Lattice Boltzmann schemes rely on the enlargement of the size of the target problem to solve PDEs in a highly parallelizable and efficient kinetic-like fashion, split into a collision and a stream phase. This structure, despite the well-known advantages from a computational standpoint, is not suitable to construct a rigorous notion of consistency with respect to the target equations and to provide a precise notion of stability. To alleviate these shortages and introduce a rigorous framework, we demonstrate that any lattice Boltzmann scheme can be rewritten as a corresponding multi-step Finite Difference scheme on the conserved variables. This is achieved by devising a suitable formalism based on commutative algebra. Therefore, the notion of consistency of the corresponding Finite Difference scheme allows us to invoke the Lax-Richtmyer theorem in the case of linear lattice Boltzmann schemes. Moreover, we show that the frequently-used von Neumann-like stability analysis for lattice Boltzmann schemes entirely corresponds to the von Neumann L^2 stability analysis of their Finite Difference counterpart. More generally, the usual tools for the analysis of Finite Difference schemes are now readily available to study lattice Boltzmann schemes. Their relevance is verified using numerical illustrations.
If the available time is enough, we will zoom on a non-linear two velocities lattice Boltzmann scheme, focusing on its L^{\infty} properties, and show how its Finite Difference formulation allows us to prove its convergence towards the weak entropic solution of a non-linear scalar conservation law. The result is obtained using monotonicity properties of the corresponding multi-step Finite Difference scheme and techniques germane to Finite Volume schemes, such as discrete entropy inequalities and total variation estimates. Still, only the over-relaxation regime is straightforwardly handled and we emphasize the need for a monotonicity theory for genuinely multi-step schemes for PDEs. This analysis should take the role of the initialization routines into account.