  We devise a functional renormalization group treatment for a chain of
interacting spinless fermions which is correct up to second order in the
interaction strength. We treat both inhomogeneous systems in real-space as well
as the translational invariant case in a k-space formalism. The strengths and
shortcomings of the different schemes as well as technical details of their
implementation are discussed. We use the method to study two proof-of-principle
problems in the realm of Luttinger liquid physics, namely reflection at
interfaces and power laws in the occupation number as a function of crystal
momentum.
