Numerical approximations of a lattice Boltzmann scheme with a family of partial differential equations

Is it possible to consider a lattice Boltzmann scheme as an approximation of a partial differential equation? For a nonhomogeneous advection problem in one spatial dimension, we propose equivalent partial differential equations at various orders. We compare the lattice Boltzmann results and a spectral approximation of the differential equations. No simple correlation is obtained for a stationary problem. For an unsteady situation, we show that the initialization scheme of the microscopic moments plays a crucial role.