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Asymptotic analysis of the lattice Boltzmann equation. (English) Zbl 1079.82013
A general methodology is presented to conduct an order-by-order consistency analysis of the lattice Boltzmann equation. The approach is based on a direct asymptotic analysis of finite difference schemes. It turns out that the basic steps in the asymptotic expansion are parallel to the approach shown by Y. Sone for the continuous Boltzmann equation which helps to point out the connection between the lattice Boltzmann equation and the finite discrete velocity model. It is demonstrated that the asymptotic analysis yields details about the accuracy of the lattice Boltzmann method and the structure of the error. The methodology presented here can be extended to analyze various boundary conditions, coupling conditions and initial layers in the lattice Boltzmann equation simulations. The asymptotic analysis presented here differs from the traditional Chapman-Enskog treatment of the lattice Boltzmann equation. This approach uses a single time scale (as opposed to the two-time-scale) expansion in the Chapman-Enskog analysis. This model leads to the incompressible Navier-Stokes equations. The asymptotic analysis is applied directly to the fully discrete lattice Boltzmann equation. The consistency analysis provides order-by-order information about the numerical solution of the lattice Boltzmann equation. The asymptotic analysis is used to evaluate the leading orders and accuracy of numerically derived quantities, for instance the vorticity. The Richardson’s extrapolation method is justified. As an illustration, the two-dimensional Taylor-vortex flow is shown to validate the analysis.

MSC:
82C40 Kinetic theory of gases in time-dependent statistical mechanics
76D05 Navier-Stokes equations for incompressible viscous fluids
76P05 Rarefied gas flows, Boltzmann equation in fluid mechanics
76M45 Asymptotic methods, singular perturbations applied to problems in fluid mechanics
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