  Recently, a solution theory for one-dimensional stochastic PDEs of Burgers
type driven by space-time white noise was developed. In particular, it was
shown that natural numerical approximations of these equations converge and
that their convergence rate in the uniform topology is arbitrarily close to
$\frac{1}{6}$. In the present article we improve this result in the case of
additive noise by proving that the optimal rate of convergence is arbitrarily
close to $\frac{1}{2}$.
