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

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.