  We apply pseudo-spectral methods to integrate functional flow equations with
high accuracy, extending earlier work on functional fixed point equations
\cite{Borchardt:2015rxa}. The advantages of our method are illustrated with the
help of two classes of models: first, to make contact with literature, we
investigate flows of the O$(N)$-model in 3 dimensions, for $N=1, 4$ and in the
large $N$ limit. For the case of a fractal dimension, $d=2.4$, and $N=1$, we
follow the flow along a separatrix from a multicritical fixed point to the
Wilson-Fisher fixed point over almost 13 orders of magnitude. As a second
example, we consider flows of bounded quantum-mechanical potentials, which can
be considered as a toy model for Higgs inflation. Such flows pose substantial
numerical difficulties, and represent a perfect test bed to exemplify the power
of pseudo-spectral methods.
