  We study the linear and nonlinear evolution of the tearing instability on
thin current sheets by means of two-dimensional numerical simulations, within
the framework of compressible, resistive magnetohydrodynamics. In particular we
analyze the behavior of current sheets whose inverse aspect ratio scales with
the Lundquist number $S$ as $S^{-1/3}$. This scaling has been recently
recognized to yield the threshold separating fast, ideal reconnection, with an
evolution and growth which are independent of $S$ provided this is high enough,
as it should be natural having the ideal case as a limit for $S\to\infty$. Our
simulations confirm that the tearing instability growth rate can be as fast as
$\gamma\approx 0.6\,{\tau_A}^{-1}$, where $\tau_A$ is the ideal Alfv\'enic time
set by the macroscopic scales, for our least diffusive case with $S=10^7$. The
expected instability dispersion relation and eigenmodes are also retrieved in
the linear regime, for the values of $S$ explored here. Moreover, in the
nonlinear stage of the simulations we observe secondary events obeying the same
critical scaling with $S$, here calculated on the \emph{local}, much smaller
lengths, leading to increasingly faster reconnection. These findings strongly
support the idea that in a fully dynamic regime, as soon as current sheets
develop, thin and reach this critical threshold in their aspect ratio, the
tearing mode is able to trigger plasmoid formation and reconnection on the
local (ideal) Alfv\'enic timescales, as required to explain the explosive
flaring activity often observed in solar and astrophysical plasmas.
