C. Wulff and C. Evans.

*
Runge-Kutta
time semidiscretizations of semilinear PDEs with non-smooth data
*

We study semilinear evolution equations $\partial_tU=AU+B(U)$ posed
on a Hilbert space $\kY$, where $A$ is normal and generates a
strongly continuous semigroup, $B$ is a
smooth nonlinearity
from $\kY_\ell = D(A^\ell)$ 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, 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 data $U^0\in \kY_\ell$,
$0\leq\ell\leq p+1$, for a method of classical order $p$, extending
a result by Brenner and Thom\'ee for linear sytems. 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.