Theory of the lattice Boltzmann method: Dispersion, dissipation, isotropy, Galilean invariance, and stability

The generalized hydrodynamics (the wave vector dependence of the transport coefficients) of a generalized lattice Boltzmann equation (LBE) is studied in detail. The generalized lattice Boltzmann equation is constructed in moment space rather than in discrete velocity space. The generalized hydrodynamics of the model is obtained by solving the dispersion equation of the linearized LEE either analytically by using perturbation technique or numerically. The proposed LEE model has a maximum number of adjustable parameters for the given set of discrete velocities. Generalized hydrodynamics characterizes dispersion, dissipation (hyperviscosities), anisotropy, and lack of Galilean invariance of the model, and can be applied to select the values of the adjustable parameters that optimize the properties of the model. The proposed generalized hydrodynamic analysis also provides some insights into stability and proper initial conditions for LEE simulations. The stability properties of some two-dimensional LEE models are analyzed and compared with each Ether in the parameter space of the mean streaming velocity and the viscous relaxation time. The procedure described in this work can be applied to analyze other LEE models. As examples, LEE models with various interpolation schemes are analyzed. Numerical results on shear how with an initially discontinuous velocity profile (shock) with or without a constant streaming velocity are shown to demonstrate the dispersion effects in the LEE model: the results compare favorably with our theoretical analysis. We also show that whereas linear analysis of the LEE evolution operator is equivalent to Chapman-Enskog analysis in the long-wavelength limit (wave vector k=0), it can also provide results for large values of k. Such results are important for the stability and other hydrodynamic properties of the LEE method and cannot be obtained through Chapman-Enskog analysis.