Comparison of analysis techniques for the lattice Boltzmann method

Abstract

a b s t r a c tWe show that the Chapman-Enskog expansion can be viewed as a special instance of a general expansion procedure which also encompasses other methods like the regular error expansion and multi-scale techniques and that any two expansions which properly describe the lattice Boltzmann solution necessarily coincide up to higher order terms. For a model problem, both the regular error expansion and the Chapman-Enskog expansion are carried out. It turns out that the classical Chapman-Enskog method leads to an unstable equation at super-Burnett order in a parameter regime for which the underlying lattice Boltzmann algorithm is stable. However, our approach naturally allows us to consider variants of the super-Burnett equation which do not suffer from instabilities. The article concludes with a detailed comparison of the Chapman-Enskog and the regular error expansion.

A comparative study of lattice Boltzmann models for incompressible flow

“…The scaling [44,45], i.e. x ∼ t, where x = dx/L and t = dt/T are the dimensionless spatial and time step, L and T are the characteristic length and time, respectively.…”

“…(A.4) originate from the basic assumption that the conserved moments (ρ and u here) are not expanded, i.e. the expansion coefficients f (n) α , n ≥ 1 have no contribution to the conserved variables [43][44][45].…”

L2Convergence of the Lattice Boltzmann Method for One Dimensional Convection-Diffusion-Reaction Equations

“…Interpreting the collision operators as block-diagonal matrices when acting on the block-vector f (n), we finally obtain the compact form of our lattice Boltzmann scheme 5) or with expanded collision operator…”

“…Consistency analysis of the lattice Boltzmann method is carried out by means of the so-called asymptotic analysis which yields important information on the structure of the numerical solution order by order in grid size h. Another possibility, where the h-dependence is somewhat less transparent, is given by the Chapman-Enskog expansion [5]. We refer to [10,13] as an example of a detailed asymptotic analysis of lattice Boltzmann algorithms including initial and boundary conditions.…”

Hydrodynamic Force Evaluation by Momentum Exchange Method in Lattice Boltzmann Simulations

“…Owing to the regular lattice, the discrete velocities and the handy density distribution functions in LBM, the momentum exchange method is convenient to implement and is highly efficient for parallel performance. In recent years, some improvements are proposed to make the method more accurate [20][21][22]. Remarkably, the Galilean invariant improvement [13] promotes the momentum exchange method to become an exact scheme for hydrodynamic force evaluation without any loss of its simplicity and efficiency.…”