# On anti bounce back boundary condition for lattice Boltzmann schemes

## TL;DR

This paper analyzes the anti bounce back boundary condition for lattice Boltzmann schemes, providing asymptotic analysis, identifying discrepancies, and proposing a new mixed boundary condition that improves flow simulation accuracy.

## Contribution

It introduces a new boundary condition combining bounce back and anti bounce back to better enforce boundary conditions in lattice Boltzmann schemes.

## Key findings

- Asymptotic analysis confirms known behavior for heat equation.
- Identifies discrepancies near boundaries in linear acoustics.
- Proposes a mixed boundary condition that improves flow simulation.

## Abstract

In this contribution, we recall the derivation of the anti bounce back boundary condition for the D2Q9 lattice Boltzmann scheme. We recall various elements of the state of the art for anti bounce back applied to linear heat and acoustics equations and in particular the possibility to take into account curved boundaries. We present an asymptotic analysis that allows an expansion of all the fields in the boundary cells. This analysis based on the Taylor expansion method confirms the well known behaviour of anti bounce back boundary for the heat equation. The analysis puts also in evidence a hidden differential boundary condition in the case of linear acoustics. Indeed, we observe discrepancies in the first layers near the boundary. To reduce these discrepancies, we propose a new boundary condition mixing bounce back for the oblique links and anti bounce back for the normal link. This boundary condition is able to enforce both pressure and tangential velocity on the boundary. Numerical tests for the Poiseuille flow illustrate our theoretical analysis and show improvements in the quality of the flow.