Une introduction au schéma de Boltzmann sur réseau

We propose an elementary introduction to the lattice Boltzmann scheme. We recall the physical (Boltzmann equation) and algorithmic (cellular automata) origins of this numerical method. For a one-dimensional example, we present in detail the two characteristic steps of the algorithm: the nonlinear collision step, local in space and the linear propagation phase with the neighbouring vertices, explicit in time. We then propose a generic Taylor-type development with the so-called equivalent partial differential equation. We obtain in this way formally a so-called Chapman-Enskog development where the small parameter is the discretization step of the scheme. At order zero, the lattice Boltzmann scheme satisfies a local thermodynamical equilibrium. At first order, it satisfies the Euler equations of gas dynamics and at second order the Navier-Stokes equations. Then we detail the classical case of the nine velocities model on a square lattice