  We study the renormalization group flow of the Luttinger-Ward functional and
of its two-particle irreducible vertex functions, given a cut-off in the
two-particle interaction. We derive a conserving approximation to the flow and
relate it to the fluctuation exchange approximation as well as to
non-conserving approximations introduced in an earlier publication ctuation
exchange approximation as well as to nonconserving approximations introducen
[J. F. Rentrop, S. G. Jakobs, and V. Meden, J. Phys. A: Math. Theor. 48, 145002
(2015)]. We apply the different approximate flow equations to the single
impurity Anderson model in thermal equilibrium at vanishing temperature.
Numerical results for the effective mass, the spin susceptibility, the charge
susceptibility, and the linear conductance reflect the similarity of the
methods to the fluctuation exchange approximation. We find the majority of the
approximations to deviate stronger from the exact results than one-particle
irreducible functional renormalization group schemes. However, we identify a
simple static two-particle irreducible flow scheme which performs remarkably
well and produces an exponential Kondo-like scale in the renormalized level
position.
