Skip to main content
SpringerLink
Log in

Asymptotic Analysis of Lattice Boltzmann Boundary Conditions

  • Published:
Journal of Statistical Physics Aims and scope Submit manuscript

Abstract

In this article, we use a general method for the analysis of finite difference schemes to investigate lattice Boltzmann algorithms for Navier–Stokes problems with Dirichlet boundary conditions. Several link based boundary conditions for commonly used lattice Boltzmann BGK models are considered. With our method, the accuracy of the algorithms can be exactly predicted. Moreover, the analytical results can be used to construct new algorithms which is demonstrated with a corrected bounce back rule that requires only local evaluations but still yields second order accuracy for the velocity. The analysis is applicable to general geometries and instationary flows

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Junk M., Yang Z. Asymptotic analysis of finite difference methods, Appl. Math. Comp. (in press)

  2. S. Jin Z. Xin (1995) ArticleTitleThe relaxation schemes for systems of conservation laws in arbitrary space dimensions Comm. Pure Appl. Math 48 235–276 Occurrence Handle96c:65134

    MathSciNet  Google Scholar 

  3. U. Frisch D. d’Humières B. Hasslacher P. Lallemand Y. Pomeau J. Rivet (1987) ArticleTitleLattice gas hydrodynamics in two and three dimensions Complex Sys. 1 649–707

    Google Scholar 

  4. R. Benzi S. Succi M. Vergassola (1992) ArticleTitleThe lattice-Boltzmann equation: Theory and applications Phys. Rep 222 145–197 Occurrence Handle10.1016/0370-1573(92)90090-M Occurrence Handle1992PhR...222..145B

    Article  ADS  Google Scholar 

  5. H. Chen S. Chen W. Matthaeus (1992) ArticleTitleRecovery of the Navier–Stokes equations using a Lattice-gas Boltzmann method Phys. Rev. A 45 5339–5342 Occurrence Handle1992PhRvA..45.5339C

    ADS  Google Scholar 

  6. S. Chen G. Doolen (1998) ArticleTitleLattice Boltzmann method for fluid flows Annu. Rev. Fluid Mech 30 329–364 Occurrence Handle10.1146/annurev.fluid.30.1.329 Occurrence Handle1998AnRFM..30..329C Occurrence Handle98m:76118

    Article  ADS  MathSciNet  Google Scholar 

  7. X. He L.-S. Luo (1997) ArticleTitleA priori derivation of the lattice Boltzmann equation Phys. Rev E 55 6333–6336 Occurrence Handle10.1103/PhysRevE.55.R6333 Occurrence Handle1997PhRvE..55.6333H

    Article  ADS  Google Scholar 

  8. X. He L.-S. Luo (1997) ArticleTitleTheory of the lattice Boltzmann method: From the Boltzmann equation to the lattice Boltzmann equation Phys. Rev E 56 6811–6817 Occurrence Handle1997PhRvE..56.6811H

    ADS  Google Scholar 

  9. Y.Sone, Asymptotic theory of a steady flow of a rarefied gas past bodies for small Knudsen numbers, in Advances in Kinetic Theory and Continuum Mechanics, Proceedings of a Symposium Held in Honour of Henri Cabannes, R.Gatignol and J.Soubbaramayer, eds. (Springer, Berlin, 1990), pp. 19–31.

  10. T. Inamuro M. Yoshino F. Ogino (1997) ArticleTitleAccuracy of the lattice Boltzmann method for small Knudsen number with finite Reynolds number Phys. Fluids 9 3535–3542 Occurrence Handle10.1063/1.869426 Occurrence Handle1997PhFl....9.3535I Occurrence Handle1478126

    Article  ADS  MathSciNet  Google Scholar 

  11. M. Junk A. Klar (2000) ArticleTitleDiscretizations for the incompressible Navier–Stokes equations based on the lattice-Boltzmann method SIAM J. Sci. Comput 22 1–19 Occurrence Handle10.1137/S1064827599357188 Occurrence Handle2001b:76068

    Article  MathSciNet  Google Scholar 

  12. M. Junk (2001) ArticleTitleA finite difference interpretation of the lattice Boltzmann method Numer. Methods Partial Differ. Eq. 17 383–402 Occurrence Handle0987.76082 Occurrence Handle2002c:65138

    MATH  MathSciNet  Google Scholar 

  13. M.Junk, A.Klar, and L.-S. Luo, Asymptotic analysis of the lattice Boltzmann equation, preprint, submitted for publication.

  14. I. Ginzbourg d’Humières. (2003) ArticleTitleThe multireflection boundary conditions for lattice Boltzmann models Phys. Rev E 68 066614 Occurrence Handle2003PhRvE..68f6614G Occurrence Handle2005a:76126

    ADS  MathSciNet  Google Scholar 

  15. R. Mei L.-S. Luo W. Shyy (1999) ArticleTitleAn accurate curved boundary treatment in the lattice Boltzmann method J. Comput. Phys 155 307–330 Occurrence Handle10.1006/jcph.1999.6334 Occurrence Handle1999JCoPh.155..307M

    Article  ADS  Google Scholar 

  16. M. Junk W.-A. Yong (2003) ArticleTitleRigorous Navier–Stokes limit of the lattice Boltzmann equation Asymp. Anal. 35 165–184 Occurrence Handle2004m:76004

    MathSciNet  Google Scholar 

  17. R.Mei, L.-S. Luo, and D.d’Humieres, Initializations of lbe simulations, Preprint.

  18. P. Skordos (1993) ArticleTitleInitial and boundary conditions for the Lattice Boltzmann method Phys. Rev E 48 4823–4842 Occurrence Handle1993PhRvE..48.4823S Occurrence Handle96j:76133

    ADS  MathSciNet  Google Scholar 

  19. M. Bouzidi M. Firdaouss P. Lallemand (2001) ArticleTitleMomentum transfer of a Boltzmann-lattice fluid with boundaries Phys. Fluids 13 3452–3459 Occurrence Handle10.1063/1.1399290 Occurrence Handle2001PhFl...13.3452B

    Article  ADS  Google Scholar 

  20. M. Rohde D. Kandhai J.J. Derksen H.E.A.V. Akker Particleden (2003) ArticleTitleImproved bounce-back methods for no-slip walls in lattice-boltzmann schemes: Theory and simulations Phys. Rev E 67 066703 Occurrence Handle10.1103/PhysRevE.67.066703 Occurrence Handle2003PhRvE..67f6703R

    Article  ADS  Google Scholar 

  21. I. Ginzbourg d’Humières. (1996) ArticleTitleLocal second-order boundary methods for lattice Boltzmann models J. Stat. Phys 84 5/6 Occurrence Handle10.1007/BF02174124

    Article  Google Scholar 

  22. O. Filippova D. Hänel (1998) ArticleTitleGrid refinement for lattice-BGK models J. Comp. Phys 147 219–228

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. FB Mathematik, Universität des Saarlandes, Postfach 15 11 50, 66041, Saarbrücken, Germany

    Michael Junk & Zhaoxia Yang

Authors
  1. Michael Junk
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Zhaoxia Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Michael Junk.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Junk, M., Yang, Z. Asymptotic Analysis of Lattice Boltzmann Boundary Conditions. J Stat Phys 121, 3–35 (2005). https://doi.org/10.1007/s10955-005-8321-2

Download citation

  • Received:

  • Accepted:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10955-005-8321-2

Keywords

Search

Navigation