A discrete kinetic approximation for the incompressible Navier-Stokes equations
ESAIM: Modélisation mathématique et analyse numérique, Tome 42 (2008) no. 1, pp. 93-112.

In this paper we introduce a new class of numerical schemes for the incompressible Navier-Stokes equations, which are inspired by the theory of discrete kinetic schemes for compressible fluids. For these approximations it is possible to give a stability condition, based on a discrete velocities version of the Boltzmann H-theorem. Numerical tests are performed to investigate their convergence and accuracy.

DOI : 10.1051/m2an:2007055
Classification : 65M06, 76M20, 76R
Mots clés : incompressible fluids, kinetic schemes, BGK models, finite difference schemes
@article{M2AN_2008__42_1_93_0,
     author = {Carfora, Maria Francesca and Natalini, Roberto},
     title = {A discrete kinetic approximation for the incompressible {Navier-Stokes} equations},
     journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique},
     pages = {93--112},
     publisher = {EDP-Sciences},
     volume = {42},
     number = {1},
     year = {2008},
     doi = {10.1051/m2an:2007055},
     mrnumber = {2387423},
     zbl = {1135.76037},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/m2an:2007055/}
}
TY  - JOUR
AU  - Carfora, Maria Francesca
AU  - Natalini, Roberto
TI  - A discrete kinetic approximation for the incompressible Navier-Stokes equations
JO  - ESAIM: Modélisation mathématique et analyse numérique
PY  - 2008
SP  - 93
EP  - 112
VL  - 42
IS  - 1
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/m2an:2007055/
DO  - 10.1051/m2an:2007055
LA  - en
ID  - M2AN_2008__42_1_93_0
ER  - 
%0 Journal Article
%A Carfora, Maria Francesca
%A Natalini, Roberto
%T A discrete kinetic approximation for the incompressible Navier-Stokes equations
%J ESAIM: Modélisation mathématique et analyse numérique
%D 2008
%P 93-112
%V 42
%N 1
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/m2an:2007055/
%R 10.1051/m2an:2007055
%G en
%F M2AN_2008__42_1_93_0
Carfora, Maria Francesca; Natalini, Roberto. A discrete kinetic approximation for the incompressible Navier-Stokes equations. ESAIM: Modélisation mathématique et analyse numérique, Tome 42 (2008) no. 1, pp. 93-112. doi : 10.1051/m2an:2007055. http://www.numdam.org/articles/10.1051/m2an:2007055/

[1] D. Aregba-Driollet and R. Natalini, Discrete kinetic schemes for multidimensional systems of conservation laws. SIAM J. Numer. Anal. 37 (2000) 1973-2004. | MR | Zbl

[2] D. Aregba-Driollet, R. Natalini and S. Tang, Explicit diffusive kinetic schemes for nonlinear degenerate parabolic systems. Math. Comp. 73 (2004) 63-94. | MR | Zbl

[3] M.K. Banda, A. Klar, L. Pareschi and M. Seaid, Compressible and incompressible limits for hyperbolic systems with relaxation. J. Comput. Appl. Math. 168 (2004) 41-52. | MR | Zbl

[4] S. Bianchini, Hyperbolic limit of the Jin-Xin relaxation model. Comm. Pure Appl. Math. 59 (2006) 688-753. | MR | Zbl

[5] Y. Brenier, R. Natalini and M. Puel, On a relaxation approximation of the incompressible Navier-Stokes equations. Proc. Amer. Math. Soc. 132 (2004) 1021-1028. | MR | Zbl

[6] B.M. Boghosian, P.J. Love, P.V. Coveney, I.V. Karlin, S. Succi and J. Yepez, Galilean-invariant Lattice-Boltzmann models with H theorem. Phys. Rev. E 68 (2003) 25103-25106.

[7] F. Bouchut, Construction of BGK models with a family of kinetic entropies for a given system of conservation laws. J. Statist. Phys. 95 (1999) 113-170. | MR | Zbl

[8] F. Bouchut, Entropy satisfying flux vector splittings and kinetic BGK models. Numer. Math. 94 (2003) 623-672. | MR | Zbl

[9] F. Bouchut, Nonlinear stability of finite volume methods for hyperbolic conservation laws, and well-balanced schemes for sources, Frontiers in Mathematics series. Birkhäuser (2004). | MR | Zbl

[10] A.J. Chorin, Numerical solution of the Navier-Stokes equations. Math. Comput. 22 (1968) 745-762. | MR | Zbl

[11] D. Donatelli and P. Marcati, Convergence of singular limits for multi-D semilinear hyperbolic systems to parabolic systems. Trans. Amer. Math. Soc. 356 (2004) 2093-2121. | MR | Zbl

[12] W. E and J.G. Liu, Projection method. I. Convergence and numerical boundary layers. SIAM J. Numer. Anal. 32 (1995) 1017-1057; Projection method. II. Godunov-Ryabenki analysis. SIAM J. Numer. Anal. 33 (1996) 1597-1621. | MR | Zbl

[13] T.Y. Hou and B.T.R. Wetton, Second-order convergence of a projection scheme for the incompressible Navier-Stokes equations with boundaries. SIAM J. Numer. Anal. 30 (1993) 609-629. | MR | Zbl

[14] M. Junk, Kinetic schemes in the case of low Mach numbers. J. Comput. Phys. 151 (1999) 947-968. | MR | Zbl

[15] M. Junk and A. Klar, Discretization for the incompressible Navier-Stokes equations based on the Lattice Boltzmann method. SIAM J. Sci. Comp. 22 (2000) 1-19. | MR | Zbl

[16] M. Junk and W.A. Yong, Rigorous Navier-Stokes limit of the Lattice Boltzmann equation. Asymptot. Anal. 35 (2003) 165-185. | MR | Zbl

[17] J. Kim and P. Moin, Application of a fractional-step method to incompressible Navier-Stokes. J. Comput. Phys. 59 (1985) 308-323. | MR | Zbl

[18] R. Natalini, A discrete kinetic approximation of entropy solutions to multidimensional scalar conservation laws. J. Diff. Equation 148 (1998) 292-317. | MR | Zbl

[19] R. Natalini and F. Rousset, Convergence of a singular Euler-Poisson approximation of the incompressible Navier-Stokes equations. Proc. Am. Math. Soc. 134 (2006) 2251-2258. | MR

[20] B. Perthame, Kinetic formulation of conservation laws, Oxford Lecture Series in Mathematics and its Applications 21. Oxford University Press, Oxford (2002). | MR | Zbl

[21] M. Reider and J. Sterling, Accuracy of discrete velocity BGK models for the simulation of the incompressible Navier-Stokes equations. Comput. Fluids 24 (1995) 459-467. | MR | Zbl

[22] S. Succi, The Lattice Boltzmann Equation. Oxford University Press, Oxford (2001). | MR

[23] R. Temam, Sur l'approximation de la solution des équations de Navier-Stokes par la méthode des pas fractionnaires. I. Arch. Ration. Mech. Anal. 32 (1969) 135-153; Sur l'approximation de la solution des équations de Navier-Stokes par la méthode des pas fractionnaires. II. Arch. Ration. Mech. Anal. 33 (1969) 377-385. | MR | Zbl

[24] B.R. Wetton, Analysis of the spatial error for a class of finite difference methods for viscous incompressible flow. SIAM J. Numer. Anal. 34 (1997) 723-755; Error analysis for Chorin's original fully discrete projection method and regularizations in space and time. SIAM J. Numer. Anal. 34 (1997) 1683-1697. | MR | Zbl

[25] D.A. Wolf-Gladrow, Lattice-gas cellular automata and Lattice Boltzmann models. An introduction, Lecture Notes in Mathematics 1725. Springer-Verlag, Berlin (2000). | MR | Zbl

Cité par Sources :