Up to this point, the issues that have been risen (against SRT, and hence for the necessity of using MRT) are: 1- Position of the wall in BB-type boundary conditions. As shown in many articles, the actual position of the BB wall is viscosity-dependent, therefore using MRT, or TRT, gives you the opportunity to fix the position of the wall regardless of the relaxation coefficient associated to viscosity (to do so one of the diagonal components of the relaxation matrix has to be set to a specific value, again given in many articles) [1-3]. 2- Stability at high Reynolds numbers; Linear stability analyses show that the blow-ups observed for the SRT method at high Re numbers are usually caused by the interaction between, kinetic, acoustic and shear modes. Therefore, relaxing each mode with a specific relaxation coefficient can help you damp out non-physical modes and therefore, stabilize your scheme -to a given extent [4-6].

Now if you're planning on doing a very well-resolved simulation, and your relaxation coefficient (related to viscosity) is not too far from 1, SRT would be fine. But if you are planning on doing under-resolved simulations, with high Re numbers and using BB boundaries, to have a stable code, MRT -or TRT, seem to be good choices. The list given here is of course non-exhaustive. For example, in the last reference you can see how the MRT scheme negatively impacts aero-acoustic simulations.

[1] Pan, Chongxun, Li-Shi Luo, and Cass T. Miller. "An evaluation of lattice Boltzmann schemes for porous medium flow simulation." Computers & fluids 35.8-9 (2006): 898-909.

[2] Ginzburg, Irina, Frederik Verhaeghe, and Dominique d’Humieres. "Two-relaxation-time lattice Boltzmann scheme: About parametrization, velocity, pressure and mixed boundary conditions." Communications in computational physics 3.2 (2008): 427-478.

[3] d’Humières, Dominique, and Irina Ginzburg. "Viscosity independent numerical errors for Lattice Boltzmann models: from recurrence equations to “magic” collision numbers." Computers & Mathematics with Applications 58.5 (2009): 823-840.

[4] Lallemand, Pierre, and Li-Shi Luo. "Theory of the lattice Boltzmann method: Dispersion, dissipation, isotropy, Galilean invariance, and stability." Physical Review E 61.6 (2000): 6546.

[5] d'Humières, Dominique. "Multiple–relaxation–time lattice Boltzmann models in three dimensions." Philosophical Transactions of the Royal Society of London A: Mathematical, Physical and Engineering Sciences 360.1792 (2002): 437-451.

[6] Marié, Simon, Denis Ricot, and Pierre Sagaut. "Comparison between lattice Boltzmann method and Navier–Stokes high order schemes for computational aeroacoustics." Journal of Computational Physics 228.4 (2009): 1056-1070.

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