  We study time integration schemes for $\dot H^1$-solutions to the
energy-(sub)critical semilinear wave equation on $\mathbb{R}^3$. We show
first-order convergence in $L^2$ for the Lie splitting and convergence order
$3/2$ for a corrected Lie splitting. To our knowledge this includes the first
error analysis performed for scaling-critical dispersive problems. Our approach
is based on discrete-time Strichartz estimates, including one (with a
logarithmic correction) for the case of the forbidden endpoint. Our schemes and
the Strichartz estimates contain frequency cut-offs.
