Equilibrium boundary conditions for vectorial multi-dimensional lattice Boltzmann schemes

TL;DR

This paper introduces a novel approach using equilibrium boundary conditions in vectorial lattice Boltzmann schemes to improve the numerical simulation of hyperbolic conservation laws, validated through theoretical proofs and numerical experiments.

Contribution

It develops a new method for boundary conditions in multi-dimensional lattice Boltzmann schemes using equilibria, with proven convergence in scalar cases and demonstrated effectiveness in vectorial problems.

Findings

Proven convergence for scalar schemes with monotone relaxation.

Effective boundary conditions capturing physical phenomena.

Numerical experiments validate the approach.

Abstract

The concept of equilibrium is a general tool to fill the gap between macroscopic and mesoscopic information, both within kinetic systems and kinetic schemes. This work explores the use of equilibria to devise numerical boundary conditions for multi-dimensional vectorial lattice Boltzmann schemes tackling systems of hyperbolic conservation laws. In the scalar case, we prove convergence for schemes with monotone relaxation to the weak entropy solution by Bardos, Leroux, and N{\'e}delec [Commun. Partial Differ. Equ., 4 (9), 1979], following the path by Crandall and Majda [Math. Comput., 34, 149 (1980)]. Numerical experiments are conducted both for scalar and vectorial problems, and demonstrate the effectiveness of equilibrium boundary conditions in capturing significant physical phenomena.