Linear hydrodynamics and stability of the discrete velocity Boltzmann equations

The discrete velocity Boltzmann equations (DVBE) underlie the attainable properties of all numerical lattice Boltzmann methods (LBM). To that regard, a thorough understanding of their intrinsic hydrodynamic limits and stability properties is mandatory. To achieve this, we propose an analytical study of the eigenvalues obtained by a von Neumann perturbative analysis. It is shown that the Knudsen number, naturally defined as a particular dimensionless wavenumber in the athermal case, is sufficient to expand rigorously the eigenvalues of the DVBE and other fluidic systems such as Euler, Navier–Stokes and all Burnett equations. These expansions are therefore compared directly to one another. With this methodology, the influences of the lattice closure and equilibrium on the hydrodynamic limits and Galilean invariance are pointed out for the D1Q3 and D1Q4 lattices, without any ansatz. An analytical study of multi-relaxation time (MRT) models warns us of the errors and instabilities associated with the choice of arbitrarily large ratios of relaxation frequencies. Importantly, the notion of the Knudsen–Shannon number is introduced to understand which physics can be solved by a given LBM numerical scheme. This number is also shown to drive the practical stability of MRT schemes. In the light of the proposed methodology, the meaning of the Chapman–Enskog expansion applied to the DVBE in the linear case is clarified.