A stability property for a mono-dimensional three velocities scheme with relative velocity

TL;DR

This paper analyzes the stability of a one-dimensional lattice Boltzmann scheme with relative velocity, focusing on conditions that ensure non-negativity of particle distributions, and demonstrates that satisfying these conditions prevents oscillations during non-smooth profile propagation.

Contribution

It provides necessary and sufficient conditions for non-negativity preservation in relative velocity schemes, including simplified criteria for zero relative velocity and numerical validation.

Findings

Derived explicit conditions for non-negativity preservation.

Identified parameter regions ensuring stability.

Numerical experiments confirm absence of oscillations under these conditions.

Abstract

In this contribution, we study a stability notion for a fundamental linear one-dimensional lattice Boltzmann scheme, this notion being related to the maximum principle. We seek to characterize the parameters of the scheme that guarantee the preservation of the non-negativity of the particle distribution functions. In the context of the relative velocity schemes, we derive necessary and sufficient conditions for the non-negativity preserving property. These conditions are then expressed in a simple way when the relative velocity is reduced to zero. For the general case, we propose some simple necessary conditions on the relaxation parameters and we put in evidence numerically the non-negativity preserving regions. Numerical experiments show finally that no oscillations occur for the propagation of a non-smooth profile if the non-negativity preserving property is satisfied.