  This is the second part of a series devoted to the singular initial value
problem for second-order hyperbolic Fuchsian systems. In the first part, we
defined and investigated this general class of systems, and we established a
well-posedness theory in weighted Sobolev spaces. This theory is applied here
to the vacuum Einstein equations for Gowdy spacetimes admitting, by definition,
two Killing fields satisfying certain geometric conditions. We recover, by more
direct and simpler arguments, the well-posedness results established earlier by
Rendall and collaborators. In addition, in this paper we introduce a natural
approximation scheme, which we refer to as the Fuchsian numerical algorithm and
is directly motivated by our general theory. This algorithm provides highly
accurate, numerical approximations of the solution to the singular initial
value problem. In particular, for the class of Gowdy spacetimes under
consideration, various numerical experiments are presented which show the
interest and efficiency of the proposed method. Finally, as an application, we
numerically construct Gowdy spacetimes containing a smooth, incomplete,
non-compact Cauchy horizon.
