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In this paper, we develop a general rectangular multiple-relaxation-timelattice Boltzmann (RMRT-LB) method for the Navier-Stokes equations (NSEs) andnonlinear convection-diffusion equation (NCDE) by extending our recent unifiedframework of MRT-LB method [Z.H. Chai and B.C. Shi, Phys. Rev. E 102, 023306(2020)], where a rectangular equilibrium distribution function (REDF) [J.H. Luet al, Phil. Trans. R. Soc. A 369, 2311-2319 (2011)] on a rectangular latticeis utilized. Due to the anisotropy of discrete velocities on a rectangularlattice, the third-order moment of REDF is inconsistent with that of popular LBmethod, and thus the previous unified framework of MRT-LB method cannot bedirectly applied to the NSEs using the REDF on the rectangular rDdQq lattice.The macroscopic NSEs can be recovered from the RMRT-LB method through thedirect Taylor expansion method by properly selecting the relaxation sub-matrix$\mathbf{S}_2$ which is related to kinetic viscosity and bulk viscosity. Whilethe rectangular lattice does not lead to the change of the zero-th, first andsecond-order moments of REDF, thus the unified framework of MRT-LB method canbe directly applied to the NCDE. It should be noted that the RMRT-LB model forNSEs can be derived on the rDdQq lattice, including rD2Q9, rD3Q19, and rD3Q27lattices, and the RMRT-LB versions (if existed) of the previous MRT-LB modelscan be obtained, including those based on raw (natural)-moment, central-moment,Hermite-moment, and central Hermite-moment, respectively.

physical review. e

Rectangular multiple-relaxation-time lattice Boltzmann method for the  Navier-Stokes and nonlinear convection-diffusion equations: General  equilibrium and some important issues