  It is well-known that there exist several free relaxation parameters in the
MRT-LBM. Although these parameters have been tuned via linear analysis, the
sensitivity analysis of these parameters and other related parameters are still
not sufficient for detecting the behaviors of the dispersion and dissipation
relations of the MRT-LBM. Previous researches have shown that the bulk
dissipation in the MRT-LBM induces a significant over-damping of acoustic
disturbances. This indicates that MRT-LBM cannot be used to obtain the correct
behavior of pressure fluctuations because of the fixed bulk relaxation
parameter. In order to cure this problem, an effective algorithm has been
proposed for recovering the linearized Navier-Stokes equations from the
linearized MRT-LBM. The recovered L-NSE appear as in matrix form with arbitrary
order of the truncation errors with respect to ${\delta}t$. Then, in
wave-number space, the first/second-order sensitivity analyses of matrix
eigenvalues are used to address the sensitivity of the wavenumber magnitudes to
the dispersion-dissipation relations. By the first-order sensitivity analysis,
the numerical behaviors of the group velocity of the MRT-LBM are first
obtained. Afterwards, the distribution sensitivities of the matrix eigenvalues
corresponding to the linearized form of the MRT-LBM are investigated in the
complex plane. Based on the sensitivity analysis and the recovered L-NSE, we
propose some simplified optimization strategies to determine the free
relaxation parameters in the MRT-LBM. Meanwhile, the dispersion and dissipation
relations of the optimal MRT-LBM are quantitatively compared with the exact
dispersion and dissipation relations. At last, some numerical validations on
classical acoustic benchmark problems are shown to assess the new optimal
MRT-LBM.
