zbMATH — the first resource for mathematics

Convergence of lattice Boltzmann methods for Navier-Stokes flows in periodic and bounded domains. (English) Zbl 1160.76038
Summary: Combining an asymptotic analysis of the lattice Boltzmann method with a stability estimate, we are able to prove some convergence results which establish a strict relation to incompressible Navier-Stokes equations. The proof applies to the lattice Boltzmann method in the case of periodic domains and for specific bounded domains if the Dirichlet boundary condition is realized with the bounce back rule.

76M28 Particle methods and lattice-gas methods
76D05 Navier-Stokes equations for incompressible viscous fluids
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
Full Text: DOI
[1] Caiazzo A.: Analysis of lattice Boltzmann initialization routines. J. Stat. Phys. 121, 37–48 (2005) · Zbl 1108.76087 · doi:10.1007/s10955-005-7010-5
[2] Elton B.: Comparison of lattice Boltzmann and finite difference methods for a two-dimensional viscous Burgers equation. SIAM J. Sci. Comp. 17, 783–813 (1996) · Zbl 0924.65082 · doi:10.1137/0917052
[3] Elton B., Levermore C., Rodrigue G.: Convergence of convective diffusive lattice Boltzmann methods. SIAM J. Numer. Anal. 32, 1327–1354 (1995) · Zbl 0845.35018 · doi:10.1137/0732062
[4] Frisch U., d’Humi√®res D., Hasslacher B., Lallemand P., Pomeau Y., Rivet J.P.: Lattice gas hydrodynamics in two and three dimensions. Complex Syst. 1, 649–707 (1987) · Zbl 0662.76101
[5] He X., Luo L.-S.: A priori derivation of the lattice Boltzmann equation. Phys. Rev. E 55, 6333–6336 (1997) · doi:10.1103/PhysRevE.55.R6333
[6] He X., Luo L.-S.: Theory of the lattice Boltzmann method: from the Boltzmann equation to the lattice Boltzmann equation. Phys. Rev. E 56, 6811–6817 (1997) · doi:10.1103/PhysRevE.56.6811
[7] Junk M., Klar A., Luo L.-S.: Asymptotic analysis of the lattice Boltzmann equation. J. Comp. Phys. 210, 676–704 (2005) · Zbl 1079.82013 · doi:10.1016/j.jcp.2005.05.003
[8] Junk M., Yang Z.: Convergence of lattice Boltzmann methods for Stokes flows in periodic and bounded domains. Comp. Math. Appl. 55, 1481–1491 (2008) · Zbl 1142.76451 · doi:10.1016/j.camwa.2007.08.002
[9] Junk M., Yang Z.: Asymptotic analysis of lattice Boltzmann boundary conditions. J. Stat. Phys. 121, 3–35 (2005) · Zbl 1107.82049 · doi:10.1007/s10955-005-8321-2
[10] Junk, M., Yong, W.A.: Weighted L 2 stability of the lattice Boltzmann method. SIAM J. Numer. Anal. (in press) · Zbl 1406.65075
[11] Junk M., Yong W.A.: Rigorous Navier–Stokes limit of the lattice Boltzmann equation. Asymptot. Anal. 35, 165–184 (2003) · Zbl 1043.76003
[12] Lallemand P., Luo L.-S.: Theory of the lattice Boltzmann method: dispersion, dissipation, isotropy, galilean invariance, and stability. Phys. Rev. E 61, 6546–6562 (2000) · doi:10.1103/PhysRevE.61.6546
[13] Mei R., Luo L.-S., Lallemand P., d’Humieres D.: Consistent initial conditions for lattice Boltzmann simulations. Comput. Fluids 35, 855–862 (2006) · Zbl 1177.76319 · doi:10.1016/j.compfluid.2005.08.008
[14] Mei R., Luo L.-S., Shyy W.: An accurate curved boundary treatment in the lattice Boltzmann method. J. Comp. Phys. 155, 307–330 (1999) · Zbl 0960.82028 · doi:10.1006/jcph.1999.6334
[15] Sterling J.D., Chen S.: Stability analysis of lattice Boltzmann methods. J. Comp. Phys. 123, 196–206 (1996) · Zbl 0840.76078 · doi:10.1006/jcph.1996.0016
[16] Strang G.: Accurate partial difference methods ii. non-linear problems. Numer. Math. 6, 37–64 (1964) · Zbl 0143.38204 · doi:10.1007/BF01386051
[17] Worthing R.A., Mozer J., Seeley G.: Stability analysis of lattice Boltzmann methods. Phys. Rev. E 56-2, 2243–2253 (1997) · doi:10.1103/PhysRevE.56.2243
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.