  The classical method for deriving the macroscopic dynamics of a lattice
Boltzmann system is to use a combination of different approximations and
expansions. Usually a Chapman-Enskog analysis is performed, either on the
continuous Boltzmann system, or its discrete velocity counterpart. Separately a
discrete time approximation is introduced to the discrete velocity Boltzmann
system, to achieve a practically useful approximation to the continuous system,
for use in computation. Thereafter, with some additional arguments, the
dynamics of the Chapman-Enskog expansion are linked to the discrete time system
to produce the dynamics of the completely discrete scheme. In this paper we put
forward a different route to the macroscopic dynamics. We begin with the system
discrete in both velocity space and time. We hypothesize that the alternating
steps of advection and relaxation, common to all lattice Boltzmann schemes,
give rise to a slow invariant manifold. We perform a time step expansion of the
discrete time dynamics using the invariance of the manifold. Finally we
calculate the dynamics arising from this system. By choosing the fully discrete
scheme as a starting point we avoid mixing approximations and arrive at a
general form of the microscopic dynamics up to the second order in the time
step. We calculate the macroscopic dynamics of two commonly used lattice
schemes up to the first order, and hence find the precise form of the deviation
from the Navier-Stokes equations in the dissipative term, arising from the
discretization of velocity space. Finally we perform a short wave perturbation
on the dynamics of these example systems, to find the necessary conditions for
their stability.
