# BOUNDARY CONDITIONS FOR HIGH-ORDER LATTICE BOLTZMANN MODELS

## Resumo

In this work, lattice Boltzmann (LB) regularization is extended to boundary conditions (BC).

Dealing with boundary conditions was ever considered a puzzling question in the LB method, especially,

when a large set of lattice vectors is required for the description of a given physical problem in high

order models. The most popular BC models are based on Ad-Hoc rules and, although these BC models

were shown to be suitable for low-order LBE, their extension to high-order LBE was shown to be a very

difficult problem and, at authors knowledge, never solved with satisfaction. In fact, the main question

to be solved is how to deal with a problem when the number of unknowns (the particle populations

coming from the outside part of the numerical domain) is greater than the number of equations we have

at each boundary site. A new boundary condition model is here proposed. The main idea is that when

we write both the equilibrium and non-equilibrium parts of the discrete populations in terms of its

equilibrium and non-equilibrium hydrodynamic moments, these moments replace the discrete

populations as unknowns, independently of the number of discrete velocities that are needed for solving

a given problem. This idea is here applied to the 2D lid-driven cavity flow problem and improved

stability properties are demonstrated.