  We consider strong approximations of $1+1$-dimensional stochastic PDEs driven
by additive space-time white noise. It has been long proposed (Davie-Gaines
'01, Jentzen-Kloeden '08), as well as observed in simulations, that
approximation schemes based on samples from the stochastic convolution, rather
than from increments of the underlying Wiener processes, should achieve
significantly higher convergence rates with respect to the temporal timestep.
The present paper proves this. For a large class of nonlinearities, with
possibly superlinear growth, a temporal rate of (almost) $1$ is proven, a major
improvement on the rate $1/4$ that is known to be optimal for schemes based on
Wiener increments. The spatial rate remains (almost) $1/2$ as it is standard in
the literature.
