Lattice and discrete Boltzmann equations for fully compressible flow

Equilibria for the common two-dimensional, nine-velocity (D2Q9) lattice Boltzmann equation are not uniquely determined by the Navier–Stokes equations. An otherwise undetermined function must be chosen to suppress grid-scale instabilities. By contrast, the Navier–Stokes–Fourier equations with heat conduction determine unique equilibria for a one-dimensional, five-velocity (D1Q5) model on an integer lattice. Although these equilibria are subject to grid-scale instabilities under the usual lattice Boltzmann streaming and collision steps, the equivalent discrete Boltzmann equation is stable when discretized using conventional finite volume schemes. For flows with substantial shock waves, stability is confined to a window for the parameter controlling the mean free path. It is constrained between needing a large enough mean free path (large enough viscosity) to provide dissipation at shocks, and a small enough mean free path to ensure valid hydrodynamic behavior.