On the stability of a relative velocity lattice Boltzmann scheme for compressible Navier-Stokes equations

TL;DR

This paper investigates the stability of a two-dimensional relative velocity lattice Boltzmann scheme for compressible Navier-Stokes equations, demonstrating how moment choices influence stability in linear and nonlinear contexts.

Contribution

It introduces a relative velocity scheme inspired by the cascaded scheme, analyzing its stability properties and the impact of moment selection on stability for small viscosities.

Findings

The cascaded scheme's moments enhance stability at low viscosities.

Usual moments in the relative velocity scheme reduce stability.

Stability is analyzed in both linear and nonlinear test cases.

Abstract

This paper studies the stability properties of a two dimensional relative velocity scheme for the Navier-Stokes equations. This scheme inspired by the cascaded scheme has the particularity to relax in a frame moving with a velocity field function of space and time. Its stability is studied first in a linear context then on the non linear test case of the Kelvin-Helmholtz instability. The link with the choice of the moments is put in evidence. The set of moments of the cascaded scheme improves the stability of the d'Humi\`eres scheme for small viscosities. On the contrary, a relative velocity scheme with the usual set of moments deteriorates the stability.