Structural stability of Lattice Boltzmann schemes

TL;DR

This paper investigates the stability of Lattice Boltzmann schemes by analyzing the emergence of spurious solitary waves, revealing potential structural instabilities that cause persistent numerical errors in simulations.

Contribution

It extends previous analysis of classical schemes to Lattice Boltzmann schemes, identifying conditions under which spurious solitary waves occur and cause instability.

Findings

Spurious solitary waves can occur in Lattice Boltzmann schemes.

Such waves lead to persistent numerical errors over time.

The study extends previous work on classical schemes to Lattice Boltzmann methods.

Abstract

The goal of this work is to determine classes of traveling solitary wave solutions for Lattice Boltzmann schemes by means of an hyperbolic ansatz. It is shown that spurious solitary waves can occur in finite-difference solutions of nonlinear wave equation. The occurence of such a spurious solitary wave, which exhibits a very long life time, results in a non-vanishing numerical error for arbitrary time in unbounded numerical domain. Such a behavior is referred here to have a structural instability of the scheme, since the space of solutions spanned by the numerical scheme encompasses types of solutions (solitary waves in the present case) that are not solutions of the original continuous equations. This paper extends our previous work about classical schemes to Lattice Boltzmann schemes.