In this paper we analyze and compare the lattice-Boltzmann equation with th
e beam scheme in detail. We notice the similarity and differences between t
he lattice Boltzmann equation and the beam scheme. We show that the accurac
y of the lattice-Boltzmann equation is indeed second order in space. We dis
cuss the advantages and limitations of the lattice-Boltzmann equation and t
he beam scheme. Based on our analysis, we propose an improved multi-dimensi
onal beam scheme.