Truncation error analysis of lattice Boltzmann methods

Abstract

A truncation error analysis is performed for models based on the lattice Boltzmann (LB) equation. This analysis involves two steps: the recursive application of the LB equation and a Taylor series expansion. Unlike previous analytical studies of LB methods, the present work does not assume an asymptotic relationship between the temporal and spatial discretization parameters or between the probability distribution function, f, and its equilibrium distribution, f eq. Effective finite difference stencils are derived for both the distribution function and the primitive variables, i.e., density and velocity. The governing partial differential equations are also recovered. The associated truncation errors are derived and the results are validated by numerical simulation of analytic flows. Analysis of the truncation errors elucidates the roles of the kinetic theory relaxation parameter, τ, and the discretization parameters, Δ x and Δ t. The effects of initial and boundary conditions are also addressed and are shown to significantly affect the overall accuracy of the method.