  Conventional lattice Boltzmann models for the simulation of fluid dynamics
are restricted by an error in the stress tensor that is negligible only for
vanishing flow velocity and at a singular value of the temperature. To that
end, we propose a unified formulation that restores Galilean invariance and
isotropy of the stress tensor by introducing an extended equilibrium. This
modification extends lattice Boltzmann models to simulations with higher values
of the flow velocity and can be used at temperatures that are higher than the
lattice reference temperature, which enhances computational efficiency by
decreasing the number of required time steps. Furthermore, the extended model
remains valid also for stretched lattices, which are useful when flow gradients
are predominant in one direction. The model is validated by simulations of two-
and three-dimensional benchmark problems, including the double shear layer
flow, the decay of homogeneous isotropic turbulence, the laminar boundary layer
over a flat plate and the turbulent channel flow.
