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It is well-known that there exist several free relaxation parameters in theMRT-LBM. Although these parameters have been tuned via linear analysis, thesensitivity analysis of these parameters and other related parameters are stillnot sufficient for detecting the behaviors of the dispersion and dissipationrelations of the MRT-LBM. Previous researches have shown that the bulkdissipation in the MRT-LBM induces a significant over-damping of acousticdisturbances. This indicates that MRT-LBM cannot be used to obtain the correctbehavior of pressure fluctuations because of the fixed bulk relaxationparameter. In order to cure this problem, an effective algorithm has beenproposed for recovering the linearized Navier-Stokes equations from thelinearized MRT-LBM. The recovered L-NSE appear as in matrix form with arbitraryorder of the truncation errors with respect to ${\delta}t$. Then, inwave-number space, the first/second-order sensitivity analyses of matrixeigenvalues are used to address the sensitivity of the wavenumber magnitudes tothe dispersion-dissipation relations. By the first-order sensitivity analysis,the numerical behaviors of the group velocity of the MRT-LBM are firstobtained. Afterwards, the distribution sensitivities of the matrix eigenvaluescorresponding to the linearized form of the MRT-LBM are investigated in thecomplex plane. Based on the sensitivity analysis and the recovered L-NSE, wepropose some simplified optimization strategies to determine the freerelaxation parameters in the MRT-LBM. Meanwhile, the dispersion and dissipationrelations of the optimal MRT-LBM are quantitatively compared with the exactdispersion and dissipation relations. At last, some numerical validations onclassical acoustic benchmark problems are shown to assess the new optimalMRT-LBM.

Sensitivity analysis and determination of free relaxation parameters for the weakly-compressible MRT-LBM schemes