Theory of the lattice Boltzmann method: discrete effects due to advection

TL;DR

This paper analyzes the discrete effects of advection in lattice Boltzmann models, focusing on inaccuracies linked to odd order space derivatives, and proposes scenarios with third-order accuracy in 2D and 3D simulations.

Contribution

It introduces a detailed analysis of advection inaccuracies in lattice Boltzmann models related to odd order derivatives, extending understanding of model precision.

Findings

Identifies inaccuracies in advection due to odd order derivatives.

Proposes 2D scenarios with third-order accuracy.

Includes 3D advection-diffusion simulation results.

Abstract

Lattice Boltzmann models are briefly introduced together with references to methods used to predict their ability for simulations of systems described by partial differential equations that are first order in time and low order in space derivatives. Several previous works have been devoted to analyzing the accuracy of these models with special emphasis on deviations from pure Newtonian viscous behaviour, related to higher order space derivatives of even order. The presentcontribution concentrates on possible inaccuracies of the advection behaviour linked to space derivatives of odd order. Detailed properties of advection-diffusion and athermal fluids are presented for two-dimensional situations allowing to propose situations that are accurate to third order in space derivatives. Simulations of the advection of a gaussian dot or vortex are presented. Similar results are discussed in appendices for three-dimensional advection-diffusion.