# Curious convergence properties of lattice Boltzmann schemes for diffusion with acoustic scaling

## TL;DR

This paper investigates the convergence behavior of the D1Q3 lattice Boltzmann scheme for diffusion with acoustic scaling, revealing unexpected discrepancies at fine mesh resolutions and deriving an acoustic PDE limit to explain this phenomenon.

## Contribution

The study introduces a new asymptotic analysis of the lattice Boltzmann scheme, uncovering its convergence breakdown and linking it to an acoustic PDE limit with complex eigenvalues.

## Key findings

- Asymptotic convergence observed when mesh is refined at fixed diffusivity.
- Convergence breaks down as mesh size tends to zero, showing qualitative discrepancies.
- A new acoustic PDE limit explains the observed convergence behavior.

## Abstract

We consider the D1Q3 lattice Boltzmann scheme with an acoustic scale for the simulation of diffusive processes. When the mesh is refined while holding the diffusivity constant, we first obtain asymptotic convergence. When the mesh size tends to zero, however, this convergence breaks down in a curious fashion, and we observe qualitative discrepancies from analytical solutions of the heat equation. In this work, a new asymptotic analysis is derived to explain this phenomenon using the Taylor expansion method, and a partial differential equation of acoustic type is obtained in the asymptotic limit. We show that the error between the D1Q3 numerical solution and a finite-difference approximation of this acoustic-type partial differential equation tends to zero in the asymptotic limit. In addition, a wave vector analysis of this asymptotic regime demonstrates that the dispersion equation has nontrivial complex eigenvalues, a sign of underlying propagation phenomena, and a portent of the unusual convergence properties mentioned above.