  Lattice Boltzmann schemes rely on the enlargement of the size of the target
problem in order 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. In order 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 operators, commutative
algebra and polynomials. Therefore, the notion of consistency of the
corresponding Finite Difference scheme allows 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 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 by means of numerical
illustrations.
