  Numerical Relativity is a mature field with many applications in
Astrophysics, Cosmology and even in Fundamental Physics. As such, we are
entering a stage in which new sophisticated methods adapted to open problems
are being developed. In this paper, we advocate the use of Pseudo-Spectral
Collocation (PSC) methods in combination with high-order precision arithmetic
for Numerical Relativity problems with high accuracy and performance
requirements. The PSC method provides exponential convergence (for smooth
problems, as is the case in many problems in Numerical Relativity) and we can
use different bit precision without the need of changing the structure of the
numerical algorithms. Moreover, the PSC method provides high-compression
storage of the information. We introduce a series of techniques for combining
these tools and show their potential in two problems in relativistic
gravitational collapse: (i) The classical Choptuik collapse, estimating with
arbitrary precision the location of the apparent horizon. (ii) Collapse in
asympotically anti-de Sitter spacetimes, showing that the total energy is
preserved by the numerical evolution to a very high degree of precision.
