The paper is devoted to the analysis of characteristic-based (CB) schemes for the simulation of the convection–diffusion equation by the lattice Boltzmann method (LBM). Numerical schemes from the first order to fourth one are considered. The stability analysis is realized by the von Neumann method. The stability domains of the schemes are constructed. It is demonstrated that the areas of the stability domains for CB schemes are larger than the domains for the schemes, constructed by the traditional approach, based on the discretization at the Cartesian axes directions. By the solution of the numerical examples with the smooth initial conditions, it is demonstrated that the practical convergence rates of the schemes are consistent with the theoretical values. As it is shown, the proposed schemes can be used for the cases of the Peclet number values, when the classical LBM is unstable.