# A discrete kinetic approximation for the incompressible Navier-Stokes equations

Maria Francesca Carfora; Roberto Natalini

ESAIM: Mathematical Modelling and Numerical Analysis (2008)

- Volume: 42, Issue: 1, page 93-112
- ISSN: 0764-583X

## Access Full Article

top## Abstract

top## How to cite

topCarfora, Maria Francesca, and Natalini, Roberto. "A discrete kinetic approximation for the incompressible Navier-Stokes equations." ESAIM: Mathematical Modelling and Numerical Analysis 42.1 (2008): 93-112. <http://eudml.org/doc/250377>.

@article{Carfora2008,

abstract = {
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.
},

author = {Carfora, Maria Francesca, Natalini, Roberto},

journal = {ESAIM: Mathematical Modelling and Numerical Analysis},

keywords = {Incompressible fluids; kinetic schemes; BGK models; finite difference schemes.; BGK model; finite difference scheme; Boltzmann H-theorem; convergence},

language = {eng},

month = {1},

number = {1},

pages = {93-112},

publisher = {EDP Sciences},

title = {A discrete kinetic approximation for the incompressible Navier-Stokes equations},

url = {http://eudml.org/doc/250377},

volume = {42},

year = {2008},

}

TY - JOUR

AU - Carfora, Maria Francesca

AU - Natalini, Roberto

TI - A discrete kinetic approximation for the incompressible Navier-Stokes equations

JO - ESAIM: Mathematical Modelling and Numerical Analysis

DA - 2008/1//

PB - EDP Sciences

VL - 42

IS - 1

SP - 93

EP - 112

AB -
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.

LA - eng

KW - Incompressible fluids; kinetic schemes; BGK models; finite difference schemes.; BGK model; finite difference scheme; Boltzmann H-theorem; convergence

UR - http://eudml.org/doc/250377

ER -

## References

top- D. Aregba-Driollet and R. Natalini, Discrete kinetic schemes for multidimensional systems of conservation laws. SIAM J. Numer. Anal.37 (2000) 1973–2004. Zbl0964.65096
- D. Aregba-Driollet, R. Natalini and S. Tang, Explicit diffusive kinetic schemes for nonlinear degenerate parabolic systems. Math. Comp.73 (2004) 63–94. Zbl1031.65093
- 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. Zbl1058.76035
- S. Bianchini, Hyperbolic limit of the Jin-Xin relaxation model. Comm. Pure Appl. Math.59 (2006) 688–753. Zbl1100.35063
- 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. Zbl1080.35064
- 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. E68 (2003) 25103–25106.
- 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. Zbl0957.82028
- F. Bouchut, Entropy satisfying flux vector splittings and kinetic BGK models. Numer. Math.94 (2003) 623–672. Zbl1029.65092
- F. Bouchut, Nonlinear stability of finite volume methods for hyperbolic conservation laws, and well-balanced schemes for sources, Frontiers in Mathematics series. Birkhäuser (2004). Zbl1086.65091
- A.J. Chorin, Numerical solution of the Navier-Stokes equations. Math. Comput.22 (1968) 745–762. Zbl0198.50103
- 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. Zbl1052.35014
- 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. Zbl0842.76052
- 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. Zbl0776.76055
- M. Junk, Kinetic schemes in the case of low Mach numbers. J. Comput. Phys.151 (1999) 947–968. Zbl0944.76056
- 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. Zbl0972.76083
- M. Junk and W.A. Yong, Rigorous Navier-Stokes limit of the Lattice Boltzmann equation. Asymptot. Anal.35 (2003) 165–185. Zbl1043.76003
- J. Kim and P. Moin, Application of a fractional-step method to incompressible Navier-Stokes. J. Comput. Phys.59 (1985) 308–323. Zbl0582.76038
- R. Natalini, A discrete kinetic approximation of entropy solutions to multidimensional scalar conservation laws. J. Diff. Equation148 (1998) 292–317. Zbl0911.35073
- 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. Zbl1220.35122
- B. Perthame, Kinetic formulation of conservation laws, Oxford Lecture Series in Mathematics and its Applications21. Oxford University Press, Oxford (2002).
- M. Reider and J. Sterling, Accuracy of discrete velocity BGK models for the simulation of the incompressible Navier-Stokes equations. Comput. Fluids24 (1995) 459–467. Zbl0845.76086
- S. Succi, The Lattice Boltzmann Equation. Oxford University Press, Oxford (2001). Zbl0990.76001
- 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. Zbl0207.16904
- 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. Zbl0886.76061
- D.A. Wolf-Gladrow, Lattice-gas cellular automata and Lattice Boltzmann models. An introduction, Lecture Notes in Mathematics1725. Springer-Verlag, Berlin (2000). Zbl0999.82054

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.