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We consider a mono-dimensional two-velocities scheme used to approximate thesolutions of a scalar hyperbolic conservative partial differential equation. Weprove the convergence of the discrete solution toward the unique entropysolution by first estimating the supremum norm and the total variation of thediscrete solution, and second by constructing a discrete kineticentropy-entropy flux pair being given a continuous entropy-entropy flux pair ofthe hyperbolic system. We finally illustrate our results with numericalsimulations of the advection equation and the Burgers equation.

A result of convergence for a mono-dimensional two-velocities lattice Boltzmann scheme