arXiv:2610.32810v1 Announce Type: new Abstract: A linear monodimensional lattice Boltzmann scheme with three velocities tackling the advection equation is investigated. We prove that it is second-order accurate and provide necessary and sufficient stability conditions through an explicit formula involving the two relaxation parameters. Relying on existing results for finite difference schemes, we prove convergence for smooth initial data with initialization of the non-conserved moments at equilibrium. Several numerical simulations illustrate these theoretical findings.