Tony Fevrier

François Dubois, Tony Fevrier, Benjamin Graille

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'Humières scheme for small viscosities. On the contrary, a relative velocity scheme with the usual set of moments deteriorates the stability.

François Dubois, Tony Fevrier, Benjamin Graille

In this contribution, a new class of lattice Boltzmann schemes is introduced and studied. These schemes are presented in a framework that generalizes the multiple relaxation times method of d'Humières. They extend also the Geier's cascaded method. The relaxation phase takes place in a moving frame involving a set of moments depending on a given relative velocity field. We establish with the Taylor expansion method that the equivalent partial differential equations are identical to the ones obtained with the multiple relaxation times method up to the second order accuracy. The method is then performed to derive the equivalent equations up to third order accuracy.

Benjamin Graille, François Dubois, Tony Fevrier

We study the formal precision of the relative velocity lattice Boltzmann schemes. They differ from the d'Humières schemes by their relaxation phase: it occurs for a set of moments parametrized by a velocity field function of space and time. We deal with the asymptotics of the relative velocity schemes for one conservation law: the third order equivalent equation is exposed for an arbitrary number of dimensions and velocities.