Approximation viscosity of one-parameter families of lattice Boltzmann equations




lattice Boltzmann method, approximation viscosity, stability


A number of properties of parametric lattice Boltzmann schemes are considered. The Chapman-Enskog method is used to derive a system of equations for hydrodynamic variables and to obtain an expression for the approximation viscosity from the differential approximation of the schemes. It is shown that there exists the numerical viscosity that should be taken into account during numerical computations. Necessary stability conditions are obtained from the nonnegativity condition for the approximation viscosity. The possibility of computations using the proposed schemes is demonstrated by the numerical solution of the lid-driven cavity flow problem when the standard lattice Boltzmann equation is inapplicable.

Author Biographies

G.V. Krivovichev

St Petersburg University
• Associate Professor

E.A. Prokhorova


  1. S. Chen and G. D. Doolen, “Lattice Boltzmann Method for Fluid Flows,” Annu. Rev. Fluid Mech. 30, 329-364 (1998).
  2. N. E. Grachev, A. V. Dmitriev, and D. S. Senin, “Simulation of Gas Dynamics with the Lattice Boltzmann Method,” Vychisl. Metody Programm. 12, 227-231 (2011).
  3. A. L. Kupershtokh, “Three-Dimensional Simulations of Two-Phase Liquid-Vapor Systems on GPU Using the Lattice Boltzmann Method,” Vychisl. Metody Programm. 13, 130-138 (2012).
  4. T. Abe, “Derivation of the Lattice Boltzmann Method by Means of the Discrete Ordinate Method for the Boltzmann Equation,” J. Comput. Phys. 131 (1), 241-246 (1997).
  5. X. He and L.-S. Luo, “Theory of the Lattice Boltzmann Method: From the Boltzmann Equation to the Lattice Boltzmann Equation,” Phys. Rev. E 56 (6), 6811-6817 (1997).
  6. J. D. Sterling and S. Chen, “Stability Analysis of Lattice Boltzmann Methods,” J. Comput. Phys. 123 (1), 196-206 (1996).
  7. G. V. Krivovichev, “Application of the Integro-Interpolation Method to the Construction of Single-Step Lattice Boltzmann Schemes,” Vychisl. Metody Programm. 13, 19-27 (2012).
  8. V. Sofonea and R. F. Sekerka, “Viscosity of Finite Difference Lattice Boltzmann Models,” J. Comput. Phys. 184 (2), 422-434 (2003).
  9. D. A. Wolf-Gladrow, Lattice-Gas Cellular Automata and Lattice Boltzmann Models: An Introduction (Springer, Berlin, 2005).
  10. L. D. Landau and E. M. Lifshitz, Course of Theoretical Physics , Vol. 6: Hydrodynamics (Fizmatlit, Moscow, 2003; Butterworth-Heinemann, Oxford, 1987).
  11. A. J. Chorin, “A Numerical Method for Solving Incompressible Viscous Flow Problems,” J. Comput. Phys. 2 (1), 12-26 (1967).
  12. T. Ohwada and P. Asinari, “Artificial Compressibility Method Revisited: Asymptotic Numerical Method for Incompressible Navier-Stokes Equations,” J. Comput. Phys. 229 (5), 1698-1723 (2010).
  13. U. Ghia, K. N. Ghia, and C. T. Shin, “High-Re Solutions for Incompressible Flow Using the Navier-Stokes Equations and a Multigrid Method,” J. Comput. Phys. 48 (3), 387-411 (1982).
  14. G. V. Krivovichev, “On the Computation of Viscous Fluid Flows by the Lattice Boltzmann Method,” Kompyut. Issled. Model. 5 (2), 165-178 (2013).



How to Cite

Кривовичев Г., Прохорова Е. Approximation Viscosity of One-Parameter Families of Lattice Boltzmann Equations // Numerical Methods and Programming (Vychislitel’nye Metody i Programmirovanie). 2017. 18. 41-52. doi 10.26089/NumMet.v18r104



Section 1. Numerical methods and applications

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