Fourth order Galilean invariance for the lattice Boltzmann method

Using the structure of a recursive asymptotic analysis we derive conditions on cumulants that guarantee a prescribed order of Galilean invariance for lattice Boltzmann models. We then apply these conditions to three different lattice Boltzmann models and obtain three models with fourth order accurate advection. One of the models uses 27 speeds on a body centered cubic lattice, one uses 33 speeds on an extended Cartesian lattice and one uses 27 speeds plus three finite differences on a Cartesian lattice. All models offer too few degrees of freedom to impose the conditions on the cumulants directly. However, the specific aliasing structure of these lattices permit fourth order accuracy for a model specific optimal reference temperature. Our theoretical derivations are confirmed by measuring the phase lag of traveling vortexes and shear waves.