  We present high-precision numerical results for time-like
Dokshitzer-Gribov-Lipatov-Altarelli-Parisi evolution in the $\bar{\rm MS}$
factorisation scheme, for the first time up to next-to-next-to-leading order
accuracy in quantum chromodynamics. First, we scrutinise the analytical
expressions of the splitting functions available in the literature, in both x
and N space, and check their mutual consistency. Second, we implement time-like
evolution in two publicly available, entirely independent and conceptually
different numerical codes, in x and N space respectively: the already existing
APFEL code, which has been updated with time-like evolution, and the new MELA
code, which has been specifically developed to perform the study in this work.
Third, by means of a model for fragmentation functions, we provide results for
the evolution in different factorisation schemes, for different ratios between
renormalisation and factorisation scales and at different final scales. Our
results are collected in the format of benchmark tables, which could be used as
a reference for global determinations of fragmentation functions in the future.
