  In this contribution, we address the numerical solutions of high-order
asymptotic equivalent partial differential equations with the results of a
lattice Boltzmann scheme for an inhomogeneous advection problem in one spatial
dimension. We first derive a family of equivalent partial differential
equations at various orders, and we compare the lattice Boltzmann experimental
results with a spectral approximation of the differential equations. For an
unsteady situation, we show that the initialization scheme at a sufficiently
high order of the microscopic moments plays a crucial role to observe an
asymptotic error consistent with the orderof approximation. For a stationary
long-time limit, we observe that the measured asymptotic error converges with a
reduced order of precision compared to the one suggested by asymptotic
analysis.
