  The functional renormalization group (fRG) approach has the property that, in
general, the flow equation for the two-particle vertex generates
$\mathcal{O}(N^4)$ independent variables, where $N$ is the number of
interacting states (e.g. sites of a real-space discretization). In order to
include the flow equation for the two-particle vertex one needs to make further
approximations if $N$ becomes too large. We present such an approximation
scheme, called the coupled-ladder approximation, for the special case of onsite
interaction. Like the generic third-order-truncated fRG, the coupled-ladder
approximation is exact to second order and is closely related to a simultaneous
treatment of the random phase approximation in all channels, i.e. summing up
parquet-type diagrams. The scheme is applied to a one-dimensional model
describing a quantum point contact.
