  We consider semilinear evolution equations for which the linear part
generates a strongly continuous semigroup and the nonlinear part is
sufficiently smooth on a scale of Hilbert spaces. In this setting, we prove the
existence of solutions which are temporally smooth in the norm of the lowest
rung of the scale for an open set of initial data on the highest rung of the
scale. Under the same assumptions, we prove that a class of implicit,
$A$-stable Runge--Kutta semidiscretizations in time of such equations are
smooth as maps from open subsets of the highest rung into the lowest rung of
the scale. Under the additional assumption that the linear part of the
evolution equation is normal or sectorial, we prove full order convergence of
the semidiscretization in time for initial data on open sets. Our results
apply, in particular, to the semilinear wave equation and to the nonlinear
Schr\"odinger equation.
