Derivation of Hydrodynamics for Multi-Relaxation Time Lattice Boltzmann Using the Moment Approach (Communications in Computational Physics, 2020)
A general analysis of the hydrodynamic limit of multi-relaxation time latticeBoltzmann models is presented. We examine multi-relaxation time BGK collision operators that are constructed similarly to those for the MRT case, however, without explicitly moving into a moment space representation. The corresponding 'moments' arederived as left eigenvectors of said collision operator in velocity space. Consequentlywe can, in a representation independent of the chosen base velocity set, generate theconservation equations. We find a significant degree of freedom in the choice of thecollision matrix and the associated basis which leaves the collision operator invariant.We explain why MRT implementations in the literature reproduce identical hydrodynamics despite being based on different orthogonalization relations. More importantly,however, we outline a minimal set of requirements on the moment base necessary tomaintain the validity of the hydrodynamic equations. This is particularly useful in thecontext of position and time-dependent moments such as those used in the contextof peculiar velocities and some implementations of fluctuations in a lattice-Boltzmannsimulation.