  We apply exponential Runge Kutta time discretizations to semilinear evolution
equations $\frac { {\rm d} U}{{\rm d} t}=AU+B(U)$ posed on a Hilbert space
${\mathcal Y}$. Here $A$ is normal and generates a strongly continuous
semigroup, and $B$ is assumed to be a smooth nonlinearity from ${\mathcal
Y}_\ell = D(A^\ell)$ to itself, and $\ell \in I \subseteq [0,L]$, $L \geq 0$,
$0,L \in I$. In particular the semilinear wave equation and nonlinear
Schr\"odinger equation with periodic boundary conditions or posed on ${\mathbb
R}^d$ fit into this framework. We prove convergence of order
$O(h^{\min(\ell,p)})$ for non-smooth initial data $U^0\in{\mathcal Y}_\ell$,
where $\ell >0$, for a method of classical order $p$. We show in an example of
an exponential Euler discretization of a linear evolution equation that our
estimates are sharp, and corroborate this in numerical experiments for a
semilinear wave equation. To prove our result we Galerkin truncate the semiflow
and numerical method and balance the Galerkin truncation error with the error
of the time discretization of the projected system. We also extend these
results to exponential Rosenbrock methods.
