  We derive an expansion of the functional renormalization (fRG) equations that
treats the frequency and momentum dependencies of the vertices in a systematic
manner. The scheme extends the channel-decomposed fRG equations to the
frequency domain and reformulates them as a series of linear integral equations
in the particle-particle, particle-hole and particle-hole exchange channels. We
show that the linearity of the equations offers numerous computational
advantages and leads to converged, stable solutions for a variety of
Hamiltonians. As the expansion is in the coupling between channels, the
truncations that are necessary to making the scheme computationally viable
still lead to equations that treat contributions from all channels equally. As
a first benchmark we apply the two-loop fRG equations to the single impurity
Anderson model. We consider the sources of error within the fRG, the
computational cost associated with each, and how the choice of regulator
affects the flow of the fRG. We then use the optimal truncation scheme to study
the Extended Hubbard Hamiltonian in one and two dimensions. We find that in
many cases of interest the fRG flow converges to a stable vertex and
self-energy from which we can extract the various correlation functions and
susceptibilities of interest.
