08.028
URL : https://hal.archives-ouvertes.fr/hal-00861282

D. I. Pullin, Direct simulation methods for compressible inviscid ideal-gas flow, Journal of Computational Physics, vol.34, issue.2, pp.231-244, 1980.
DOI : 10.1016/0021-9991(80)90107-2

R. D. Reitz, One-dimensional compressible gas dynamics calculations using the Boltzmann equation, Journal of Computational Physics, vol.42, issue.1, pp.108-123, 1981.
DOI : 10.1016/0021-9991(81)90235-7

P. Renard, A. Genty, and F. Stauffer, Laboratory determination of the full permeability tensor, Journal of Geophysical Research: Solid Earth, vol.84, issue.4, pp.106-117, 2001.
DOI : 10.1029/2001JB000243

S. Schochet, Fast Singular Limits of Hyperbolic PDEs, Journal of Differential Equations, vol.114, issue.2, pp.476-512, 1994.
DOI : 10.1006/jdeq.1994.1157

A. Shah and L. Yuan, Flux-difference splitting-based upwind compact schemes for the incompressible Navier-Stokes equations, International Journal for Numerical Methods in Fluids, vol.167, issue.2, pp.552-568, 2009.
DOI : 10.1002/fld.1965

A. Shah, L. Yuan, and A. Khan, Upwind compact finite difference scheme for time-accurate solution of the incompressible Navier???Stokes equations, Applied Mathematics and Computation, vol.215, issue.9, pp.215-3201, 2010.
DOI : 10.1016/j.amc.2009.10.001

A. Sohankar, C. Norberg, and L. Davidson, Low-Reynolds-number flow around a square cylinder at incidence: study of blockage, onset of vortex shedding and outlet boundary condition, International Journal for Numerical Methods in Fluids, vol.89, issue.1, pp.39-56, 1998.
DOI : 10.1002/(SICI)1097-0363(19980115)26:1<39::AID-FLD623>3.0.CO;2-P

S. Succi, The lattice Boltzmann equation for fluid dynamics and beyond, Numerical Mathematics and Scientific Computation, 2001.

S. Succi, E. Foti, and F. , Three-Dimensional Flows in Complex Geometries with the Lattice Boltzmann Method, Europhysics Letters (EPL), vol.10, issue.5, pp.433-438, 1989.
DOI : 10.1209/0295-5075/10/5/008

S. Succi, I. V. Karlin, and H. Chen, theorem in lattice Boltzmann hydrodynamic simulations, Reviews of Modern Physics, vol.74, issue.4, pp.1203-1220, 2002.
DOI : 10.1103/RevModPhys.74.1203

L. Talon, D. Bauer, N. Gland, S. Youssef, H. Auradou et al., Assessment of the two relaxation time Lattice-Boltzmann scheme to simulate Stokes flow in porous media, Water Resources Research, vol.75, issue.5, pp.48-04526, 2012.
DOI : 10.1029/2011WR011385

W. A. Yong and L. S. Luo, Nonexistence of H Theorem for Some Lattice Boltzmann Models, Journal of Statistical Physics, vol.50, issue.1-2, pp.91-103, 2005.
DOI : 10.1007/s10955-005-5958-9