  In this contribution, we study the theoretical and numerical stability of a
bidimensional relative velocity lattice Boltzmann scheme. These relative
velocity schemes introduce a velocity field parameter called "relative
velocity" function of space and time. They generalize the d'Humi\`eres multiple
relaxation times scheme and the cascaded automaton. This contribution studies
the stability of a four velocities scheme applied to a single linear advection
equation according to the value of this relative velocity. We especially
compare when it is equal to 0 (multiple relaxation times scheme) or to the
advection velocity ("cascaded like" scheme). The comparison is made in terms of
L1 and L2 stability. The L1 stability area is fully described in terms of
relaxation parameters and advection velocity for the two choices of relative
velocity. These results establish that no hierarchy of these two choices exists
for the L1 notion. Instead, choosing the parameter equal to the advection
velocity improves the numerical L2 stability of the scheme. This choice cancels
some dispersive terms and improve the numerical stability on a representative
test case. We theoretically strengthen these results with a weighted L2 notion
of stability.
