Stability of a bidimensional relative velocity lattice Boltzmann scheme

Abstract: 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.