  In this paper, we describe in detail a scheme for the construction of highly
accurate numerical solutions to Einstein's field equations in five and six
spacetime dimensions corresponding to non-uniform black strings. The scheme
consists of a sophistically adapted multi-domain pseudo-spectral method which
incorporates a detailed understanding of the solution's behavior at the domain
boundaries and at critical points. In particular, the five-dimensional case is
exceedingly demanding as logarithmic terms appear which need to be treated with
special care. The scheme resolves these issues and permits the investigation of
unprecedentedly strong deformations of the black string horizon. As a
consequence, we are able to study in detail the critical regime in phase
diagrams displaying characteristic thermodynamic quantities such as mass and
entropy. Our results show typical spiral curves in such diagrams which provides
a strong support of previous numerical works.
