  We study semilinear evolution equations $ \frac {{\rm d} U}{{\rm d}
t}=AU+B(U)$ posed on a Hilbert space ${\cal Y}$, where $A$ is normal and
generates a strongly continuous semigroup, $B$ is a smooth nonlinearity from
${\cal Y}_\ell = D(A^\ell)$ to itself, and $\ell \in I \subseteq [0,L]$, $L
\geq 0$, $0,L \in I$. In particular the one-dimensional semilinear wave
equation and nonlinear Schr$\"o$dinger equation with periodic, Neumann and
Dirichlet boundary conditions fit into this framework. We discretize the
evolution equation with an A-stable Runge-Kutta method in time, retaining
continuous space, and prove convergence of order $O(h^{p\ell/(p+1)})$ for
non-smooth initial data $U^0\in {\cal Y}_\ell$, where $\ell\leq p+1$, for a
method of classical order $p$, extending a result by Brenner and Thom$\'e$e for
linear systems. Our approach is to project the semiflow and numerical method to
spectral Galerkin approximations, and to balance the projection error with the
error of the time discretization of the projected system. Numerical experiments
suggest that our estimates are sharp.
