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

Lattice Boltzmann methods (LBMs) are very efficient for computational fluid dynamics, and for capturing the dynamics of weak acoustic fluctuations. It is known that multi-relaxation-time lattice Boltzmann method (MRT-LBM) appears as a very robust scheme with high precision. 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 is still not sufficient for describing the behavior 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 the classical MRT-LBM is not best suited to recover the correct behavior of pressure fluctuations. 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 an effective algorithm of recovering linearized Navier-Stokes equations (L-NSEs) from linearized MRT-LBM (L-MRT-LBM), we propose some simplified optimization strategies to determine the free relaxation parameters of 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.