Equivalent partial differential equations of a lattice Boltzmann scheme

We show that when we formulate the lattice Boltzmann equation with a small-time step Δt and an associated space scale Δx, a Taylor expansion joined with the so-called equivalent equation methodology leads to establishing macroscopic fluid equations as a formal limit. We recover the Euler equations of gas dynamics at the first order and the compressible Navier–Stokes equations at the second order.