Consistency and stability of boundary conditions for a two-velocities lattice Boltzmann scheme

TL;DR

This paper analyzes boundary conditions for a simple two-velocities lattice Boltzmann scheme, using theoretical methods to ensure consistency and stability, and develops kinetic boundary conditions for linear advection.

Contribution

It introduces a rigorous framework for analyzing and designing boundary conditions in lattice Boltzmann schemes, combining finite difference mapping with stability analysis techniques.

Findings

Developed kinetic boundary conditions for inflow and outflow.

Established the trade-off between accuracy and stability.

Provided spectral analysis explanations for low-resolution effects.

Abstract

We theoretically explore boundary conditions for lattice Boltzmann methods, focusing on a toy two-velocities scheme to tackle a linear one-dimensional advection equation. By mapping lattice Boltzmann schemes to Finite Difference schemes, we facilitate rigorous consistency and stability analyses. We develop kinetic boundary conditions for inflows and outflows, highlighting the trade-off between accuracy and stability, which we successfully overcome. Consistency analysis relies on modified equations, whereas stability is assessed using GKS (Gustafsson, Kreiss, and Sundstr{\"o}m) theory and -- when this approach fails on coarse meshes -- spectral and pseudo-spectral analyses of the scheme's matrix that explain effects germane to low resolutions.