  A expansion to fourth-order for lattice Boltzmann methods is presented. This
expansion provides an easy model for finding fourth-order corrections to
lattice Boltzmann methods for various physical systems. The fourth-order terms
can give rise to improved results over traditional second-order lattice
Boltzmann implementations. Although, this manuscript solely deals with
fourth-order expansions, this expansion is easily extended to arbitrary order.
We present examples of how this expansion is utilized and provide basic
analysis to show how the fourth-order methods differ from lower order models
for both diffusive systems and phase separating systems.
