  In this paper we demonstrate for the first time that it is possible to solve
numerically the Cauchy problem for the linearisation of the general conformal
field equations near spacelike infinity, which is only well-defined in
Friedrich's cylinder picture. We have restricted ourselves here to the "core"
of the equations - the spin-2 system - propagating on Minkowski space. We
compute the numerical solutions for various classes of initial data, do
convergence tests and also compare to exact solutions. We also choose initial
data which intentionally violate the smoothness conditions and then check the
analytical predictions about singularities. This paper is the first step in a
long-term investigation of the use of conformal methods in numerical
relativity.
