Third order equivalent equation for the relative velocity lattice Boltzmann schemes with one conservation law

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.