On the cubic velocity deviations in lattice Boltzmann methods

The macroscopic equations derived from the lattice Boltzmann equation are not exactly the Navier?Stokes equations. Here the cubic deviation terms and the methods proposed to eliminate them are studied. The most popular two- and three-dimensional models (D2Q9, D3Q15, D3Q19, D3Q27) are considered in the paper. It is demonstrated that the compensation methods provide only partial elimination of the deviations for these models. It is also shown that the compensation of Qian and Zhou (1998 Europhys. Lett. 42 359) using the compensation parameter K = 1 in a D2Q9 or D3Q27 model can eliminate all the cross terms perfectly, but the deviation terms ?x?u3x, ?y?u3y and ?z?u3z still survive the compensation.